1 Introduction
An important feature of Teichmüller space, seen as a connected component of the space of representations of the fundamental group of a closed connected orientable surface S of genus at least
$2$
in
$\mathsf {PSL}_2(\mathbb R)$
, is that it consists entirely of representations which are discrete and faithful. These representations are moreover quasi-isometries from
$\pi _1(S)$
to
$\mathsf {PSL}_2(\mathbb R)$
. This situation does not extend to the case of any semi-simple group, notably for simply connected complex ones, where the representation variety is irreducible as an algebraic variety [Reference Rapinchuk, Benyash-Krivetz and Chernousov36].
However, this phenomenon was shown to happen for some groups of higher rank. Two families of representation varieties of the fundamental group of S have been singled out as they contain connected components consisting entirely of discrete and faithful representations:
-
• Hitchin components when
$\mathsf G$
is a real split group [Reference Fock and Goncharov17] (the Hitchin components were defined in [Reference Hitchin29]; the case of
$\mathsf {SL}_3(\mathbb {R})$
was treated in [Reference Choi and Goldman13] and the case of
$\mathsf {SL}_n(\mathbb {R})$
in [Reference Labourie31]), -
• spaces of maximal representations, which are defined when
$\mathsf G$
is Hermitian [Reference Burger, Iozzi and Wienhard12] ([Reference Goldman19] proves the case of
$\mathsf {SL}_2( \mathbb {R})$
, [Reference Burger, Iozzi, Labourie and Wienhard10] studies maximal representations into symplectic groups).
When
$\mathsf G$
is
$\mathsf {PSL}_2(\mathbb R)$
, the Hitchin component and the space of maximal representations both agree with the Teichmüller space.
The study of these two families is closely related to the theory of Anosov representations as introduced in [Reference Labourie31, Reference Guichard and Wienhard25]. Being Anosov is a notion defined for any reductive Lie group and with respect to a choice of a parabolic subgroup. Every Anosov representation is in particular faithful, discrete, and a quasi-isometric embedding [Reference Labourie32, Reference Guichard and Wienhard25, Reference Delzant, Guichard, Labourie and Mozes15].
Representations in the Hitchin components as well as maximal representations can be characterized in terms of equivariant curves from the boundary at infinity of
$\pi _1(S)$
into an appropriate flag variety, which preserve some positivity. In [Reference Labourie31] Labourie established the Anosov property for representations in Hitchin components for
$\mathsf {SL}_n(\mathbb R)$
and showed that they furthermore admit hyperconvex boundary curves. In combination with [Reference Guichard23] this gives a characterization of Hitchin components in terms of hyperconvex maps. The link to positivity was first made for split real groups in work of Fock and Goncharov [Reference Fock and Goncharov17]; in their groundbreaking work, they develop a new way to study moduli spaces of local systems, by looking at configuration spaces of (decorated) flags. They introduce new cluster coordinate systems, which are closely related to Lusztig’s total positivity [Reference Lusztig34], and open new ways for quantizations. The positive points of their cluster varieties give rise to higher rank Teichmüller spaces, and, building on insight of [Reference Labourie31], lead to a characterization of Hitchin components in terms of positive boundary maps.
The characterization of maximal representations in terms of positivity is given in work of Burger, Iozzi, and Wienhard [Reference Burger, Iozzi and Wienhard12], where the notion of positivity is based on the maximality of the Maslov index and related to Lie semigroups in
$\mathsf G$
.
In [Reference Guichard and Wienhard26, Reference Guichard and Wienhard24], Guichard and Wienhard introduced the notion of
$\Theta $
-positivity. This notion extends Lusztig’s total positivity to generalized flag manifolds associated with the parabolic defined by a set
$\Theta $
of simple roots. They classified all possible simple Lie groups that admit a positive structure relative to some
$\Theta $
. These include real split Lie groups, for which
$\Theta $
-positivity is Lusztig’s total positivity, Hermitian Lie groups of tube type, where
$\Theta $
-positivity is given by the maximality of the Maslov index, but also two other families of Lie groups, namely the family of classical groups
$\mathsf {SO}(p,q)$
– with
$p\not =q$
– and an exceptional family consisting of the real rank
$4$
form of
$\mathsf F_4$
,
$\mathsf E_6$
,
$\mathsf E_7$
, and
$\mathsf E_8$
respectively. They conjectured that
$\Theta $
-positivity provides the right underlying algebraic structure for the existence of components made solely of discrete and faithful representations [Reference Wienhard38, Conjecture 19].
A positive structure on
$\mathsf G$
relative to
$\Theta $
implies in particular the existence of a positive semigroup in the unipotent radical of the parabolic group
$\mathsf P_\Theta $
, which then leads to the notions of positive triples and positive quadruples (as well as positive tuples) in the flag variety
$\mathbf {F}_\Theta \simeq \mathsf G/\mathsf P_\Theta $
. In the basic example of
$\mathsf G=\mathsf {PSL}_2(\mathbb R)$
and
$\mathbf {F}_\Theta ={\mathbf P}^1(\mathbb {R})$
, a triple is positive if it consists of pairwise distinct points and a quadruple is positive if it is cyclically ordered.
Let us give a geometric picture of positivity in the flag variety
$\mathbf {F}_\Theta $
. For this let a and b be two points in
$\mathbf {F}_\Theta $
which are transverse to each other. Then
$\Theta $
-positivity provides the existence of preferred connected components of the set of all points in
$\mathbf {F}_\Theta $
that are transverse to both a and b. These preferred components are called diamonds (with extremities a and b). They are several, at least two, disjoint diamonds with given extremities. The semigroup property alluded to before translates into a nesting property of diamonds: if c is a point in a diamond
$V(a,b)$
with extremities a and b, then there is exactly one diamond
$V(c,b)$
(with extremities c and b) included in
$V(a,b)$
. These nesting properties of diamonds play an important role in our arguments.
If a and b are transverse, and c belongs to a diamond with extremities a and b, we say the triple
$(a,b,c)$
is positive. Similarly, one can define positive quadruples using configurations of diamonds (see Figure 2 and Definition 2.10). We show in Section 3 that being positive is invariant under all permutations for a triple, and invariant under the dihedral group for a quadruple.
We define a map
$\xi $
from a cyclically ordered set A to
$\mathbf {F}_\Theta $
to be positive if
$\xi $
maps triples of pairwise distinct points to positive triples and cyclically ordered quadruples to positive quadruples.
This allows us to define the notion of a
$\Theta $
-positive representation: A representation
$\rho \colon \pi _1(S) \to \mathsf G$
is
$\Theta $
-positive if there exist a nonempty subset A of
$\partial _\infty \pi _1(S)$
, invariant by
$\pi _1(S)$
, and a
$\rho $
-equivariant positive boundary map from A to
$\mathbf {F}_\Theta $
.
We prove
Theorem A. Let
$\mathsf G$
be a semi-simple Lie group that admits a positive structure relative to
$\Theta $
. Let
$\rho $
be a
$\Theta $
-positive representation from
$\pi _1(S)$
to
$\mathsf G$
. Then
$\rho $
is a
$\Theta $
-Anosov representation.
As a direct consequence we obtain that a
$\Theta $
-positive representation is faithful with discrete image, its orbit map into the symmetric space is a quasi-isometric embedding and the boundary map extends uniquely to a Hölder map [Reference Labourie32, Reference Guichard and Wienhard25, Reference Delzant, Guichard, Labourie and Mozes15, Reference Bridgeman, Canary, Labourie and Sambarino9].
Theorem A provides a general proof of the Anosov property for all Hitchin representations and all maximal representations. This is especially relevant for the case of the Hitchin component of
$\mathsf {SO}(p,p)$
and of
$\mathsf {F}_4$
,
$\mathsf {E}_6$
,
$\mathsf {E}_7$
, and
$\mathsf {E}_8$
and the case of maximal representations into the exceptional Hermitian Lie group of tube type, which cannot be tightly embedded into
$\mathsf {Sp}_{2n}(\mathbb R)$
[Reference Burger, Iozzi and Wienhard11, Reference Hamlet27, Reference Hamlet28]. The Anosov property was established for all maximal representations which tightly embed into
$\mathsf {Sp}_{2n}(\mathbb R)$
in [Reference Burger, Iozzi, Labourie and Wienhard10] and for the Hitchin component of
$\mathsf {SL}_n(\mathbb {R})$
in [Reference Labourie31], from which follows the Anosov property for the Hitchin components of
$\mathsf {Sp}_{2n}(\mathbb {R})$
,
$\mathsf {SO}(p,p+1)$
, and
$G_2$
. Fock and Goncharov established a related key property: for every Hitchin representation, there exists a continuous, transverse (and positive) boundary map [Reference Fock and Goncharov17, Theorem 7.2]; from this, the Anosov property can be established for Zariski dense Hitchin representations using for example [Reference Guichard and Wienhard25, Theorem 4.11].
Using their work on amalgamation of Anosov representations, Dey and Kapovich [Reference Dey and Kapovich16, Section 6] established also the Anosov property for all Hitchin components for all real split groups.
Using the openness of the set of
$\Theta $
-Anosov representations, a further consequence of Theorem A is the following
Corollary B. The set of
$\Theta $
-positive representations
$\operatorname {Hom}_{\Theta \textrm {-pos}}(\pi _1(S),\mathsf G)$
is an open subset in the set of all homomorphisms
$\operatorname {Hom}(\pi _1(S),\mathsf G)$
.
To show that the set of
$\Theta $
-positive representations indeed give rise to higher Teichmüller spaces it remains to prove that the set of
$\Theta $
-positive representations is closed. We establish essential steps in this direction. For this we consider the set
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
of homomorphisms
$\rho $
of
$\pi _1(S)$
in
$\mathsf G$
such that the image of
$\rho $
contains a
$\Theta $
-loxodromic element (i.e., an element having both attracting and repelling fixed points in the flag variety
$\mathbf {F}_\Theta $
associated to
$\Theta $
). Observe that Proposition 6.3 clarifies the relation with the Zariski closure, and in particular the set
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
contains the representations with Zariski dense images. We establish in Proposition 6.1 that
$\operatorname {Hom}_{\Theta \textrm {-pos}}(\Gamma ,\mathsf G)$
is a subset of
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
. We show
Theorem C. The set of
$\Theta $
-positive representations
$\operatorname {Hom}_{\Theta \textrm {-pos}}(\pi _1(S),\mathsf G)$
is a nonempty union of connected components of
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
.
In the case when
$\mathsf G$
is locally isomorphic to
$\mathsf {SO}(p,q)$
,
$p\leqslant q$
, Beyrer and Pozzetti [Reference Beyrer and Pozzetti4] recently proved the closedness of the set of
$\Theta $
-positive Anosov representations in
$\operatorname {Hom}(\pi _1(S),\mathsf G)$
, thus by Theorem A also the closedness of the set of
$\Theta $
-positive representations. They derive this as a consequence of a family of collar lemmas and fine properties of the boundary maps they establish. In [Reference Beyrer, Guichard, Labourie, Pozzetti and Wienhard3], together with Beyrer and Pozzetti, we prove collar lemmas in full generality for
$\Theta $
-positivity, which in combination with Theorem C establishes that the set of
$\Theta $
-positive representations is closed in
$\operatorname {Hom}(\pi _1(S),\mathsf G)$
. Thus with Corollary B and Theorem A
$\Theta $
-positive representations give rise to connected components consisting entirely of discrete and faithful representations.
Note that special
$\Theta $
-positive representations arise from positive embeddings of
$\mathsf {SL}_2(\mathbb {R})$
into
$\mathsf G$
. These positive embeddings of
$\mathsf {SL}_2(\mathbb {R})$
can be produced explicitly using specific “positive” nilpotent element in the Lie algebra of
$\mathsf G$
. They have the property that the embedding induces a positive map from
${\mathbf P}^1(\mathbb {R})$
into
$\mathbf {F}_\Theta $
. We call the image of such a map a positive circle. Every group
$\mathsf G$
admitting a positive structure relative to
$\Theta $
contains a special (conjugacy class of)
$\Theta $
-principal
$\mathsf {SL}_2(\mathbb {R})$
. The circles associated to this
$\Theta $
-principal
$\mathsf {SL}_2(\mathbb {R})$
play an important role in some of our arguments. Precomposing a positive embedding of
$\mathsf {SL}_2(\mathbb {R})$
into
$\mathsf G$
with a discrete embedding of
$\pi _1(S)$
into
$\mathsf {SL}_2(\mathbb {R})$
, we obtain a
$\Theta $
-positive representation.
Recently, Bradlow, Collier, García-Prada, Gothen, and Oliveira [Reference Bradlow, Collier, García-Prada, Gothen and Oliveira7] developed the theory of magical
$\mathfrak {sl}_2$
-triples, which is very closely related to the theory of
$\Theta $
-positivity. In fact a real simple Lie group is associated to a magical
$\mathfrak {sl}_2$
-triple if and only if it admits a positive structure relative to
$\Theta $
. Using methods from the theory of Higgs bundles, they parametrize special connected components
$\mathcal {P}_e(S,\mathsf G)$
, called Cayley components. We expect these connected components
$\mathcal {P}_e(S,\mathsf G)$
to consist entirely of
$\Theta $
-positive representations, and furthermore to coincide with the set of
$\Theta $
-positive representations. We discuss the relation between
$\mathcal {P}_e(S,\mathsf G)$
and
$\Theta $
-positive representations in Section 7.
Outline of the paper: In Section 2, we recall the necessary algebraic material from [Reference Guichard and Wienhard26, Reference Guichard and Wienhard24] and introduce the main definitions: diamonds, positive configurations, positive circles, and positive maps. In Section 3, we prove three propositions concerning combinatorial properties of configurations, proper inclusion of diamonds, and extension of positive maps – some of the properties proved here are also in [Reference Guichard and Wienhard24], but the proofs in the present paper are geometric, while those in [Reference Guichard and Wienhard24] are algebraic. In Section 4, we introduce the diamond metric on diamonds and establish its properties. With these preparations we prove Theorem A and Corollary B in Section 5, Theorem C in Section 6. In Section 7 we discuss the connection with the Cayley components introduced in [Reference Bradlow, Collier, García-Prada, Gothen and Oliveira7].
2 Definitions
2.1 Lie algebra notations
Let
$\mathsf G$
be a semi-simple Lie group.
The roots of
$\mathsf G$
are the nonzero weights under the adjoint action of a Cartan subspace
$\mathfrak {a}$
on the Lie algebra
$\mathfrak {g}$
of
$\mathsf G$
. They form a root system
$\Sigma \subset \mathfrak {a}^*$
(nonreduced in some cases) and the choice of a linear ordering on
$\mathfrak {a}^*$
gives rise to the set
$\Sigma ^+$
of positive roots and to the set
$\Delta $
of simple roots. The
$\alpha $
-weight space will be denoted by
$\mathfrak {g}_\alpha \subset \mathfrak {g}$
.
The parabolic subgroups of
$\mathsf G$
are the subgroups conjugated to one of the standard parabolic subgroups
$\mathsf {P}_\Theta $
(for
$\Theta $
varying in the subsets of
$\Delta $
); namely
$\mathsf {P}_\Theta $
is the normalizer in
$\mathsf G$
of the Lie algebra 
. The unipotent radical of
$\mathsf {P}_\Theta $
is the subgroup
$\mathsf {U}_\Theta = \exp ( \mathfrak {u}_\Theta )$
. A parabolic subgroup is its own normalizer so that the space
$\mathbf {F}_\Theta $
of parabolic subgroups conjugated to
$\mathsf {P}_\Theta $
is isomorphic to
$\mathsf G/\mathsf {P}_\Theta $
.
The space
$\mathbf {F}_\Theta $
is also naturally
$\mathsf G$
-isomorphic to the
$\mathsf G$
-orbit (for the adjoint action) of
$\mathfrak {u}_\Theta $
in the space
$\mathbf {L}$
of Lie subalgebras of
$\mathfrak g$
. The group
$\operatorname {Aut}(\mathfrak g)$
of automorphisms of
$\mathfrak g$
also acts on
$\mathbf {L}$
and the actions of
$\mathsf G$
and of
$\operatorname {Aut}(\mathfrak g)$
on this space are related via the adjoint action seen as a homomorphism
$\mathsf G \to \operatorname {Aut}(\mathfrak g)$
. For
$\psi $
in
$\operatorname {Aut}(\mathfrak g)$
and for
$\mathfrak {u}$
in the
$\mathsf G$
-orbit of
$\mathfrak {u}_\Theta $
(i.e.,
$\mathfrak {u}$
belongs to
$\mathbf {F}_\Theta $
), the algebra
$\psi (\mathfrak {u})$
may not belong to
$\mathbf {F}_\Theta $
; in fact
$\psi (u)$
belongs to
$\mathbf {F}_{\psi _*(\Theta )}$
where
$\psi _*\colon \Delta \to \Delta $
denotes the action of
$\psi $
on the set of simple roots (or on the Dynkin diagram). There is thus a subgroup
$\operatorname {Aut}_0(\mathfrak g)$
of
$\operatorname {Aut}(\mathfrak g)$
that acts (transitively) on
$\mathbf {F}_\Theta $
. This group
$\operatorname {Aut}_0(\mathfrak g)$
has better transitive properties than
$\mathsf G$
, for example, it will act transitively on the diamonds that are introduced later. We will therefore use several times
$\operatorname {Aut}_0(\mathfrak g)$
instead of G.
Two parabolic subgroups
$\mathsf {P}$
and
$\mathsf {P}'$
are called transverse or opposite if their intersection
$\mathsf {P} \cap \mathsf {P}'$
is a reductive subgroup (i.e., the unipotent radical of this intersection is trivial); this is equivalent to having
$\operatorname {UniRad}( \mathsf {P}) \cap \mathsf {P}'=\{1\}$
. In that case, there exists
$\Theta \subset \Delta $
such that the pair
$( \mathsf {P}, \mathsf {P}')$
is conjugated to
$( \mathsf {P}_{\Theta }, \mathsf {P}^{\mathrm {opp}}_{\Theta })$
where
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
is the normalizer of
$\bigoplus _{\alpha \in \Sigma ^+ \smallsetminus \operatorname {span}(\Delta \smallsetminus \Theta )} \mathfrak {g}_{-\alpha }$
. The intersection 
is a Levi factor of
$\mathsf {P}_{\Theta }$
(and of
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
).
We will always work with a parabolic subgroup
$\mathsf {P}\simeq \mathsf {P}_\Theta $
such that
$\mathsf {P}_\Theta $
is conjugated to its opposite
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
; in this situation it makes sense to look at transverse elements in
$\mathbf {F}_\Theta \simeq \mathsf G/\mathsf {P}_\Theta $
. In particular we will use the following notation, for x in
$\mathbf {F}_\Theta $
,
We will sometimes use that, if a and b are transverse points, then 
is a Levi factor of
$\mathsf {P}_a$
and
$\mathsf {P}_b$
. Recall that
$\Omega _x$
is an open orbit of
$\mathsf U_x$
and that
$S_x$
is a proper algebraic subvariety of
$\mathbf {F}_\Theta $
.
Given a point a in
$\mathbf {F}_\Theta $
, a unipotent pinning, or U-pinning at a, is an identification s of
$\mathsf U_\Theta $
with
$\mathsf U_a$
that exponentiates an isomorphism from
$\mathfrak {u}_\Theta $
to
$\mathfrak {u}_a$
which itself is induced by the restriction of an automorphism of the Lie algebra
$\mathfrak {g}$
(i.e., an element of
$\operatorname {Aut}_0(\mathfrak g)$
). Observe that there are finitely many U-pinnings up to the action of
$\mathsf L_\Theta $
.
2.2 Cones and semigroup
Definition 2.1 [Reference Guichard and Wienhard24, Theorem 12.2].
A positive structure with respect to
$\mathbf {F}_\Theta $
(or a positive structure relative to
$\Theta $
) is a semigroup
$\mathsf {N}$
of
$\mathsf {U}_{\Theta }$
such that, denoting x and y the points of
$\mathbf {F}_\Theta $
whose stabilizers are
$\mathsf {P}_\Theta $
and
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
respectively,
$\mathsf {N}\cdot y$
is a connected component of
$\Omega _x \cap \Omega _y$
.
In this case,
$\mathsf {N}$
is invariant by conjugation by the connected component
$\mathsf {L}^{\circ }_{\Theta }$
of
$\mathsf {L}_\Theta $
and is a sharp semigroup: for any h, k in
$\overline {\mathsf N}$
, if
$hk=1$
, then
$h=k=1$
(i.e., the only invertible element in
$\overline {\mathsf {N}}$
is the identity element).
We shall see that given a and b transverse to each other in
$\mathbf {F}_\Theta $
and an identification of
$\mathsf U_\Theta $
with
$\mathsf U_a$
(i.e., a U-pinning) which sends
$\mathsf N$
to a subgroup
$\mathsf N_a$
of
$\mathsf U_a$
, then
$\mathsf N_a\cdot b$
is a connected component of
$\Omega _a\cap \Omega _b$
.
In [Reference Guichard and Wienhard24] it is proved that, up to the action of
$\operatorname {Aut}_0(\mathfrak g)$
, the semigroup
$\mathsf {N}$
in the definition is unique.
We first present some conclusions of the construction of the semigroup
$\mathsf {N}$
that we are going to use in this paper, then concentrate on the notions of diamonds and positive configurations that play a crucial role in this paper.
2.2.1 The parametrization of the positive semigroup
Theorem 4.5 of [Reference Guichard and Wienhard26] and Theorem 1.3 of [Reference Guichard and Wienhard24] give a precise description of the possible parametrizations of the semigroup
$\mathsf {N}$
. We recall here the material necessary for our purpose.
Fact 2.2. There exist
$N\geqslant 1$
and
$\mathsf {C}$
a
$\mathsf {L}^{\circ }_{\Theta }$
-invariant cone in
$( \mathfrak {u}_\Theta )^N$
such that the map
$$ \begin{align*} (\mathfrak{u}_\Theta)^N & \longrightarrow \mathsf{U}\\ (x_1, \dots, x_N) & \longmapsto \exp(x_1)\cdots \exp(x_N) \end{align*} $$
induces by restriction a
$\mathsf {L}^{\circ }_{\Theta }$
-equivariant diffeomorphism
$$ \begin{align*} \Psi\colon \mathsf{C}& \longrightarrow \mathsf{N}. \end{align*} $$
Furthermore the stabilizer in
$\mathsf {L}^{\circ }_{\Theta }$
of any point h in
$\mathsf {C}$
, and therefore of any point n in
$\mathsf {N}$
, is a compact subgroup of
$\mathsf {L}^{\circ }_{\Theta }$
.
The closure
$\overline {\mathsf {C}}$
is also
$\mathsf {L}^{\circ }_{\Theta }$
-invariant and Definition 2.1 implies that the cone
$\overline {\mathsf {C}}$
is salient, i.e., the intersection of
$\overline {\mathsf {C}}$
and
$-\overline {\mathsf {C}}$
is reduced to
$\{0\}$
.
Remark 2.3. More precisely, for every
$\alpha $
in
$\Theta $
an
$\mathsf {L}^{\circ }_{\Theta }$
-invariant cone
$C_\alpha $
has been chosen in the
$\mathsf {L}$
-irreducible factor of
$\mathfrak {u}_\Theta $
corresponding to
$\alpha $
(and
$C_\alpha $
is open in that factor) and we have that
$\mathsf {C}=C_1 \times C_2 \times \cdots \times C_N$
where N is the length of the longest element in a finite Coxeter group associated with
$\Theta $
and, for every
$i=1,\dots , N$
,
$C_i$
is one of the cones
$C_\alpha $
[Reference Guichard and Wienhard24, Theorem 1.3].
2.3 Diamonds
Let a and b be two transverse points in
$\mathbf {F}_\Theta $
.
Definition 2.4. A diamond with extremities a and b, associated with a U-pinning
$s_a$
at a, is the subset
$$ \begin{align*} s_a(\mathsf N)\cdot b. \end{align*} $$
The terminology diamond was coined in [Reference Labourie and Toulisse33] in the context of
$\mathsf G=\mathsf {SO}(2,n)$
. To give an idea, in that context
$\mathbf {F}_\Theta $
is covered by charts which are identified with the Minkowski space
$\mathbb R^{1,n-1}$
. Then a diamond is, in a suitable chart, the intersection of the future time cone
$F^+$
of a, with the past time cone
$F^-$
of b.
In that case there are precisely two diamonds with given extremities. More generally, from [Reference Guichard and Wienhard24, Corollary 13.5], it follows that the number of diamonds with given extremities is
$2^{\sharp \Theta }$
.
Remark 2.5. We observe that diamonds are semi-algebraic sets and make sense over a real closed field.
We list some first properties of diamonds that are direct consequences of the definition or are proved in [Reference Guichard and Wienhard24, Section 13].
Proposition 2.6.
-
1. A diamond with extremities a and b is a connected component of
$\Omega _a\cap \Omega _b$
. -
2. Given a diamond
$s_a(\mathsf N)\cdot b$
, there exists a U-pinning
$s_b$
at b such that
$$ \begin{align*} s_a(\mathsf N)\cdot b=s_b(\mathsf N)\cdot a. \end{align*} $$
-
3. Given any diamond
$V(a,b)=s_a( \mathsf {N})\cdot b$
then a belongs to the closure of
$V(a,b)$
.
Proof. The first item is a consequence of [Reference Guichard and Wienhard24, Theorems 1.3 and 1.4]. The second item is a consequence of [Reference Guichard and Wienhard24, Proposition 13.1].
The third item follows from the fact that the identity belongs to the closure of
$\mathsf N$
.
We also remark that
Proposition 2.7. Given a diamond V there is a unique diamond
$V^*$
satisfying the following property: given any U-pinning
$s_b$
at b, if
$V=s_b(\mathsf N)\cdot a$
then
$V^*=s_b(\mathsf N^{-1})\cdot a$
. The diamond
$V^*$
is called the opposite diamond to V (one says also that the diamond
$V^*$
is opposite to V). A diamond and its opposite are disjoint; more precisely any point in V is transverse to any point in
$V^*$
.
Proof. We just have to remark that the definition of the opposite diamond does not depend on the choices. More precisely, given two U-pinnings
$s_a$
and
$s_b$
, if
$$\begin{align*}V=s_b(\mathsf N)\cdot a=s_a(\mathsf N)\cdot b , \end{align*}$$
then
$$\begin{align*}s_b(\mathsf N^{-1})\cdot a=s_a(\mathsf N^{-1})\cdot b\; \end{align*}$$
this holds by [Reference Guichard and Wienhard24, Section 13].
The last point comes from [Reference Guichard and Wienhard26, Remark 4.9] and from [Reference Guichard and Wienhard24, Section 13.6]. In particular, if
$x\in V$
, then
$x=s_b(n)\cdot a$
with
$n\in \mathsf N$
, while if
$y\in V^*$
, then
$y=s_b(m^{-1})\cdot a$
with
$m\in \mathsf N$
. Thus
$$ \begin{align*} x=s_b(nm)\cdot y. \end{align*} $$
Since
$\mathsf {N}$
is a semigroup, this means that x belongs to a diamond with extremities y and b. By the first point of Proposition 2.6, x is transverse to y.
As a consequence of the proposition, if c is an element in a diamond with extremities a and b, we will denote by
$$\begin{align*}V_c(a,b)\end{align*}$$
the unique diamond containing c with extremities a and b.
Note that, for any d in
$V_c(a,b)$
, one has
$V_d(a,b)=V_c(a,b)$
; also
$V_c(b,a) = V_c(a,b)$
.
In addition,
$V^*_c(a,b)$
is the diamond opposite to the diamond containing c.
As an immediate consequence of the semigroup property we obtain the following result that we shall use freely:
Lemma 2.8 (Nesting property).
Let c be a point in a diamond with extremities a and b.
-
1. Then there exists a unique diamond
$V(a,c)$
with extremities a and c such that
$$ \begin{align*} V(a,c)\subset V_c(a,b) . \end{align*} $$
Furthermore there is a neighborhood U of a in
$\mathbf {F}_\Theta $
such that
$$ \begin{align*}U\cap V(a,c)=U\cap V_c(a,b).\end{align*} $$
-
2. Moreover, if
$V(c,b)$
is the unique diamond with extremities c and b included in
$V_c(a,b)$
then
$$ \begin{align*} V(a,c)\cap V(c,b)= \emptyset. \end{align*} $$
-
3. Finally a belongs to the opposite diamond
$V^*(c,b)$
and the diamond
$V(a,c)$
is contained in
$V^*(c,b)$
.
Figure 1 The nesting of
$V(c,b)$
in
$V_c(a,b)$
.
Figure 2 A positive
$5$
-configuration and some diamonds.
The proof of Lemma 2.8 will use the following statement.
Lemma 2.9. Let a and b be two transverse points of
$\mathbf {F}_\Theta $
and
$V_0$
a diamond with extremities a and b. Then, there exists a basis
$\mathcal {B}$
of neighborhoods of a such that for every U in
$\mathcal {B}$
, the intersection
$U\cap V_0$
is connected and nonempty.
Proof. Up to acting by an element of
$\mathsf G$
, we can assume that the stabilizer of b is
$\mathsf {P}_\Theta $
and that the stabilizer of a is
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
. The map from
$\mathfrak {u}_\Theta $
to
$\Omega _b$
given by
$ x \mapsto \exp (x)\cdot a$
is a
$\mathsf {L}_\Theta $
-equivariant diffeomorphism.
Consider the decomposition
$\mathfrak {u}_\Theta = \bigoplus _{i} V_i$
into
$\mathsf {L}_\Theta $
-irreducible factors. Let us fix an auxiliary Euclidean norm
$\Vert \cdot \Vert $
on
$\mathfrak {u}_\Theta $
such that the previous decomposition is orthogonal. There is a one-parameter subgroup
$\Lambda =\{ \lambda _t\}_{t\in \mathbb {R}}$
of
$\mathsf {L}_\Theta $
such that, for all i and for all v in
$V_i$
,
$\lambda _t\cdot v = e^{n_i t}v$
for some positive numbers
$n_i$
.
Let S be the unit sphere in
$\mathfrak {u}_\Theta $
for
$\Vert \cdot \Vert $
. Then the map from
$S\times \mathbb {R}$
to
$ \Omega _b\smallsetminus \{a\}$
, given by
$$ \begin{align*}g\colon (v,t)\mapsto \exp(\lambda_t \cdot v) \cdot a ,\end{align*} $$
is a diffeomorphism satisfying that for all v in S, all real numbers t and s
$$ \begin{align*}g(v,t+s)=\lambda_s \cdot g(v,t).\end{align*} $$
Thus since
$\Omega _a\cap \Omega _b$
is
$\Lambda $
-invariant we have the following property: for all v in S and all t,
$t'$
in
$\mathbb {R}$
,
$g(v,t)$
belongs to
$\Omega _a\cap \Omega _b$
if and only if
$g(v,t')$
belongs to
$\Omega _a\cap \Omega _b$
.
Thus, there is a connected open
$\Omega _0$
in S, such that the diamond
$V_0$
– being a connected component of
$\Omega _a\cap \Omega _b$
by the first item of Proposition 2.6 – is the image of
$\Omega _0\times \mathbb {R}$
by the map g. Let finally
$O_t$
be the images of
$S\times (-\infty , t)$
by g and
$U_t=O_t\cup \{a\}$
. Then
$\{U_t\}_{t\in \mathbb R}$
is a family of neighborhoods of a with the wanted property.
Proof of Lemma 2.8.
Let us first construct diamonds
$V^0(c,b)$
and
$V^0(a,c)$
included in
$V_c(a,b)$
. Let us write
$V_c(a,b)=\mathsf N_b\cdot a= \mathsf N_a\cdot b$
and consider the diamonds
$$ \begin{align*} V^0(c,b)=\mathsf N_b\cdot c, \ V^0(a,c)= \mathsf N_a\cdot c. \end{align*} $$
By construction
$c=n_b\cdot a= n_a\cdot b$
with
$n_b\in \mathsf N_b$
and
$n_a\in \mathsf N_a$
. By the semigroup property
$$ \begin{align*} \mathsf N_b\cdot n_b\subset \mathsf N_b, \ \mathsf N_a\cdot n_a\subset \mathsf N_a, \end{align*} $$
which leads to the inclusions
$$ \begin{align*} V^0(c,b)\subset V_c(a,b) , \ \ V^0(a,c)\subset V_c(a,b) . \end{align*} $$
We now prove that these specific diamonds are disjoint. By the construction and the inclusion above both
$V^0(a,c)$
and
$V^0(b,c)$
are connected components of
$V_c(a,b)\smallsetminus S_c$
. It follows that they are either equal or disjoint. By the sharpness property of
$\mathsf {N}$
, the identity element does not belong to the closure of
$\mathsf N\cdot n_a$
. Let thus O be an open set in
$\mathsf U_a$
containing the identity and with trivial intersection with
$\mathsf N\cdot n_a$
. Then
$O\cdot b$
is a neighborhood of b that does not intersect
$\mathsf N_a\cdot c=\mathsf N_an_a\cdot b$
. Thus b does not belong to the closure of
$V^0(a,c)$
. From the last item of Proposition 2.6,
$V^0(a,c)$
is hence different from
$V^0(c,b)$
and by the above discussion they are disjoint:
$$\begin{align*}V^0(a,c)\cap V^0(b,c)=\emptyset. \end{align*}$$
This concludes item (2) of the lemma.
Let us prove next the existence of the neighborhood U. Denote for any open set V, 
and denote
$V^c$
the complementary of V and recall that
$$ \begin{align*} \partial(V\cap W)\subset\ &\overline{V}\cap \overline{W}\smallsetminus (V\cap W)\\ =\ & \overline{V}\cap \overline{W} \cap (W^c \cup V^c)\\ =\ & (\overline{V}\cap \overline{W} \cap W^c) \cup (\overline{V}\cap \overline{W} \cap V^c)\\ =\ &(\partial V\cap \overline{W})\cup (\partial W\cap \overline{V}). \end{align*} $$
Let
$V(a,c)$
be any diamond with extremities a and c included in
$V_c(a,b)$
. Let U be a neighborhood of a such that
-
• the intersection of
$\overline {U}$
with
$S_c\cup S_b$
is empty, -
•
$V_c(a,b)\cap U$
is connected and nonempty.
The existence of this open set U is guaranteed by Lemma 2.9. From the first item we have that
$$ \begin{align*} \partial (V(a,c)\cap U)\subset((\partial V(a,c))\cap \overline{ U}) \cup (\overline{V}(a,c)\cap\partial U) \ \subset& \ (S_a\cup\partial U). \end{align*} $$
From the inclusion
$V(a,c)\subset V_c(a,b)$
we have
$$ \begin{align*} \partial (V(a,c)\cap U)\subset& \ \overline{V_c(a,b)\cap U}. \end{align*} $$
Since
$S_a\cup \partial U$
is included in the complementary of
$V_c(a,b)\cap U$
we furthermore have
$$ \begin{align*} (S_a\cup\partial U)\cap (\overline{V_c(a,b)\cap U})\ \subset\ & \partial (V_c(a,b)\cap U). \end{align*} $$
Thus combining these inclusions, we get
$$ \begin{align*} \partial (V(a,c)\cap U)\subset (S_a\cup\partial U) \cap (\overline{V_c(a,b)\cap U}) \subset \partial (V_c(a,b)\cap U) . \end{align*} $$
Now a simple connectedness argument shows that if A and B are two open sets, with B connected,
$A\subset B$
and
$\partial A\subset \partial B$
, then
$A=B$
. Thus, in our case,
$$ \begin{align*}V(a,c)\cap U=V_c(a,b)\cap U\not=\emptyset .\end{align*} $$
Since this is true for all diamonds with extremities a and c included in
$V_c(a,b)$
and since diamonds with the same extremities are either disjoint of equal, we finally conclude that there is a unique diamond with extremities a and c included in
$V_c(a,b)$
. This concludes item (1) of the lemma.
For the last item, observe that
$$ \begin{align*}a= n_b^{-1}c\in \mathsf{N}_b^{-1}c=V^*(c,b).\end{align*} $$
Since a belongs to the closure of
$V(a,c)$
, we have hence
$V(a,c)\cap V^*(c,b)\neq \emptyset $
. Furthermore
${V(a,c)\subset \Omega _c}$
and
$V(a,c)\subset V(a,b)\subset \Omega _b$
; this means that the connected set
$V(a,c)$
is contained in
$\Omega _a\cap \Omega _b$
. Therefore
$V^*(b,c)$
is the connected component of
$\Omega _a\cap \Omega _b$
containing
$V(a,b)$
: this is the sought for inclusion.
2.4 Positive configurations
The following definition plays a central role in this article:
Let
$p\geqslant 3$
and equip
$\{1,\ldots , p\}$
with the usual cyclic order.
Definition 2.10 (Positive configuration).
We say that a configuration
$(a_1,\ldots , a_p)$
in
$\mathbf {F}^{p}_{\Theta }$
is positive, if there exist diamonds
$V_{i,j}$
with extremities
$a_i$
and
$a_j$
for all
$i\not =j$
such that
-
1.
$V_{i,j}=V^*_{j,i}$
, -
2.
$a_j$
belongs to
$V_{i,k}$
, if
$(i,j,k)$
is cyclically oriented, -
3. we have
$V_{i,j}\subset V_{i,k}$
and
$V_{j,k}\subset V_{i,k}$
, if
$(i,j,k)$
is cyclically oriented.
Proposition 3.1 will give easier criteria to understand positive triples and quadruples and will show that the definition is equivalent to the definition given in the introduction.
Observe that, by properties (2) and (1) above, the choice of
$V_{i,k}$
among diamonds with extremities
$a_i$
and
$a_k$
is forced by the cyclic ordering. Furthermore, the fact that
$V_{i,k}$
does not depend on the index j between i and k involves the positivity of a subquadruple. It thus follows that if
$(a_1,\ldots ,a_p)$
is such that every cyclically oriented subquadruple is positive then
$(a_1,\ldots ,a_p)$
is positive.
By construction, every subconfiguration of a positive configuration is positive. On the real projective line, a configuration of p points with
$p>3$
is positive exactly if it is cyclically oriented, and a triple is positive if it consists of pairwise distinct elements.
Moreover
Proposition 2.11. Positivity of configurations is invariant under cyclic permutation and under the order reversing permutation. In particular
-
1. to be positive for a triple is invariant under all permutations,
-
2. to be positive for a quadruple is invariant under the dihedral group.
Proof. The definition is invariant under cyclic transformations. If
$\sigma _0$
is the reverse ordering, we choose the new diamonds
$V^\circ _{i,j}=V^*_{\sigma _0(i),\sigma _0(j)}$
.
2.5 Positive circles and
$\mathsf {PSL}_2({\mathbb R})$
Let
$\mathsf H$
be a subgroup in
$\mathsf G$
locally isomorphic to
$\mathsf {PSL}_2(\mathbb R)$
. An
$\mathsf H$
-circle in
$\mathbf {F}_\Theta $
is a closed
$\mathsf H$
-orbit, it can be parametrized by a circle map which is a
$\mathsf H$
-equivariant map from
${{\mathbf P}^1(\mathbb {R})}$
to
$\mathbf {F}_\Theta $
. The group
$\mathsf H$
is proximal if it contains a proximal element in
$\mathbf {F}_\Theta $
, that is, an element having an attracting fixed point on
$\mathbf {F}_\Theta $
.
Proposition 2.12 (
$\mathsf H$
-circle).
Given a positive structure, there exists
$\mathcal H$
, an
$\operatorname {Aut}_0(\mathfrak g)$
-orbit of pairs
$(\mathsf H,C)$
such that
$\mathsf H$
is a subgroup of
$\mathsf G$
locally isomorphic to
$\mathsf {PSL}_2(\mathbb R)$
, C is an
$\mathsf H$
-circle, satisfying the following properties
-
1.
$\mathsf H$
has a compact centralizer in
$\mathsf G$
; -
2. Any two distinct points on C are transverse;
-
3. Given a diamond V with extremities a and b, there exists
$(\mathsf H,C)$
in
$\mathcal H$
with C containing a and b, and such that C intersects the diamond V. Furthermore-
• If c is a point in C different from a and b, then
$(a,c,b)$
is a positive triple and are the two connected components of
$$ \begin{align*}V_c (a,b)\cap C\ \text{ and } V^*_c (a,b)\cap C ,\end{align*} $$
$C\smallsetminus \{a,b\}$
.
-
• If d belongs to the connected component of
$C\smallsetminus \{c,b\}$
not containing a, then
$$ \begin{align*}V_d(b,c)\subset V_d(a,b).\end{align*} $$
-
-
4. Given any three pairwise distinct points a, b, and c in
$\mathbf {F}_\Theta $
. Then there is at most one element
$(\mathsf {H},C)$
of
$\mathcal {H}$
such that C contains a, b, and c.
Note that in point (3) V needs to be equal to
$V_c(a,b)$
or
$V_{c}^{*}(a,b)$
.
Proof. Let
$s_b$
be a U-pinning at b such that
$V= s_b( \mathsf {N})\cdot a$
. One just picks the Lie subgroup associated with an
$\mathfrak {sl}_2$
-triple given by the Jacobson–Morozov theorem applied to a nilpotent element x chosen as in [Reference Guichard and Wienhard24, Section 7] so that
$\mathsf N_2=s_b(\exp (\mathbb R_{>0} x))$
is included in
$\mathsf N$
. The corresponding
$\mathfrak {sl}_2$
-triple is the
$\Theta $
-principal
$\mathfrak {sl}_2$
-triple introduced in [Reference Guichard and Wienhard24, Section 7]. Item (1) follows now from Fact 2.2.
Item (2) follows from [Reference Guichard and Wienhard24, Lemma 7.2].
For item (3), the existence is immediate by
$\operatorname {Aut}_0(\mathfrak g)$
-transitivity. The two connected components of
$C\smallsetminus \{a,b\}$
are
$\mathsf N_2\cdot a$
and
${\mathsf N}_{2}^{-1}\cdot a$
and are thus included in diamonds opposite to each other.
Moreover, for the last statement in item (3), let us write
$c=n\cdot a$
with n in
$\mathsf N_2$
. Observe that, by a deformation argument
$$ \begin{align*}V_{n\cdot d}(c,b)=V_d(c,b)\ \text{and}\ V_d(a,b)=V_c(a,b). \end{align*} $$
Then
$V_d(a,b)=\mathsf {N}\cdot a$
and
$$ \begin{align*}V_d(c,b)=V_{n\cdot d}(c,b)= n V_d(a,b)= n\mathsf N\cdot a\subset \mathsf{N}\cdot a=V_d(a,b), \end{align*} $$
where the inclusion holds by the semigroup property.
For the item (4), let us consider Z the stabilizer of a, b, and c. Then Z is precisely the stabilizer in
$\mathsf {L}_{a,b}$
of the element n in
$\mathsf {N}_2$
such that
$c=n\cdot a$
. Since
$\mathsf H$
is determined by
$\mathsf N_2$
, this implies that Z is in fact the centralizer of
$\mathsf {H}$
. This concludes the proof.
Remark 2.13. More detail on the construction of the
$\Theta $
-principal
$\mathfrak {sl}_2$
-triple can be found in [Reference Guichard and Wienhard24, Section 7]. Note that there are others
$\mathfrak {sl}_2$
-triples which induce positive maps from
${{\mathbf P}^1(\mathbb {R})}$
to
$\mathbf {F}_\Theta $
. For example, if
$\mathsf G$
is a split real Lie group, the principal
$\mathfrak {sl}_2$
-triple gives rise to such a map.
We fix once and for all such an
$\operatorname {Aut}_0(\mathfrak g)$
-orbit
$\mathcal H$
.
As an important example of positive configuration, we have
Proposition 2.14. Let
$(\mathsf {H}, C)$
be in
$\mathcal {H}$
. Any cyclically ordered configuration of points on C is positive.
Proof. It is enough to prove the results for triples and quadruples.
Let first
$(a_0,a_1,a_2)$
be a triple on C. By item (3) in Proposition 2.12,
$a_{i+1}$
belongs to a diamond with extremities
$a_i$
and
$a_{i+2}$
. Let us define (where indices are taken modulo
$3$
)
Then the properties of Definition 2.10 are obviously satisfied, and the triple is positive.
Let us now consider
$(a_0,a_1,a_2,a_3)$
a quadruple on C, such that
$a_{i+1}$
and
$a_{i+3}$
belongs to different components of
$C\smallsetminus \{a_{i},a_{i+2}\}$
. Observe that by a deformation argument we have
$$ \begin{align*}V_{a_{i+2}} (a_i,a_{i+3}) =V_{a_{i+1}}(a_i,a_{i+3}). \end{align*} $$
We now define
It then follows from item (3) of Proposition 2.12 that
$V_{a_{i+3}}(a_i,a_{i+2})=V^*_{a_{i+1}}(a_i,a_{i+2})$
, and thus that
$V_{i+2,i} = V_{i,i+2}^{*}$
.
From the equality
$V_{i,i+1}=V_{i+1,i}^*$
and from item (3) of Proposition 2.12,
$V_{i+1,i+2}\cap C$
is the connected component of
$C\smallsetminus \{a_{i+1},a_{i+2}\}$
not containing
$a_{i}$
and
$a_{i+3}$
. Let d be in
$V_{i+1,i+2}\cap C$
, then
$$ \begin{align*}V_{i+1,i+2}=V_d(a_{i+1},a_{i+2})\subset V_d(a_{i},a_{i+3})=V_{i,i+3}, \end{align*} $$
where, for the inclusion, we applied twice the last part of the item (3) of Proposition 2.12.
This concludes the proof.
2.6 Positive maps
Let S be a cyclically ordered set containing at least three points.
Definition 2.15 (Positive map).
A map f from S to
$\mathbf {F}_\Theta $
is positive if the image of every cyclically ordered quadruple is a positive quadruple, and the image of every cyclically ordered triple is a positive triple.Footnote
1
Observe then that the image of every cyclically ordered configuration by a positive map is a positive configuration.
By Proposition 2.14, for any
$(\mathsf {H}, C)$
in
$\mathcal {H}$
, C – seen as a map from
${\mathbf P}^1(\mathbb {R})$
to
$\mathbf {F}_\Theta $
– is positive.
3 Properties of positivity
We prove in this section, three main propositions concerning positivity:
-
• The first one, Proposition 3.1, gives various combinatorial properties of positive triples, quadruples, and configurations;
-
• The second one, Proposition 3.9, gives information about the limit of diamonds included in a given diamond;
-
• The last one, Proposition 3.14, shows that positive maps share the property of monotone maps: they coincide on a dense subset with a left-continuous positive map.
We also establish that certain elements in
$\mathsf G$
are
$\mathbf {F}_\Theta $
-proximal using positivity.
Several of the combinatorial properties of positive configurations have been addressed in [Reference Guichard and Wienhard24] with a more algebraic approach, for reader’s convenience, we provide here geometric proofs using the nesting properties of diamonds.
3.1 Combinatorics of positivity
The next proposition gives fundamental properties of positive triples and quadruples.
-
• The first one gives an easy criterion for positivity of triples, while the second and third concern quadruples. In particular, this shows that the definition of positivity given in the introduction is equivalent to Definition 2.10.
-
• The fourth one gives a recursive way to build positive configuration.
-
• The fifth and sixth give “exclusion” properties that are important in the study of positivity though they are not used in this paper.
We are going to prove this proposition and its corollary in the context of a group defined over
$\mathbb R$
, although by Tarski–Seidenberg Theorem, the statements will be true over every real closed field.
Proposition 3.1 (Combinatorial properties).
-
1. Assume that a and b are transverse and that c belongs to a diamond with extremities a and b, then
$(a,b,c)$
is positive. -
2. Assuming
$(a,x_0,b)$
and
$(a,y_0,b)$
are positive then
$(a,x_0,b,y_0)$
is positive if and only if
$V_{x_0}(a,b)=V^*_{y_0}(a,b)$
. -
3. Assuming
$(a,c,b)$
is positive and d belongs to
$V^*_a(c,b)$
, then
$(a,c,d,b)$
is positive. -
4. More generally, assume that
$(x_0,x_1,\ldots ,x_p)$
is a positive configuration and that
$y\in V^*_{x_2}(x_0,x_1)$
then is a positive configuration.
$$\begin{align*}(x_0,y,x_1,\ldots,x_p), \end{align*}$$
-
5. If
$(a,b,c,d)$
is positive, then
$(a,c,b,d)$
is not positive. -
6. Let
$x_0$
,
$x_1$
, and
$x_2$
be three points such that
$(a,x_i,b)$
is positive (
$i=0,1,2$
), then the three quadruples
$(a,x_0,b,x_1)$
,
$(a,x_1,b,x_2)$
, and
$(a,x_2,b,x_0)$
cannot all be positive.
Finally we have,
Corollary 3.2 (Necklace property).
Let
$(a,b,c)$
be a positive triple. Let
$\alpha $
,
$\beta $
, and
$\gamma $
be elements of
$V^*_a (b,c)$
,
$V^*_b(a,c)$
, and
$V^*_c(a,b)$
respectively. Then the triple
$(\alpha ,\beta ,\gamma )$
is positive.
The proof of this proposition and of Corollary 3.2 will be given in Section 3.1.3. It is important to remark that all these properties are true for configurations in
${\mathbf P}^1(\mathbb {R})$
. As an immediate consequence of Proposition 3.1, we also have that the intersections of diamonds arising from positive configurations of points are diamonds:
Corollary 3.3 (Intersection of diamonds).
Let
$(a,b,c,d)$
be a positive quadruple. Then the following equality holds
$$\begin{align*}V^{*}_{a}(b,c) = V^{*}_{d}(a,c) \cap V^{*}_{a}(b,d).\end{align*}$$
Proof. The inclusion
$V^{*}_{a}(b,c) \subset V^{*}_{d}(a,c) \cap V^{*}_{a}(b,d)$
is a direct consequence of the nesting properties of diamonds (point (3) of Lemma 2.8).
Let us prove the reverse inclusion. Let x be in
$V^{*}_{d}(a,c) \cap V^{*}_{a}(b,d)$
. This means that the quadruples
$(a,b,x,d)$
and
$(a,x,c,d)$
are positive. From the first one we get that b belongs to
$V_{d}^{*}(a,x)$
and from the second one we get that
$V_{d}^{*}(a,x) = V_{c}^{*}(a,x)$
. Point (4) of Proposition 3.1 applied with
$y=b$
and
$(x_0,x_1, x_2)=(a,x,c)$
gives that
$(a,b,x,c)$
is positive and hence that x belongs to
$V^{*}_{a}(b,c)$
.
3.1.1 Triples and quadruples
Before addressing the proof of Proposition 3.1, we establish a number of preliminary statements.
Lemma 3.4. A triple
$(a_0,a_1,a_2)$
is positive if and only if
$a_0$
,
$a_1$
,
$a_2$
belong to diamonds with extremities
$(a_1,a_2)$
,
$(a_2,a_0)$
and
$(a_0,a_1)$
respectively.
Proof. We just need to prove the “if” part. Let, for
$i=0,1,2$
(indices are taken modulo
$3$
)
Observe that
$$ \begin{align*} V_{i,i+1} =V^*_{i+1,i}. \end{align*} $$
Then Lemma 2.8(3) provides all the necessary inclusions needed to prove that the triple is positive.
The following lemma gives a way to go from positive triples to positive quadruples.
Lemma 3.5. Let
$(a_0,a_1,a_2,a_3)$
be a quadruple. Assume that all subtriples are positive. Then the quadruple
$(a_0,a_1,a_2,a_3)$
is positive, if and only if, for all i (indices are taken modulo
$4$
)
$$ \begin{align} a_i&\in V^*_{a_{i+2}}(a_{i+1},a_{i+3}), \end{align} $$
$$ \begin{align} a_{i+2}&\in V_{a_{i+1}}(a_{i},a_{i+3}). \end{align} $$
Proof. The “only if” part follows from the definition. It remains to prove the “if” part. Let
where in the second line we used the hypothesis (1), while in the first and last lines we used the hypothesis (2) and the fact that if d belongs to
$V_a(b,c)$
then
$V_d(b,c)=V_a(b,c)$
. Hence by definition
$$ \begin{align*} V_{i,i+1}=V^*_{i+1,i}, V_{i,i+2}=V^*_{i+2,i}. \end{align*} $$
It thus follows that for all i and j,
$$ \begin{align} V_{i,j}=V^*_{j,i}. \end{align} $$
From the positivity of the subtriple
$(a_i,a_{i+1},a_{i+2})$
, we get the inclusions
$$ \begin{align} V_{i,i+1}\subset V_{i,i+2}, \ \ V_{i+1,i+2}\subset V_{i,i+2}. \end{align} $$
From the positivity of the triple
$(a_{i},a_{i+1},a_{i+3})$
we get the inclusions
$$ \begin{align}\ \ V_{i,i+1}=V^*_{a_{i+3}}(a_i,a_{i+1})&\subset V_{a_{i+1}}(a_i,a_{i+3})=V_{i,i+3}, \end{align} $$
$$ \begin{align} V_{i+1,i+3}=V^*_{a_i}(a_{i+1},a_{i+3})&\subset V_{a_{i+1}}(a_{i},a_{i+3})=V_{i,i+3}. \end{align} $$
Similarly the positivity of the triple
$(a_{i},a_{i+2},a_{i+3})$
yields
$$ \begin{align} V_{i+2,i+3}=V^*_{a_{i}}(a_{i+2},a_{i+3})&\subset V_{a_{i+2}}(a_i,a_{i+3})=V_{i,i+3}, \end{align} $$
$$ \begin{align} \ \ V_{i,i+2}=V^*_{a_{i+3}}(a_{i},a_{i+2})&\subset V_{a_{i+2}}(a_{i},a_{i+3})=V_{i,i+3}. \end{align} $$
All together the equation (3) as well as the inclusions (4), (5), (6), (7), and (8) prove that
$(a_0,a_1,a_2,a_3)$
is a positive quadruple.
3.1.2 Deformation lemmas
We need to prove some deformation lemmas.
Lemma 3.6 (Deforming triples).
Let
$a(t)$
,
$b(t)$
, and
$c(t)$
be continuous arcs from
$[0,1]$
to
$\mathbf {F}_\Theta $
such that
-
1. for all t in
$[0,1]$
,
$a(t)$
,
$b(t)$
, and
$c(t)$
are pairwise transverse, -
2. the triple
$(a(0),b(0),c(0))$
is positive.
Then, for all t,
$(a(t),b(t),c(t))$
is a positive triple.
Proof. The hypothesis tells us that
$$ \begin{align*} c(t)\in\Omega_{a(t)}\cap\Omega_{b(t)}, \ \ a(t)\in\Omega_{c(t)}\cap\Omega_{b(t)}, \ \ b(t)\in\Omega_{a(t)}\cap\Omega_{c(t)}. \end{align*} $$
By hypothesis, there are diamonds
$V(a(0),b(0))$
,
$V(c(0),b(0))$
, and
$V(c(0),a(0))$
such that
$$ \begin{align*} c(0) \in V(a(0),b(0)),\ \ a(0) \in V(c(0),b(0)),\ \ b(0) \in V(c(0),a(0)). \end{align*} $$
We can find continuous maps
$t\mapsto V(a(t),b(t))$
,
$t\mapsto V(c(t),b(t))$
, and
$t\mapsto V(c(t),a(t))$
in the space of diamonds (one always has that
$V(e,d)$
is a diamond with extremities e and d). We now use the fact that a diamond with extremities e and d is a connected component of
$\Omega _e\cap \Omega _d$
(Proposition 2.6). Then by continuity, for all t
$$ \begin{align*} c(t)\in V(a(t),b(t)),\ \ a(t) \in V(c(t),b(t)),\ \ b(t) \in V(c(t),a(t)). \end{align*} $$
Thus the result follows from Lemma 3.4.
Similarly
Lemma 3.7 (Deforming quadruples).
Let
$\gamma $
and
$\eta $
be continuous arcs from
$[0,1]$
to
$\mathbf {F}_\Theta $
such that there exist a and b in
$\mathbf {F}_\Theta $
satisfying
-
1. for all t in
$[0,1]$
,
$a,\gamma (t),b, \eta (t)$
are pairwise transverse, -
2. the quadruple
$(a,\gamma (0),b,\eta (0))$
is positive.
Then, for all t,
$(a,\gamma (t),b,\eta (t))$
is a positive quadruple.
Proof. By applying Lemma 3.6, we obtain that all the subtriples of
$(a,\gamma (t),b, \eta (t))$
are positive. By Lemma 3.5, we only need to check that
$$ \begin{align*} a\in V_{b}^{*}(\gamma(t),\eta(t)) & ,\ b\in V_{a}^{*}(\gamma(t),\eta(t)),\\\gamma(t)\in V^{*}_{\eta(t)}(a,b) & ,\ \eta(t)\in V^{*}_{\gamma(t)}(a,b),\\\gamma(t)\in V_{a}(\eta(t),b) & ,\ b\in V_{\gamma(t)}(a,\eta(t)),\\\eta(t)\in V_b(\gamma(t),b) & , \ a\in V_{\eta(t)}(b,\gamma(t)). \end{align*} $$
Using again the fact that a diamond with extremities c and d is a connected component of
$\Omega _c\cap \Omega _d$
(Proposition 2.6), the statement follows by continuity.
Finally we also have as an immediate consequence of the connectedness of the positive cone:
Lemma 3.8 (Connectedness).
Let a and b be two transverse points.
-
1. Assume c is so that
$(a,c,b)$
is positive. Then there is
$(\mathsf {H}, C)$
in
$\mathcal {H}$
such that a and b belong to C, and there is a path
$t\mapsto c(t)$
from
$[0,1]$
to
$V_c(a,b)$
connecting
$c=c(0)$
to
$c(1)$
so that
$(a,c(1),b)$
is a positive triple on C. -
2. Assume furthermore that d belongs to
$V^*_a(c,b)$
then there are
$(\mathsf {H}, C)$
in
$\mathcal {H}$
, a path
$t\mapsto c(t)$
as in the previous item, and a path
$t\mapsto d(t)$
from
$[0,1]$
to
$V_c(a,b)$
, so that
$d(t)\in V^*_a(c(t),b)$
and
$(a,c(1),d(1),b)$
are on C.
Proof. Using a U-pinning at b, we identify
$\mathsf N$
with a positive semigroup
$\mathsf N_b$
in
$\mathsf U_b$
such that we have
$V_c(a,b)=\mathsf N_b\cdot a$
. The first point just follows from the connectedness of the positive semigroup
$\mathsf N_b$
. For use in the second point we take a path
$t \mapsto c(t)$
which is constant for
$t>1/2$
.
Recall that
$d=m_0\cdot c$
, with
$m_0\in \mathsf N_b$
. Let us define, for
$t\in [0,1/2]$
,
$$ \begin{align*}d(t)=m_0\cdot c(t),\end{align*} $$
then we have by the semigroup property
$d(t)\in V^*_{a}(c(t),b)$
. Observe also that
$d(0)=d$
. Then for
$t\in [1/2,1]$
, we have
$c(t)=c(1/2)$
, and we choose, using that C contains elements of
$V^*_a(c(1/2),d)$
(cf. Proposition 2.12.(3)), a path
$t\mapsto d(t)$
with
$d(t)\in V^*_a(c(1/2),d)$
, and such that
$d(1)$
belongs to C.
3.1.3 Proof of the combinatorial properties
Proof of item (1) of Proposition 3.1.
Assume
$(a,b,c)$
satisfies the hypothesis. Let
$(\mathsf {H}, C)$
in
$\mathcal {H}$
and
$t\mapsto c(t)$
obtained in Lemma 3.8. On C, a triple is positive if and only if the three points are pairwise distinct, the result thus follows from Lemma 3.6.
Proof of item (2) of Proposition 3.1.
The “only if” part follows from the definition. Then for the “if” part we find, by Lemma 3.8,
$(\mathsf {H}, C)$
in
$\mathcal {H}$
and paths
$t\mapsto x(t)$
and
$t\mapsto y(t)$
in
$V_{x_0}(a,b)$
and
$V_{y_0}(a,b)$
respectively, such that
$(x(0),y(0))=(x_0,y_0)$
and
$x(1)$
,
$y(1)$
are on C, the
$\mathsf H$
-circle passing through a and b. Then
$(a,x(1),b,y(1))$
is positive and so is
$(a,x_0,b,y_0)$
by Lemma 3.7, since
$x(t)$
and
$y(t)$
are transverse thanks to Proposition 2.7.
Proof of item (3) of Proposition 3.1.
From the connectedness Lemma 3.8 we obtain
$(\mathsf {H},C)$
in
$\mathcal {H}$
and paths
$t\mapsto c(t)$
,
$t\mapsto d(t)$
such that
$a,c(t),d(t),b$
are pairwise transverse,
$c(0)=c$
,
$d(0)=d$
,
$(a,c(1),d(1),b)$
on C and
$d(1)\in V_{a}(c(1),b)$
. In particular
$(a,c(1),d(1),b)$
is positive and thus by the deformation Lemma 3.7,
$(a,c,d,b)$
is positive.
Proof of item (4) of Proposition 3.1.
This is an immediate consequence of item (3) and the fact that in order to check the positivity of a configuration one only needs to check the positivity of subtriples and subquadruples.
Proof of item (5) of Proposition 3.1.
If
$(a,b,c,d)$
is positive, we have the strict inclusion
$V_a(b,d)\subset V_a(c,d)$
and if
$(a,c,b,d)$
is positive, we have the strict inclusion
$V_a(c,d)\subset V_a(b,d)$
. Hence a contradiction.
Proof of item (6) of Proposition 3.1.
Assume that
$(a,x_0,b,x_1)$
is positive. Then
$V= V_{x_0}(a,b)$
and
$ V_{x_1}(a,b)$
are opposite diamonds. If both
$(a,x_1,b,x_2)$
and
$(a,x_0,b,x_2)$
are positive then we get that
$x_2$
belongs to both V and
$V^*$
, which is a contradiction.
Proof of the necklace property (Corollary 3.2).
Let us first remark that from item (4) of Proposition 3.1, applied three times (and using cyclic invariance of positivity) the configuration
$$ \begin{align*}(a,\gamma, b,\alpha, c,\beta), \end{align*} $$
is positive. Thus
$(\gamma ,\alpha ,\beta )$
is positive.
3.2 Inclusion of diamonds
Proposition 3.9 (Boundedness property).
Let
$(a,b,d)$
be a positive triple and let
$c\in V_b(a,d)$
. Assume that there exist sequences
$ \{b_m\}_{m\in \mathbb N}$
and
$ \{c_m\}_{m\in \mathbb N}$
, converging respectively to b and c and such that, for all m,
$(a,b_m,c_m,d)$
is a positive quadruple. Then the sequence
$(\overline {V}{}^{*}_d(b_m,c_m))$
converges in the Hausdorff topology and
$$ \begin{align*} \lim_{m\rightarrow\infty}\left(\overline{V}{}^{*}_d(b_m,c_m)\right)\ \subset V_c(a,d). \end{align*} $$
In particular,
Corollary 3.10 (Inclusion).
Let
$(a,b,d,e)$
be a positive quadruple in
$\mathbf {F}_\Theta $
. Then
$$ \begin{align*} \overline{V}{}^{*}_e(b,d)\subset V_b(a,e). \end{align*} $$
Let
$(a,b,c,d,e)$
be a positive quintuple of points in
$\mathbf {F}_\Theta $
. then
$$ \begin{align*} \overline{V}{}_c(b,d)\subset V_c(a,e). \end{align*} $$
Proposition 3.9 will be proved in Section 3.2.2. Corollary 3.10 will be proved in Section 3.2.1 as the consequence of an intermediate statement.
3.2.1 Preliminaries on circles
Let
$V(a,d)$
be a diamond with extremities a and b, and let
$(\mathsf {H},C)$
be in
$\mathcal {H}$
such that a and d belong to C and
$C\cap V(a,d)\neq \emptyset $
.
-
• Let
$ \delta =\{\delta {_t}\mid t\in \mathbb R\}$
, be the
$1$
-parameter group in
$\mathsf H$
for which d is the attracting fixed point and a is the repelling fixed point for the element
$\delta _t$
(
$t>0$
). The corresponding basins of attraction/repulsion are denoted by -
• Let
$\gamma =C\smallsetminus \{a,d\}$
. -
• Let F be the set of fixed points of
$\delta $
in
$\mathbf {F}_\Theta $
.
The result of this section is
Proposition 3.11. For any e in
$\gamma $
, we have
$\overline {V}{}^{*}_a(e,d)\subset \Omega _a$
.
This proposition implies Corollary 3.10:
Proof of the Corollary 3.10.
Applying Proposition 3.11, we get that
$$ \begin{align*} \overline{V}{}^{*}_a(b,d)\subset \overline{V}{}^{*}_a(b,e)\cap \overline{V}{}^{*}_e(d,a)\subset \Omega_e\cap\Omega_a. \end{align*} $$
Since
$V_d({a,e})$
is a connected component of
$\Omega _e\cap \Omega _a$
containing
$V^*_a(b,d)$
it follows that
$$ \begin{align*} \overline{V}{}^{*}_a(b,d)\subset V_{d}(a,e). \end{align*} $$
This proves the first part of the corollary.
Suppose now that
$(a,b,c,d,e)$
is positive. Then the equalities
$V_b(a,e)=V_c(a,e)$
and
$V_c(b,d) = V_{e}^{*}(b,d)$
together with the first part imply the inclusion
$\overline {V}{}^{*}_e(b,d)\subset V_c(a,e)$
.
Recall that
$S_a$
is complementary to
$\Omega _a$
. In order to prove Proposition 3.11, we introduce the following sets for e in
$\gamma $
We will first prove that the sets
$J(e)$
and
$F(e)$
are empty. We first prove the following lemma
Lemma 3.12. Let
$\gamma _1$
be one of the two connected components of
$\gamma $
. For any e in
$\gamma _1$
,
-
1.
$J(e)$
is invariant by the semigroup , that is,
$\delta _t( J(e))\subset J(e)$
for all
$t>0$
, -
2.
$F(e)$
is independent of the choice of e in
$\gamma _1$
, -
3. if
$J(e)$
is not empty, so is
$F(e)$
, -
4. for all c and b in
$\mathbf {F}_\Theta $
such that
$(a,c,d,b)$
is a positive quadruple and then
$$ \begin{align*}V_{a}^{*}(c,d)\cap\gamma_1\neq \emptyset , \end{align*} $$
$$ \begin{align*} F(e)\subset S_c\cup S_b. \end{align*} $$
, that is, 
Proof. We prove the first point. By Proposition 2.14 for
$t>0$
$$ \begin{align*} V^*_a(\delta_t(e),d)\subset V^*_a(e,d). \end{align*} $$
This implies that
$V^*_a(e,d)$
is invariant by
$\delta ^+$
and so is
$J(e)$
.
The second point is a consequence that F is pointwise fixed by
$\delta _t$
:
$$ \begin{align*} J(\delta_t(e))\cap F=\delta_t(J(e))\cap F=J(e)\cap F. \end{align*} $$
The third point is a consequence of the hyperbolic nature of the subgroup
$\delta $
: indeed, using a linear representation of
$\mathsf G$
we can assume that
$\delta $
is a one-parameter subgroup of diagonal matrices and that
$J(e)$
is a nonempty closed subset of the projective space invariant by the semigroup
$\delta ^+$
; in this case, every ray
$( \delta _t ( x))_{t\geqslant 0}$
(for x in
$J(e)$
) has a limit that is a fixed point of
$\delta $
.
Let us prove now the last point. Let
$(a,c,d,b)$
be as in the hypothesis. Thus
$V_{a}^{*}(c,d)\cap \gamma _1\not =\emptyset $
and by point (2) we can choose e in this intersection. By Proposition 3.1.(4),
$(a,c,e,d,b)$
is a positive configuration and hence
$V^{*}_{a}(e,d) \subset V^{*}_{a}(c,b)$
. From this we get
$$ \begin{align} F(e)\subset J(e)=\left(\overline{V}{}^{*}_a(e,d)\cap S_a\right)\subset\left( \overline{V}{}^{*}_a(c,b)\cap S_a\right). \end{align} $$
Now, we remark that since
$(a,c,d,b)$
is positive, we have
$$ \begin{align*}V^{*}_{a}(c,b)\subset\Omega_a.\end{align*} $$
Thus
$$ \begin{align} \overline{V}{}_a^{*}(c,b)\cap S_a \subset \overline{V}{}_a^{*}(c,b)\smallsetminus V_{a}^{*}(c,b). \end{align} $$
But since
$V^{*}_{a}(c,b)$
is a connected component of the open set
$$ \begin{align*}\Omega_c\cap\Omega_b=\mathbf{F}_\Theta \smallsetminus \left(S_c\cup S_b\right),\end{align*} $$
we get
$$ \begin{align} \left(\overline{V}{}^{*}_a(c,b)\smallsetminus V^{*}_{a}(c,b)\right)\subset \left(S_c\cup S_b\right). \end{align} $$
Combining inclusions (9), (10) and (11), we get that
$$ \begin{align*} F(e)\subset \left(S_c\cup S_b\right).\\[-37pt] \end{align*} $$
We can now prove Proposition 3.11, in other words that
$J(e)$
is empty. By item (3) of Lemma 3.12, it suffices to show that
$F(e)$
is empty. The fact that
$F(e)$
is empty follows from item (4) of Lemma 3.12 and the following result.
Lemma 3.13. Let Q be a subset of
$\mathbf {F}_\Theta $
. Assume that there exist nonempty open sets U and V such that for all c in U, and for all b in V,
$$ \begin{align*} Q\subset S_c\cup S_b, \end{align*} $$
then Q is empty.
Proof. Let q be in Q and set 
. Then Z is a proper closed Zariski subset of
$\mathbf {F}_\Theta $
. Observe that if
$u\not \in Z$
, then
$$ \begin{align*} q\notin S_u. \end{align*} $$
On the other hand we can find u in the nonempty set
$U\smallsetminus Z$
and v in the nonempty set
$V\smallsetminus Z$
, and by hypothesis
$q\in S_u\cup S_v$
. This shows that
$q\neq q$
and concludes the proof.
3.2.2 Proof of the boundedness Proposition 3.9.
We use the notation of the previous paragraph.
First note that
$\overline {V}{}^{*}_a(b_m,c_m) = \overline {V}{}^{*}_d(b_m,c_m) = \overline {V}{}^{*}_a(b_m,d) \cap \overline {V}{}^{*}_d(a,c_m)$
(cf. Corollary 3.3). Since the sequences
$ \{b_m\}_{m\in \mathbb N}$
and
$ \{c_m\}_{m\in \mathbb N}$
converge, the sequences of closures of diamonds
$(\overline {V}{}^{*}_a(b_m,d)) $
and
$(\overline {V}{}^{*}_d(a,c_m))$
also converge and thus the sequence
$(\overline {V}{}^{*}_a(b_m,c_m))$
converges. Let C be an
$\mathsf H$
-circle through a and d such that 
is not empty. Since being positive is an open condition for quadruples, we can find e and f in
$\gamma $
so that
$(e,b_m,c_m,f)$
is positive for m large enough as well as
$(a,e,f,d)$
. Thus
$$ \begin{align*}V^{*}_{a}(b_m,c_m)\subset V^{*}_{a}(e,f). \end{align*} $$
Applying Proposition 3.11, we get that
$$ \begin{align*}\overline{V}{}^{*}_a(b_m,c_m)\subset \overline{V}{}^{*}_a(e,f)\subset V_c(a,d), \end{align*} $$
which easily implies the result.
3.3 Left and right limits of positive maps
Our main result is
Proposition 3.14 (Existence of left and right limits).
Let S be a totally ordered set and
$\phi $
be a positive map from S to
$\mathbf {F}_\Theta $
.
Let
$\{b_n\}_{n\in \mathbb N}$
be a sequence in S such that there exist a, b, and c in S with
$a< b_n\leqslant b_{n+1}\leqslant b<c$
, for all n. Then
$\{\phi (b_n)\}_{n\in \mathbb N}$
converges to a point y in
$V_b(a,c)$
.
Symmetrically, let
$\{a_n\}_{n\in \mathbb N}$
be a sequence of points such that
$c<a\leqslant a_{n+1}\leqslant a_n<b$
. Then
$\{\phi (a_n)\}_{n\in \mathbb N}$
converges to a point y in
$V_a(c,b)$
.
Remark 3.15. In the case of Lusztig’s total positivity, this statement was proved in [Reference Fock and Goncharov17, Theorem 7.4], this is also proved for
$\mathsf {SL}_n(\mathbb R)$
in [Reference Labourie31, Section 5].
As an immediate corollary, we show that positive maps defined on dense subsets extend to positive maps. More precisely:
Corollary 3.16 (Extension of positive maps).
Let A be dense subset in
$[0,1]$
. Assume that we have a positive map
$\xi $
from A to
$\mathbf {F}_\Theta $
. Then there exist
-
• a unique left-continuous positive map
$\xi _+$
from
$[0,1]$
to
$\mathbf {F}_\Theta $
such that
$\xi $
coincide with
$\xi _+$
on a dense subset of A, -
• a unique right-continuous positive map
$\xi _-$
from
$[0,1]$
to
$\mathbf {F}_\Theta $
such that
$\xi $
coincide with
$\xi _-$
on a dense subset of A.
Moreover,
-
• for any ordered quadruple
$(x,y,z,t)$
of pairwise distinct points in
$[0,1]$
is a positive quadruple for any choice of
$$ \begin{align*}(\xi_\epsilon(x),\xi_\eta(y),\xi_\nu(z),\xi_\beta(t)), \end{align*} $$
$\epsilon $
,
$\eta $
,
$\nu $
, and
$\beta $
in
$\{+,-\}$
,
-
• if
$ \{x_m\}_{m\in \mathbb N}$
,
$ \{z_m\}_{m\in \mathbb N}$
are sequences in
$[0,1]$
converging to y, with for all n,
$x_m<y<z_m$
, then for any sequences
$$ \begin{align*}\lim_{m\to\infty}\xi_{\epsilon_m}(x_m)=\xi_+(y), \ \lim_{m\to\infty}\xi_{\eta_m}(z_m)=\xi_-(y), \end{align*} $$
$ \{\epsilon _m\}_{m\in \mathbb N}$
and
$ \{\eta _m\}_{m\in \mathbb N}$
in
$\{+,-\}$
.
Proof of Proposition 3.14.
Let us define
$x_n=\phi (b_n)$
. We can write
$x_n=m_n\cdot x_{n-1}$
, with
$m_n\in {\mathsf N}_{\phi (c)}$
. Thus by induction we have
$$ \begin{align*} x_n=m_n\cdots m_1\cdot x_0. \end{align*} $$
But we know that
$V^{*}_{\phi (c)}(x_0, \phi (b))$
is a relatively compact region of
$\Omega _{\phi (c)}$
by Proposition 3.9. Thus
$\{x_n\}_{n\in \mathbb N}$
is relatively compact in
$\Omega _{\phi (c)}$
. It follows that
$$ \begin{align*} \pi_n=m_n\cdots m_1, \end{align*} $$
is a bounded sequence in
${\mathsf N}_{\phi (c)}$
. We now prove that this sequence converges. Assume that we have subsequences that converge to different limits u and v. After extracting further subsequence, we may find subsequences
$$ \begin{align*} q_i=\pi_{n_i},\ \ p_i=\pi_{m_i},\text{ with } n_i\leqslant m_i .\end{align*} $$
such that
$\{q_i\}_{i\in \mathbb N}$
converges to u and
$\{p_i\}_{i\in \mathbb N}$
converges to v. It follows that
$u=w_1\cdot v$
with
$w_1\in \overline {\mathsf N}_{\phi (c)}$
. Symmetrically,
$v=w_0\cdot u$
with
$w_0\in \overline {\mathsf N}_{\phi (c)}$
. It follows that
$w_0\cdot w_1=1$
, thus
$w_0$
and
$w_1$
are invertible in the closed semigroup
$\overline {\mathsf N}_{\phi (c)}$
, hence equal to the identity. In particular
$u=v$
and
$\{x_n\}_{n\in \mathbb N}$
converges.
The proof for the sequence
$\{a_n\}_{n\in \mathbb N}$
is symmetric.
3.3.1 Positivity and continuity
In some cases, it suffices to show that the image of every triple is positive
Proposition 3.17 (Triples suffice).
Let
$\phi $
be a continuous map from an interval I to
$\mathbf {F}_\Theta $
such that the image of every ordered triple is positive, then
$\phi $
is positive.
Proof. Since
$[x,y]$
is connected and
$\phi $
is continuous,
$V_{\phi (z)}(\phi (x),\phi (y))$
does not depend on z in
$]x,y[$
, and we denote it by
$W(x,y)$
.
Let
$V(t,y)$
be the unique diamond with extremities
$\phi (t)$
and
$\phi (y)$
obtained in Lemma 2.8 so that
$$ \begin{align} V(t,y)\subset W(x,y). \end{align} $$
Main step: We first prove that if
$x<t<y$
, then
$$ \begin{align} W(t,y)=V(t,y). \end{align} $$
Let us consider
$$ \begin{align*}U=\{t\in ]x,y[\mid W(t,y)=V(t,y)\}. \end{align*} $$
Let us write
$W(x,y)=\mathsf N \cdot \phi (x)$
, where
$\mathsf N$
is an open semigroup in
$\mathsf U_{\phi (y)}$
. We can thus write
$\phi (t)=n_t\cdot \phi ( x)$
, with
$t\mapsto n_t$
a continuous map defined on
$]x,y[$
with values in
$\mathsf N$
; the limit of
$n_t$
when t tends to x is equal to
${\operatorname {id}}$
. Then
$V(t,y)=n_t W(x,y)$
. We now proceed to the proof and show that U is open, nonempty, and closed.
-
1. The set U is also the set of t for which there exists s, with
$t<s<y$
such that
$\phi (s)$
is in
$V(t,y)$
. In other words,
$n_t^{-1}n_{s}$
belongs to
$\mathsf N$
. Thus U is open. -
2. Since
$\mathsf N$
is open, given s, for all t close enough to x we have
$n_t^{-1} n_s$
is in
$\mathsf {N}$
. Thus
$n_s\in n_t\mathsf N$
, hence
$\phi (s)\in V(t,y)$
. Therefore U is nonempty. -
3. Let t be in the closure of U. Let s be
$t$
and let
$ \{t_m\}_{m\in \mathbb N}$
be a sequence in U converging to t. Thus
$ \{n_{t_m}^{-1} n_s\}_{m\in \mathbb N}$
converges to
$n_{t}^{-1} n_{s}$
. Since, for m big enough, we have
$s>t_m$
, the element
$n_{t_m}^{-1} n_s$
belongs to
$\mathsf N$
; hence
$n_{t}^{-1} n_{s}$
belongs to
$\overline {\mathsf N}$
, that is,
$\phi (s)$
belongs to
$\overline {V}(t,y)$
. As the map
$\phi $
is transverse,
$\phi (t)$
is transverse to
$\phi (s)$
and this implies that
$\phi (s)$
belongs to
$V(t,y)$
and
$n_{t}^{-1} n_{s}$
belongs to
${\mathsf N}$
. Therefore t belongs to U, and we have completed the proof that U is closed.
The proof of the Equation (13) is now complete.
Conclusion: Let
$(a,b,c,d)$
so that
$a<b<c<d$
, with all subtriples of
$(\phi (a),\phi (b),\phi (c),\phi (d))$
positive. By item (3) of Proposition 3.1, we only have to prove that
$$ \begin{align*}V_{\phi(d)}(\phi(a), \phi(b))=V^{*}_{\phi(c)}( \phi(a), \phi(b)), \end{align*} $$
Observe that
$$ \begin{align*}W(a,b)\subset W(a,d)=V_{\phi(b)}(\phi(a),\phi(d)),\end{align*} $$
by Equation (13). Thus
$\phi (d)$
does not belong to
$W(a,b)$
and hence belongs to
$W^{*}(a,b)$
by Lemma 2.8. We thus have
$$ \begin{align*}V_{\phi(d)}(\phi(a),\phi(b))=W^*(a,b)=V^{*}_{\phi(c)}(\phi(a),\phi(b)).\end{align*} $$
This completes the proof that the quadruple
$(\phi (a),\phi (b),\phi (c),\phi (d))$
is positive, hence of the proposition.
3.4 Proximal elements
In this section we show that positive equivariant maps give rise to proximal elements.
We first prove the following proposition for elements in
$\mathsf G$
:
Proposition 3.18. Let g be in
$\mathsf G$
and let
$g^-$
,
$g^+$
, and a be in
$\mathbf {F}_\Theta $
such that
$g^-$
and
$g^+$
are fixed by the action of g and that the quadruple
$(g^-, a, g\cdot a, g^+)$
is positive.
Then the action of g on
$\mathbf {F}_\Theta $
is proximal, its attracting fixed point is
$g^+$
, and its repelling fixed point is
$g^-$
.
Proof. Up to the action of
$\operatorname {Aut}_0(\mathfrak g)$
we can assume that
$g^-$
is the point in
$\mathbf {F}_\Theta $
with stabilizer equal to
$\mathsf {P}_\Theta $
,
$g^+$
is the point with stabilizer
$\mathsf {P}_{\Theta }^{\mathrm {opp}}$
, and that
$a=n\cdot g^+$
with n in
$\mathsf {N}\subset \mathsf {U}_\Theta $
. One then has g belonging to
$\mathsf {L}_\Theta $
and
$g\cdot a = n'\cdot g^+$
with
$n'$
in
$\mathsf {N}$
equal to
$g ng^{-1}$
.
For every
$\alpha $
in
$\Theta $
, we denote by
$\pi _\alpha \colon \mathfrak {u}_\Theta \to \mathfrak {u}_\alpha $
the
$\mathsf {L}_\Theta $
-equivariant projection and by
$C_\alpha $
the salient invariant open convex cone in
$\mathfrak {u}_\alpha $
defining positivity (cf. Remark 2.3).
Let
$\alpha $
be in
$\Theta $
. The positivity of the quadruple
$(g^-, a, g\cdot a, g^+)$
implies that the elements
$x=\pi _\alpha ( \log n)$
and
$x'=\pi _\alpha ( \log n')$
are both in
$C_\alpha $
and that
$x-x'$
also belongs to
$C_\alpha $
. Let A be the automorphism of
$\mathfrak {u_\alpha }$
given by the restriction of the adjoint action of g to the subspace
$\mathfrak {u}_\alpha $
; one has thus
$A(C_\alpha )=C_\alpha $
and
$A(x)=x'$
. This implies that the set 
is sent into
$(-x'+C_\alpha )\cap (x'-C_\alpha )$
by A (where
$x-C_\alpha =\{ x-y \mid y\in C_\alpha \}$
). Therefore the element A is contracting for the norm on
$\mathfrak {u}_\alpha $
whose unit ball is the open convex set B.
We deduce from this that all the eigenvalues of the adjoint action of g on
$\bigoplus _{\alpha \in \Theta } \mathfrak {u}_\alpha $
are of modulus less than
$1$
. Since this subspace generates
$\mathfrak {u}_\Theta $
[Reference Kostant30], we have also that all the eigenvalues of g on
$\mathfrak {u}_\Theta $
are of modulus
$<1$
. This means precisely that
$g^+$
is an attracting fixed point for the action of g on
$\mathbf {F}_\Theta $
and thus g is proximal. For the same reasons,
$g^-$
is the repelling fixed point of g.
From this, we immediately get:
Proposition 3.19. Let
$\gamma $
be a homeomorphism of the circle
$S^1$
having one attracting fixed point
$\gamma ^+$
and one repelling fixed
$\gamma ^-$
in
$S^1$
. Let
$S\subset S^1$
be an infinite
$\gamma $
-invariant set containing
$\gamma ^+$
and
$\gamma ^-$
and let
$\xi $
be positive map from S to
$\mathbf {F}_\Theta $
. Assume that there exists an element g in
$\mathsf G$
such that, for all s in S,
$$\begin{align*}\xi(\gamma(s))=g\cdot \xi(s), \end{align*}$$
Then g is proximal and
$\xi (\gamma ^+)$
is the attracting fixed point of g and similarly
$\xi (\gamma ^-)$
is the repelling fixed point of g.
4 Triples, Tripods and metrics
In this section, we construct for every positive triple
$(a,b,c)$
a complete metric on the diamond
$V_c(a,b)$
(Definition 4.9). We also show that this family of metrics satisfies contraction properties (Propositions 4.11 and 4.12).
We first do it for triples of a special type that we call tripods.
4.1 Tripods and metrics
Recall that in Proposition 2.12, we fixed
$\mathcal {H}$
a class of pairs
$(\mathsf {H}, C)$
where the subgroups
$\mathsf H$
are isogenic to
$\mathsf {PSL}_2(\mathbb R)$
and C is an
$\mathsf H$
-orbit, isomorphic to
${\mathbf P}^1(\mathbb {R})$
and called an
$\mathsf H$
-circle.
Definition 4.1. A tripod is a triple of pairwise distinct points on C for some
$(\mathsf {H},C)$
in
$\mathcal {H}$
.
A tripod is always positive. If
$\tau =(x,t,y)$
is a tripod, we write
More generally, for a positive triple
$t=(a,b,c)$
, we write
$t^-=a$
,
$t^0=b$
,
$t^+=c$
and 
.
Let
$\mathcal T_0$
be the set of tripods. Observe that
$\operatorname {Aut}_0(\mathfrak g)$
acts transitively on the left on the space of tripods, and that the positive circle containing a tripod is unique.
The stabilizer of any tripod is compact (cf. Section 4.1.1 below), in particular
$\operatorname {Aut}_0(\mathfrak g)$
acts properly on the space of tripods, and we can then fix d an
$\operatorname {Aut}_0(\mathfrak g)$
-invariant Riemannian metric on the set of tripods
$\mathcal T_0$
.
4.1.1 Tripods and the parametrization
Let us consider (as in Paragraph 2.2.1) the convex cone
$\mathsf {C}$
and the parametrization
$\Psi \colon \mathsf C\to \mathsf N$
equivariant with respect to
$\mathsf {L}^{\circ }_{\Theta }$
(
$\Psi $
is given by the product of exponential maps). Note that
$\Psi $
extends continuously to a map
$\overline {\mathsf {C}}\to \mathsf {U}_\Theta $
that is also
$\mathsf {L}^{\circ }_{\Theta }$
-equivariant.
Let h be the element of
$\mathsf C$
corresponding to the unipotent associated to the preferred
$\mathsf {SL}_2(\mathbb R)$
– see the proof of Proposition 2.12. Let
$\mathsf K_h$
be the stabilizer of h in
$\mathsf L^{\circ }_{\Theta }$
. Since the stabilizer of a positive triple is compact, it follows that
$\mathsf K_h$
is compact.
If now x and y are transverse points in
$\mathbf {F}_\Theta $
and
$\sigma \colon \mathsf {U}_\Theta \to \mathsf {U}_y$
is a U-pinning at y, then the map
$$ \begin{align*} \Psi^\sigma \colon \mathsf{C}\mapsto \mathbf{F}_\Theta, \ u\mapsto \sigma\circ\Psi(u)\cdot x, \end{align*} $$
is a parametrization of the diamond
$V_t(x,y)$
with 
. We then define
Definition 4.2. Given a tripod
$\tau =(x,t,y)$
a
$\tau $
-parametrization of the diamond
$V_\tau $
, is a map
$\Psi _\tau $
of the form
$\Psi ^\sigma $
so that
$\Psi ^\sigma (h)=t$
.
From the definition follows
Proposition 4.3. Given a tripod
$\tau $
, a
$\tau $
-parametrization of the diamond exists and is unique up to postcomposition by the stabilizer of
$\tau $
(equivalently up to precomposition by
$\mathsf K_h$
).
The next proposition is crucial; it insures that a sequence of parametrizations of diamonds associated with tripods converges to the constant map as soon as one sequence in the image converges, precisely
Proposition 4.4 (Contraction in corners).
Let
$ \{\tau _m\}_{m\in \mathbb N}$
be a sequence of tripods, with
$\tau _m=(x_m,t_m,y)$
, and
$\Psi _{\tau _m}$
a
$\tau _m$
-parametrization for all m.
Assume that
$ \{x_m\}_{m\in \mathbb N}$
converges to a point x transverse to y. Assume that there exists a converging sequence
$ \{k_m\}_{m\in \mathbb N}$
in the cone
$\mathsf C$
, and such that
$$ \begin{align} \lim_{m\to\infty}\Psi_{\tau_m}(k_m)=x. \end{align} $$
Then for any sequence
$ \{k^{\prime }_m\}_{m\in \mathbb N}$
in
$\mathsf C$
that is bounded in
$\overline {\mathsf {C}}$
,
$$ \begin{align} \lim_{m\to\infty}\Psi_{\tau_m}(k^{\prime}_m)=x. \end{align} $$
Proof. Since x is transverse to y, by replacing
$\tau _m$
by
$u_m\tau _m$
where
$ \{u_m\}_{m\in \mathbb N}$
is a converging sequence in
$\mathsf U_y$
, we may as well assume that
$ \{x_m\}_{m\in \mathbb N}$
is constant and equal to x.
Using the fact that
$\operatorname {Aut}_0(\mathfrak g)$
acts transitively on tripods, let us write
$t_m=g_m\cdot t_0$
, with
$g_m$
fixing x and y. Thus
$$ \begin{align*} \Psi_{\tau_m}(h)= g_m \cdot t_0= g_m\cdot \Psi_{\tau_0}(h) =g_m \sigma( \Psi(h))\cdot x . \end{align*} $$
Note that the U-pinning
$\sigma \colon \mathsf {U}_\Theta \to \mathsf {U}_y$
comes from an element
$\sigma $
of
$\operatorname {Aut}_0(\mathfrak g)$
. Denote
$g_{m}^{0} = \sigma ^{-1}\circ g_m \circ \sigma $
; it is an element of
$\operatorname {Aut}_0(\mathfrak g)$
stabilizing the standard unipotent subalgebras
$\mathfrak {u}_\Theta $
and
$\mathfrak {u}_{\Theta }^{\mathrm {opp}}$
. Up to maybe precomposing by an element of
$\mathsf {K}_h$
, we may assume that
$\Psi _{\tau _m}$
is the map
$k\mapsto \sigma ( \Psi ( g_{m}^{0}\cdot k)) \cdot x$
.
Therefore we have, for any
$ \{\ell _m\}_{m\in \mathbb N}$
in
$\overline { \mathsf {C}}$
, that
$ \{ \Psi _{\tau _m}(\ell _m)\}_{m\in \mathbb N}$
converges to x if and only if the sequence
$ \{ g_{m}^{0}\cdot \ell _m\}_{m\in \mathbb N}$
converges to
$0$
in
$\overline { \mathsf {C}}$
.
For any y in
$\overline {\mathsf {C}}$
, let
Since
$\overline { \mathsf {C}}$
is salient,
$K(y)$
is compact for any y.
From the previous discussion, we get that the sequence
$ \{ c_m=g_{m}^{0}\cdot k_m\}_{m\in \mathbb N}$
converges to
$0$
. Thus, using again the fact that
$\overline { \mathsf {C}}$
is salient, for every positive real R, the sequence of compact sets
$ \{K(R\cdot c_m)\}_{m\in \mathbb N}$
converges to
$\{0\}$
.
Let now
$ \{ k_{m}^{\prime }\}_{m\in \mathbb N}$
be a sequence in
$\mathsf {C}$
, bounded in
$\overline {\mathsf {C}}$
. Since by hypothesis
$ \{ k_{m}\}_{m\in \mathbb N}$
converges in
$\mathsf {C}$
, there exists a positive real R such that, for all m,
$R\cdot k_m -k_{m}^{\prime }$
belongs to
$\overline {\mathsf {C}}$
. In other words:
$k_{m}^{\prime }$
belongs to
$R\cdot k_{m} -\overline {\mathsf {C}}$
. Thus, for all m,
$g_{m}^{0}\cdot k_{m}^{\prime }$
belongs to
$K( R\cdot c_m )$
. Hence the sequence
$ \{ g_{m}^{0}\cdot k_{m}^{\prime }\}_{m\in \mathbb N}$
converges to
$0$
. This means that the sequence
$ \{ \Psi _{\tau _m}( k_{m}^{\prime })\}_{m\in \mathbb N}$
converges to x as wanted.
4.1.2 Diamond metrics for tripods
We choose once and for all a Euclidean distance
$d_0$
on the convex cone
$\mathsf {C}$
, associated with the Riemannian
$g_0$
induced by a
$\mathsf K_h$
-invariant scalar product on
$\mathfrak {u}_\Theta $
. This distance
$d_0$
is
$\mathsf K_h$
-invariant and extends to
$\overline { \mathsf {C}}$
.
Definition 4.5. Given a tripod
$\tau =(x,t,y)$
, let
$\Psi _\tau $
be a
$\tau $
-parametrization of
$V_\tau $
, let
as well as
$d^+_\tau $
,
$d^-_\tau $
, and
$d_\tau $
the associated distances so that
$$ \begin{align*}d^\pm_\tau\leqslant d_\tau\leqslant d^+_\tau+d^-_{\tau} . \end{align*} $$
The metric
$g_\tau $
is the diamond metric (for the tripod
$\tau $
) on
$V_\tau $
, while
$d_\tau $
is the diamond distance.
The terminology is justified by
Proposition 4.6 (Uniqueness and Completeness).
The diamond metric is independent of the choice of the
$\tau $
-parametrization and only depends on
$\tau $
. Moreover,
$d_\tau $
is complete and proper on
$V_\tau $
.
There exists a function
$F\colon \mathbb {R}_{>0} \to \mathbb {R}_{>0}$
with
$\lim _0 F=1$
and such that the following holds: For any tripods
$\tau $
and
$\tau '$
with the same extremities
$\tau _-$
and
$\tau _+$
, if
$d_\tau ( \tau ^0, \tau ^{\prime 0})\leqslant \varepsilon $
then
$$ \begin{align*}F(\varepsilon)^{-1}d_{\tau'} \leqslant d_{\tau}\leqslant F(\varepsilon)\ d_{\tau' } . \end{align*} $$
Proof. The independence on the parametrization is a consequence of Proposition 4.3 and the fact that
$d_0$
itself is invariant under the group
$\mathsf K_h$
.
Let us now prove completeness. Let
$ \{u_m\}_{m\in \mathbb N}$
be a Cauchy sequence for
$d_\tau $
, then it is a Cauchy sequence for both
$d^+_\tau $
and
$d^-_\tau $
. It follows that the sequences
$ \{v_m\}_{m\in \mathbb N}$
and
$ \{w_m\}_{m\in \mathbb N}$
defined by
$$ \begin{align*} v_m=\Psi_\tau^{-1}(u_m) , \ w_m=\Psi_{\overline\tau}^{-1}(u_m), \end{align*} $$
are both Cauchy sequences. Since
$\overline {\mathsf { C}}$
is complete with respect to the metric
$d_0$
, there exist v and w in
$\overline {\mathsf {C}}$
such that
$$ \begin{align*} \lim_{m\to\infty}v_m=v , \ \lim_{m\to\infty}w_m=w. \end{align*} $$
Since by construction
$\Psi _{\tau }$
and
$\Psi _{\overline \tau }$
extend continuously to the closure of
$\mathsf {C}$
,
Obviously u belongs to
$\overline {V}_\tau $
. By construction
$u=n_y\cdot x$
and
$u=n_x\cdot y$
where
$n_y$
belongs to
$\mathsf U_y$
and
$n_x$
belongs to
$\mathsf U_x$
. Since x and y are transverse, it follows that u is transverse to both y and x. Thus u belongs to
$V_\tau $
. Hence
$d_\tau $
is complete.
Let us prove the last part. Observe now by hypothesis, there exists
$\ell $
in
$\mathsf L_\Theta $
such that
$\Psi _{\tau '}=\Psi _\tau \circ \ell $
. Thus
$$ \begin{align*}g^+_{\tau'}=(\Psi_{\tau})_* (\ell_* g_0). \end{align*} $$
Recall that
$g_0$
is induced by a scalar product on
$\mathfrak u_\Theta $
, and that
$\mathsf L_\Theta $
acts linearly on
$\mathfrak u_\Theta $
, it then follows that there is a function
$F_0$
on
$\mathsf L_\Theta $
such that
$$ \begin{align*}F_0(\ell)g_0\leqslant \ell_*(g_0)\leqslant F_0(\ell) g_0, \end{align*} $$
and with
$F_0(\ell )\underset {\ell \to \mathsf {K}_\Theta }{\longrightarrow } 1$
where
$\mathsf {K}_\Theta $
is a maximal compact subgroup
$\mathsf {L}_\Theta $
. Pushing forward by
$\Psi _\tau $
we have
$$ \begin{align*}F_0(\ell)g^+_\tau\leqslant g^+_{\tau'}\leqslant F_0(\ell) g^+_{\tau}. \end{align*} $$
The same holds for
$g^{-}_{\tau }$
and
$g^{-}_{\tau '}$
. Hence the same inequality holds for
$g_\tau =g_{\tau }^{+}+g_{\tau }^{-}$
and
$g_{\tau '}$
; this concludes the proof with the remark that
$d_\tau ( \tau ^0, \tau ^{\prime 0}) \longrightarrow 0$
implies that
$\ell \longrightarrow \mathsf {K}_\Theta $
. Precisely, we can define
$$ \begin{align*}F(\epsilon)=\sup\{F_0(h)\mid d_\tau( \tau^0, h(\tau^{0}))\leqslant \epsilon\}, \end{align*} $$
and observing that by equivariance F does not depend on the choice of
$\tau $
.
Proposition 4.7 (Contraction for tripods).
Let
$ \{\tau _m\}_{m\in \mathbb N}$
be a sequence of tripods. Assume that, for all m in
$\mathbb {N}$
,
$V_{\tau _{m+1}}\subset V_{\tau _m}$
and that
$$ \begin{align} \bigcap_{m\in\mathbb N}V_{\tau_m}=\{z\}. \end{align} $$
For any positive R, let
$V_{\tau _m}(R)$
be the ball of radius R and center
$\tau ^0_m$
with respect to
$d_{\tau _m}$
. Then on
$V_{\tau _m}(R)$
, we have
$$ \begin{align*} g_{\tau_0}\leqslant k_m \cdot g_{\tau_m}, \end{align*} $$
with
$ \{k_m\}_{m\in \mathbb N}$
converging to zero.
Proof. Since
$\operatorname {Aut}_0(\mathfrak g)$
acts transitively on the space of tripods
$\mathcal T_0$
, it follows that
$\tau _m=h_m\cdot \tau _0$
, for some
$h_m$
in
$\operatorname {Aut}_0(\mathfrak g)$
. Since the construction of the tripod metrics is
$\operatorname {Aut}_0(\mathfrak g)$
-equivariant, we observe that
$g_{\tau _m}=h_{m}^{*}g_{\tau _0}$
. Then, we take
$$ \begin{align} k_m&=\sup\left\{ \frac{g_{\tau_0}(w,w)}{g_{\tau_m}(w,w)} \mathrel{\Big|} w\in \mathsf{T} V_{\tau_m}(R) \right\}\end{align} $$
$$ \begin{align} &= \sup\left\{\frac{g_{\tau_0}(w,w)}{g_{\tau_0}(\mathsf T h_m^{-1}w, \mathsf T h_m^{-1} w)} \mathrel{\Big|} w\in \mathsf{T} V_{\tau_m}(R) \right\} \end{align} $$
$$ \begin{align} &=\sup\left\{\frac{g_{\tau_0}(\mathsf T h_m (v),\mathsf T h_m (v))}{g_{\tau_0}(v,v)} \mathrel{\Big|} v\in \mathsf T V_{\tau_0}(R) \right\}. \end{align} $$
The hypothesis (16) says that
$ \{h_m\}_{m\in \mathbb N}$
, seen as a sequence of diffeomorphisms of
$\mathbf {F}_\Theta $
converges uniformly on every compact set of
$V_{\tau _0}$
to the constant map. It follows that
$ \{h_m\}_{m\in \mathbb N}$
also converges
$C^1$
to the constant map on any compact set in
$V_{\tau _0}$
and hence
$ \{\mathsf T h_m\}_{m\in \mathbb N}$
converges to zero uniformly on every compact set of
$V_{\tau _0}$
. Thus, equality (19) shows that
$ \{k_m\}_{m\in \mathbb N}$
converges to zero.
4.2 Positive triples, tripods and metrics
Our goal is to construct a complete metric on the diamond associated with a positive triple and to prove a generalization of the contraction properties (Propositions 4.11 and 4.12).
4.2.1 Approximating triples: the tripod defect
We will first approximate in a rough sense positive triples by tripods. For any positive triple
$t=(x,z,y)$
, let
We call
$\mathrm { K}(t)$
the tripod defect.
Observe that
$\mathrm {K}(t)$
depends continuously on t, and that the tripod defect vanishes for tripods. Let also
Proposition 4.8.
-
1. Given
$K_0\geqslant {\mathrm {K}(t})$
, the set
$D(t,K_0)$
is compact and nonempty. -
2.
${\mathrm { K}(t)}=0$
if and only if t is a tripod. -
3. For any
$K_0$
, there exists a constant
$A=A(K_0)$
such that if
$t=(a,b,c)$
is a positive triple with
$\mathrm {K}(t)\leqslant K_0$
, then for every
$\tau _0$
and
$\tau _1$
in
$D(t, K_0)$
, we have, on
$V_{b}(a,c)$
$$\begin{align*}g_{\tau_0}\leqslant A g_{\tau_1}. \end{align*}$$
Furthermore
$A(K_0)$
tends to
$1$
as
$K_0$
goes to
$0$
.
Proof. Let
$t=(a,b,c)$
be a positive triple. Let
$ \{\tau _m\}_{m\in \mathbb N}$
be a sequence of tripods such that
$(\tau ^-_m,\tau ^+_m)=(a,c)$
and
$$ \begin{align*}\{d_{\tau_m}(b,\tau^0_m)\}_{m\in\mathbb N}, \end{align*} $$
is bounded. Let
$ \{g_m\}_{m\in \mathbb N}$
be a sequence of elements in
$\operatorname {Aut}_0(\mathfrak g)$
, stabilizing a and c and such that
$ \{g^{-1}_m(\tau ^0_m)\}_{m\in \mathbb N}$
is constant and let
$\tau ^0$
be this constant. Let 
. It follows that
$$ \begin{align*}\{d_{\tau}(g_m^{-1}(b),\tau^0)\}_{m\in\mathbb N} , \end{align*} $$
is bounded. Since
$d_\tau $
is a proper metric (i.e., every bounded set is relatively compact), the sequence
$ \{g_m^{-1}(b)\}_{m\in \mathbb N}$
– after extracting a subsequence – converges to e with
$(a,e,c)$
positive. Since
$\operatorname {Aut}_0(\mathfrak g)$
acts properly on the space of tripods, it follows that
$ \{g_m\}_{m\in \mathbb N}$
is bounded. Thus after taking a subsequence
$ \{\tau _m\}_{m\in \mathbb N}$
converges to a tripod
$\tau _\infty $
, with
$\tau ^0_\infty $
in
$V_t$
. This proves that, for all
$K_0$
, the set
$D(t,K_0)$
is compact. Since
$D(t,K_0)$
is nonempty for
$K_0>\mathrm {K}(t)$
, it follows that the decreasing intersection
$$ \begin{align*}D(t)=\bigcap_{K_0>\mathrm{ K}(t)} D(t, K_0). \end{align*} $$
is not empty.
The second assertion is an immediate consequence of the first.
The third follows from the first as a consequence of the second part of Proposition 4.6.
4.2.2 The diamond metric for triples
The following definition is one of the goals of this section.
Definition 4.9. Let t be a positive triple. The diamond metric (for the triple t)
$g_t$
is the Riemannian metric on
$V_t$
defined as follows: for every x in
$V_t$
, the unit ball of
$g_{t,x}$
is the John ellipsoid of the union of the unit balls of
$g_{\tau ,x}$
for
$\tau $
varying in
$D(t)$
.
The associated distance is the diamond metric
$d_t$
.
Explicitly, one has
$g_{\tau ,x}\leqslant g_{t,x}$
for every
$\tau $
in
$D(t)$
and
$g_{t,x}\leqslant g$
for every Euclidean scalar product g on
$T_x V_t$
such that
$g_{\tau ,x}\leqslant g$
for every
$\tau $
in
$D(t)$
. Furthermore
$g_{t,x}$
is the unique minimizer among the Euclidean scalar products g satisfying the previous condition. It also follows immediately from point (3) of Proposition 4.8 that
$g_t\leqslant A g_\tau $
for every
$\tau $
in
$D(t)$
with
$A=A(K(t))$
.
When t is a tripod, this definition agrees with the one of the previous paragraph thanks to the second item of Proposition 4.8.
As an immediate corollary of Proposition 4.8 and Proposition 4.6, we have
Corollary 4.10. The diamond metric is complete. Moreover if a sequence of positive triples
$ \{t_m\}_{m\in \mathbb N}$
converges to a tripod
$\tau $
, then
$ \{g_{t_m}\}_{m\in \mathbb N}$
converges to
$g_\tau $
on every compact of the diamond
$V_\tau $
.
The following Propositions 4.11 and 4.12 are two contraction properties of the diamond metrics that we shall use in the sequel.
Proposition 4.11 (Contraction).
Let
$ \{t_m\}_{m\in \mathbb N}$
be a sequence of positive triples, with
$t_m=(a_m,b_m,c_m)$
. Assume that the sequence
$ \{\mathrm { K}(t_m)\}_{m\in \mathbb N}$
of tripod defects is bounded. Assume that
$V_{t_{m+1}}\subset V_{t_m}$
and that
$$ \begin{align} \bigcap_{m\in\mathbb N}V_{t_m}=\{z\}. \end{align} $$
For any positive R, let
$V_{t_m}(R)$
be the ball of radius R and center
$a_m$
with respect to
$d_{t_m}$
. Then on
$V_{t_m}(R)$
, we have
$$ \begin{align*}g_{t_0}\leqslant k_m \cdot g_{t_m}, \end{align*} $$
with
$ \{k_m\}_{m\in \mathbb N}$
converges to zero.
Proof. By Definition 4.9 of the diamond metric for triples, and Proposition 4.8 it follows that we can find a constant A, such that for all m, we can find a tripod
$\tau _m$
with the same extremities as
$t_m $
and with
$$ \begin{align*}d_{\tau_m}(\tau^0_m,b_m)\leqslant A, \ \frac{1}{A} g_{\tau_m}\leqslant g_{t_m}\leqslant A g_{\tau_m}. \end{align*} $$
The result now follows from the corresponding proposition for tripods: Proposition 4.7.
Proposition 4.12 (Contraction in corners).
Let
$ \{t_m\}_{m\in \mathbb N}$
be a sequence of positive triples, where
${t_m=(t_{m}^{-},t_{m}^{0},t_{m}^{+})}$
. Assume that
-
1. the sequence
$ \{\mathrm {K}(t_m)\}_{m\in \mathbb N}$
of tripod defects is bounded; -
2. the sequences
$ \{t_{m}^{-}\}_{m\in \mathbb N}$
and
$ \{t_{m}^{+}\}_{m\in \mathbb N}$
converge to transverse points a and c respectively; -
3. There exists
$ \{u_m\}_{m\in \mathbb N}$
a sequence of elements of
$\mathbf {F}_\Theta $
, such that
$u_m$
belongs to
$V_{t_m}$
, the sequence
$ \{d_{t_m}(t_{m}^{0},u_m)\}_{m\in \mathbb N}$
is uniformly bounded, and
$\lim _{m\to \infty } u_m=a$
.
Then
$\lim _{m\to \infty }t_{m}^{0}=a$
.
Proof. By the first hypothesis and Proposition 4.8, we can find a constant A, a sequence of tripods
$ \{\tau _m\}_{m\in \mathbb N}$
with
$\tau _m^\pm =t_m^\pm $
and such that
$$ \begin{align*}d_{\tau_m}\leqslant A\ d_{t_m}. \end{align*} $$
In particular, we have that
$ \{d_{\tau _m}(t^{0}_{m},\tau ^{0}_{m})\}_{m\in \mathbb N}$
and
$ \{d_{\tau _m}(u_m,\tau ^{0}_{m})\}_{m\in \mathbb N}$
are uniformly bounded. The result now follows by applying twice Proposition 4.4. Indeed, since
$ \{d_{\tau _m}(t^{0}_{m},\tau ^{0}_{m})\}_{m\in \mathbb N}$
is uniformly bounded, it follows that
$t^{0}_{m} = \Psi _{\tau _m}(k_m)$
with
$ \{k_m\}_{m\in \mathbb N}$
bounded. Hence by Proposition 4.4 (applied to any converging subsequence of
$ \{k_m\}_{m\in \mathbb N}$
), with
$k^{\prime }_m=h$
, yields that
$$ \begin{align*}\lim_{m\to\infty}\tau^{0}_{m}=a. \end{align*} $$
Applying again Proposition 4.4 to
$ \{k^{\prime }_m\}_{m\in \mathbb N}$
with
$\Psi _{\tau _m}(k^{\prime }_m)=u_m$
yields that
$$ \begin{align*}\lim_{m\to\infty} u_m=a. \end{align*} $$
This concludes the proof.
5 Positive representations are Anosov
In this section we introduce the notion of positive representations of a surface group. We then show that any
$\Theta $
-positive representation is
$\Theta $
-Anosov, establishing Theorem A and Corollary B from the introduction. As in the introduction, S is a connected oriented closed surface of genus at least
$2$
.
Definition 5.1 (Positive representations).
Let
$\mathsf G$
be a semi-simple Lie group admitting a positive structure relative to
$\Theta $
. A representation
$\rho \colon \pi _1(S) \to \mathsf G$
is
$\Theta $
-positive if there exist a nonempty
$\pi _1(S)$
-invariant subset A of
$\partial _\infty \pi _1(S)$
and a positive
$\rho $
-equivariant map
$\xi $
from A to
$\mathbf {F}_\Theta $
.
The set A is necessarily dense since the action of
$\pi _1(S)$
on
$\partial _\infty \pi _1(S)$
is minimal. We will often say that a representation is positive if it is
$\Theta $
-positive.
5.1 Anosov representations
Let us recall at this stage the definition of a
$\Theta $
-Anosov representation from [Reference Labourie31]. For simplicity we restrict ourselves to the case of representations of
$\pi _1(S)$
. Let us equip the surface S with an auxiliary hyperbolic metric. Let
${\mathsf U}S$
be the unit tangent bundle of S equipped with its geodesic flow
$\phi _t$
. Let us also freely identify the space of cyclically oriented triples of
$\partial _\infty \pi _1(S)$
with the unit tangent bundle
$\mathsf U{\mathbf H}^2$
of the universal cover of S.
Let
$\rho $
be a representation of 
in
$\mathsf G$
. Let
${\mathcal F}_\Theta $
be the flat
$\mathbf {F}_\Theta $
-bundle over
${\mathsf U}S$
associated with
$\rho $
, and
$\Phi _t$
the parallel transport on
${\mathcal F}_\Theta $
along
$\phi _t$
.
The representation
$\rho $
is
$\Theta $
-Anosov if there exists a
$\rho $
-equivariant continuous map, called the limit map.
$$ \begin{align*} \xi\colon \partial_\infty\Gamma&\to \mathbf{F}_\Theta, \end{align*} $$
such that the corresponding section
$\Xi $
of
${\mathcal F}_{\Theta }$
(which is constant along the leaves of the weakly unstable foliation) satisfies the following contraction property: there exist an open neighborhood
$\mathcal V$
of the image of
$\Xi $
that is a fiber bundle over
${\mathsf U}S$
with fiber
$\mathcal {V}_x$
for x in
${\mathsf U}S$
, a continuous family of Riemannian metric
$g_x$
on the fibers
$\mathcal {V}_x$
, and some positive number T such that
$\Phi _{-T}(\mathcal V) \subset \mathcal V$
, and, for all x in
$\mathcal V$
,
$$ \begin{align*} (\Phi_T)^* g_x\leqslant \frac{1}{2} g_{\phi_T(x)}, \end{align*} $$
where
$g_x$
is g restricted to
$\mathcal V_x$
.
Note that there is another section
$\Xi ^*$
, which is constant along the leaves of the weakly stable foliation and is contracted under
$\Phi _{-T}$
.
Observe that, in general, the existence of a continuous equivariant map, even sending distinct points to transverse points, is weaker than the condition of being Anosov.
To establish the Anosov property for positive representations, we first extend the positive boundary map to a left-continuous boundary map and to a right-continuous boundary map using Corollary 3.16. We prove then that these extensions are continuous (and thus coincide), and then deduce the Anosov property using the contraction property of the diamond metrics (Proposition 4.11).
5.2 Properness
The following definition will be used several times in the sequel: a map f defined on a subset A of a topological set X, with values in some topological set Y is bounded if for every compact set K in X,
$f(A\cap K)$
is relatively compact.
Lemma 5.2. Let A be a dense set in the circle. Let
$\phi _\pm $
be positive maps from A to
$\mathbf {F}_\Theta $
. We assume that, for all cyclically oriented quadruples
$(x,y,z,t)$
in A and for any choice of
$\varepsilon $
,
$\eta $
,
$\nu $
, and
$\beta $
in
$\{+,-\}$
, the quadruple
$$ \begin{align*}(\phi_\varepsilon(x),\phi_\eta(y),\phi_\nu(z),\phi_\beta(t))\end{align*} $$
is positive. Let
$A^3_+$
be the set of triples of pairwise distinct elements of A.
Then, for any
$\varepsilon $
,
$\eta $
, and
$\nu $
in
$\{+,-\}$
,
$\phi _\varepsilon \times \phi _\eta \times \phi _\nu $
is bounded as a map from
$A^3_+$
to the space
$\mathcal T$
of positive triples in
$\mathbf {F}_\Theta $
.
Proof. Let
$\chi =(x_1,x_2,y_1,y_2,z_1,z_2)$
be a cyclically oriented sextuplet in
$S^1$
. Let
$I_\chi $
be the subset of
$(S^1)^3$
given by
$$\begin{align*}I_\chi=\{(X,Y,Z)\mid x_1<X<x_2<y_1<Y<y_2<z_1<Z<z_2\}. \end{align*}$$
Observe that
$I_\chi $
consists of cyclically oriented triples. Let
$$\begin{align*}K = \phi_\epsilon \times \phi_\eta \times \phi_\nu \left(I_\chi\cap A_+^3\right). \end{align*}$$
It is enough to show that
$\overline K\subset \mathcal T$
, where the closure is taken in
$\mathbf {F}_\Theta ^3$
.
Let us fix, by density,
$a_0$
,
$a_1$
,
$a_2$
,
$b_0$
,
$b_1$
,
$b_2$
,
$c_0$
,
$c_1$
, and
$c_2$
in A such that
$$\begin{align*}(x_1,x_2, a_0,a_1,a_2, y_1,y_2, b_0,b_1,b_2, z_1,z_2,c_0,c_1,c_2) \end{align*}$$
is cyclically oriented. To lighten notation, we set
$\alpha _i=\phi _\epsilon (a_i)$
,
$\beta _i=\phi _\eta (b_i)$
, and
$\gamma _i=\phi _\nu (c_i)$
for
$i=0,1,2$
.
From the positivity of the maps and thus of the image of the
$15$
-tuple defined above, it follows that if
$(x,y,z)$
belongs to K, then
$$ \begin{align*}x\in V^*_{\alpha_1}(\gamma_2,\alpha_0), \ y\in V^*_{\beta_1}(\alpha_2,\beta_0), \ z\in V^*_{\gamma_1}(\beta_2,\gamma_0). \end{align*} $$
Thus if
$(a,b,c)$
belongs to
$\overline K$
, then
$$\begin{align*}a\in \overline{V}{}^{*}_{\alpha_1}(\gamma_2,\alpha_0), \ b\in \overline{V}{}^{*}_{\beta_1}(\alpha_2,\beta_0), \ c\in \overline{V}{}^{*}_{\gamma_1}(\beta_2,\gamma_0). \end{align*}$$
Using the Inclusion Corollary 3.10, we get
$$\begin{align*}a\in V^*_{\alpha_1}(\gamma_1,\alpha_1), \ b\in V^*_{\beta_1}(\alpha_1,\beta_1), \ c\in V^*_{\gamma_1}(\beta_1,\gamma_1). \end{align*}$$
By the Necklace Corollary 3.2,
$(a,b,c)$
is a positive triple, that is, it belongs to
$\mathcal T$
. This concludes the proof.
Proposition 5.3. Let
$\rho \colon \pi _1(S) \to \mathsf G$
be a representation. Let
$\xi $
be a positive
$\pi _1(S)$
-invariant map from
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
.
Let
$\mathcal T_{\pi _1(S)}$
be the set of triples of pairwise distinct elements in
$\partial _\infty \pi _1(S)$
, and let
$\mathcal T$
be the set of positive triples in
$\mathbf {F}_\Theta $
. Let
$\Xi $
be the map from
$\mathcal T_{\pi _1(S)}$
to
$\mathcal T/\mathsf G$
, defined by
Then the image of
$\Xi $
is relatively compact.
Proof. The map
$\Xi $
is invariant by the diagonal action of
$\pi _1(S)$
. The result follows then from Lemma 5.2 using the fact that
$\pi _1(S)$
acts cocompactly on
$\mathcal T_{\pi _1(S)}$
.
5.3 An a priori bound on the tripod defect
For any positive triple t, let
$\mathrm {K}(t)$
be the tripod defect introduced in Paragraph 4.2. Then Proposition 5.3 implies an a priori bound on the tripod defect.
Proposition 5.4. Let
$\rho\! : \pi _1(S)$
be a representation
$\mathsf G$
. Let
$\xi $
be a
$\rho $
-equivariant positive map from a
$\pi _1(S)$
-invariant dense subset A of
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
. Then there exists a constant
$K_0$
such that for all triple t of pairwise distinct points in the closure of
$\xi (A)$
, we have
$$ \begin{align*} \mathrm{K}(t)\leqslant K_0 . \end{align*} $$
Proof. This is an immediate consequence of Proposition 5.3 and the fact that
$\mathrm {K}$
is a continuous function on
$\mathcal T$
.
5.4 Continuity of equivariant positive maps
Let
$\rho $
be a
$\Theta $
-positive representation, A a nonempty
$\pi _1(S)$
-invariant subset of
$\partial _\infty \pi _1(S)$
and
$\xi \colon A \to \mathbf {F}_\Theta $
the positive
$\rho $
-equivariant boundary map. Then, by Corollary 3.16, there exist a unique right-continuous
$\rho $
-equivariant boundary map
$\xi _+\colon \partial _\infty \pi _1(S) \to \mathbf {F}_\Theta $
and a unique left-continuous
$\rho $
-equivariant boundary map
$\xi _-\colon \partial _\infty \pi _1(S) \to \mathbf {F}_\Theta $
, coinciding with the map
$\xi $
on a dense subset.
Let
$\mathcal T_{\pi _1(S)}$
be the set of triples of pairwise distinct points of
$\partial _\infty \pi _1(S)$
. For
$t=(x,y,z)$
in
$\mathcal T_{\pi _1(S)}$
, let us define
$$ \begin{align*}\tau(t)=(\xi_+(x),\xi_+(y),\xi_+(z)). \end{align*} $$
Lemma 5.5. The
$\pi _1(S)$
-invariant function f defined by
$$ \begin{align*}f(x,y,z)=d_{\tau(t)}(\xi_+(y),\xi_-(y)) \end{align*} $$
is bounded; there is a constant D such that, for all
$(x,y,z)$
in
$\mathcal {T}_{\pi _1(S)}$
,
$f(x,y,z)\leqslant D$
.
Proof. Let
$\mathcal {Q}$
be the set of quadruples
$(a,b,c,d)$
in
$\mathbf {F}_\Theta $
such that there exists a diamond V with extremities a and d and containing both b and c. Using Lemma 5.2, we see that the map
$$ \begin{align*}(x,y,z)\mapsto (\xi_+(x), \xi_+(y),\xi_-(y), \xi_+(z)),\end{align*} $$
from
$\mathcal {T}_{\pi _1(S)}$
to
$\mathcal {Q}$
is bounded. As the real valued function on
$\mathcal {Q}$
sending a quadruple
$(a,b,c,d)$
to
$d_{(a,b,d)}(b,c)$
is continuous, we get the result.
Lemma 5.6. The map
$\xi _+$
is continuous.
Proof. Since
$\xi _+$
is right-continuous we only have to prove that it is left-continuous. Let x and y be in
$\partial _\infty \pi _1(S)$
, and let
$ \{x_m\}_{m\in \mathbb N}$
be a sequence in
$\partial _\infty \pi _1(S)$
, such that
$(x_m,x,y)$
is cyclically oriented with respect to the orientation on
$\partial _\infty \pi _1(S)$
, and that
$ \{x_m\}_{m\in \mathbb N}$
converges to x. Let
$t_m=(\xi _+(x_m),\xi _+(x),\xi _+(y))$
.
Recall that by Corollary 3.16,
$ \{\xi _+(x_m)\}_{m\in \mathbb N}$
converges to
$\xi _-(x)$
.
We now apply Proposition 4.12 to the following setting:
$$ \begin{align*} t_{m}^{-}=\xi_+(x_m), \ t_{m}^{0}=\xi_+(x) , \ u_m=\xi_-(x), \ t_{m}^{+}=\xi_+(y). \end{align*} $$
Since
$$ \begin{align*}\{d_{t_m}(t_{m}^{0},u_m)\}_{m\in\mathbb N}= \{d_{t_m}(\xi_+(x),\xi_-(x))\}_{m\in\mathbb N}\end{align*} $$
is bounded by Lemma 5.5 and
$ \{\mathrm {K}(t_m)\}_{m\in \mathbb N}$
is bounded by Proposition 5.4, we get that
$$ \begin{align*}\lim_{m\to\infty}\xi_+(x_m)=\xi_+(x). \end{align*} $$
This proves that
$\xi _+$
is left-continuous.
As a consequence we obtain
Proposition 5.7. Let
$\rho $
from
$\pi _1(S)$
to
$\mathsf G$
be a positive representation and
$\xi $
the positive
$\rho $
-invariant boundary map from a
$\pi _1(S)$
-invariant dense subset of
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
. Then
$\xi $
extends to a
$\rho $
-equivariant positive continuous map from
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
.
The extended map
$\xi $
from
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
will be called the positive boundary map of
$\rho $
.
5.5 The Anosov property
We are now in position to prove Theorem A from the introduction. More precisely we show
Proposition 5.8. Let
$\rho $
from
$\pi _1(S)$
to
$\mathsf G$
be a positive representation and
$\xi $
from
$\partial _\infty \pi _1(S)$
to
$\mathbf {F}_\Theta $
be the
$\rho $
-equivariant continuous positive boundary map. Then
$\rho $
is
$\Theta $
-Anosov and its boundary map is
$\xi $
.
Let us start with a general lemma
Lemma 5.9. Let
$ \{b^0_m\}_{m\in \mathbb N}$
and
$ \{b^1_m\}_{m\in \mathbb N}$
be two sequences in
$\mathbf {F}_\Theta $
converging to c. Let
$d_0$
and
$d_1$
be in
$\mathbf {F}_\Theta $
such that
$(d_0,c, d_0)$
is a positive triple and assume that, for all m in
$\mathbb {N}$
,
$$ \begin{align*}(d_0,b^0_m,b^1_m, d_1) \end{align*} $$
is a positive quadruple. Let
$V_m$
be the unique diamond with extremities
$b^0_m$
and
$b^1_m$
contained in the diamond
$V_{c}(d_0,d_1)$
. Then
$$ \begin{align*} \lim_{m\to\infty} V_m=\{c\}. \end{align*} $$
Proof. Let
$a_0$
be in
$V_{c}^*(d_0, d_1)$
and
$a_1$
be in
$V_{c}^{*}(a_0, d_1)$
so that
$(a_0, d_0, c, d_1, a_1)$
is a positive quintuple and, for all big enough m,
$(a_0, d_0,b^0_m,b^1_m, d_1, a_1)$
is a positive configuration.
Let
$z_m$
belong to
$V_m$
, we want to prove that
$$ \begin{align*}\lim_{m\to\infty} z_m=c. \end{align*} $$
Let p in
$\overline {V}_{c}(d_0,d_1)$
be an accumulation point of the sequence
$ \{z_m\}_{m\in \mathbb N}$
. Up to extracting a subsequence we may assume
$$ \begin{align*}\lim_{m\to\infty} z_m=p, \end{align*} $$
By Corollary 3.10, p belongs to
$V_c(a_0, a_1)$
and in particular it belongs to
$\Omega _{a_0}\cap \Omega _{a_1}$
. From the fact that
$z_m$
belongs to
$V_{d_0}(a_0, b_{m}^{1})$
we get that p belongs to
$\overline {V}_{d_0}(a_0,c)$
; similarly p belongs to
$\overline {V}_{d_1}(a_1,c)$
. Therefore
$$ \begin{align*}p\in \overline{V}_{d_0}(a_0,c) \cap \overline{V}_{d_1}(a_1,c)\cap \Omega_{a_0}\cap\Omega_{a_1}. \end{align*} $$
Let
$V=V_{d_0}(a_0,c)$
, and recall that by Lemma 2.8,
$$ \begin{align*}V_{d_1}(a_1,c)\subset V^*.\end{align*} $$
Finally remark that
$$ \begin{align*}\overline{V}\cap \Omega_{a_1}=\overline{\mathsf N}_{a_1} \cdot c, \overline{V}{}^{*}\cap \Omega_{a_1}=\overline{\mathsf N}^{-1}_{a_1} \cdot c, \end{align*} $$
for the (positive) semigroup
$\mathsf N_{a_1}$
in
$\mathsf U_{a_1}$
. Since
$$ \begin{align*}\overline{\mathsf N}_{a_1}\cap \overline{\mathsf N}^{-1}_{a_1}=\{{\operatorname{id}}\}, \end{align*} $$
one has
$p=c$
, which is what we wanted to prove.
Proof of Proposition 5.8.
The chosen hyperbolization of S defines a
$\pi _1(S)$
-invariant cross-ratio on
$\partial _\infty \pi _1(S)\cong {\mathbf P}^1(\mathbb {R})$
. Let us also fix an orientation on
$\partial _\infty \pi _1(S)$
. For any cyclically oriented triple
${t=(x,y,z)}$
, let us consider the harmonic (with respect to the cross-ratio) quadruple
$(x,y,z,w)$
, and let then
By construction
$Y_t$
is an open neighborhood of
$\xi (z)$
. Moreover if
$(x,y_1,y_0,z)$
is a cyclically oriented quadruple,
$$ \begin{align} Y_{(x,y_0,z)}\subset Y_{(x,y_1,z)}. \end{align} $$
Finally, since
$\xi $
is continuous, by Lemma 5.9
$$ \begin{align} \lim_{y\to z}Y_{(x,y,z)}=\{\xi(z)\}. \end{align} $$
We now deduce the Anosov property from Assertion (22).
Recall that the chosen uniformization of the surface enables us to identify the space of triples in the boundary at infinity with the unit tangent bundle
$\mathsf U{\mathbf H}^2$
of the universal cover of S. Let
$\{\phi _s\}_{s\in \mathbb R}$
be the geodesic flow on
$\mathsf U{\mathbf H}^2$
. Let
$\mathcal F$
be the trivial bundle
$\mathbf {F}_\Theta \times \mathsf U{\mathbf H}^2$
. The actions of
$\pi _1(S)$
on
$\mathsf U{\mathbf H}^2$
and on
$\mathbf {F}_\Theta $
–through
$\rho $
– give rise to an action of
$\pi _1(S)$
on
$\mathcal F$
.
Let
$\mathcal U$
be the subbundle of
$\mathcal {F}$
with open fibers given by
$$ \begin{align*}\mathcal U=\{(x,v)\in \mathcal F\mid v\in \mathsf U{\mathbf H}^2,\ \ x\in Y_v\}. \end{align*} $$
The bundle
$\mathcal U$
is invariant by the
$\pi _1(S)$
-action, moreover it has a canonical section
$\sigma _0$
given by
$$ \begin{align*}\sigma_0(x,y,z)=\xi(z). \end{align*} $$
Let us lift the flow
$\{\phi _s\}_{s\in \mathbb R}$
to a flow
$\{\Phi _s\}_{s\in \mathbb R}$
on
$\mathcal F$
acting trivially on the first factor. By assertion (21), for all positive s
$$ \begin{align*}\Phi_{-s}(\mathcal U)\subset\mathcal U. \end{align*} $$
Moreover the section
$\sigma _0$
is invariant by
$\{\Phi _s\}_{s\in \mathbb R}$
.
The diamond metric
$g_t$
and the diamond distance
$d_t$
on each
$Y_t$
give a metric on each fiber of
$\mathcal U$
which depends continuously on the base and is equivariant under the action of
$\pi _1(S)$
.
For any R, let
$\mathcal U(R)$
be the neighborhood of the image of the section
$\sigma _0$
, given by
$$ \begin{align*}\mathcal U(R)=\{(x,v)\in \mathcal U\mid v\in \mathsf U{\mathbf H}^2,\ \ d_v(x,\sigma_0(v))\leqslant R\}. \end{align*} $$
It now follows from assertion (22) and Proposition 4.11, that for every u in
$\mathsf U{\mathbf H}^2$
, there is
$s_u$
such that, for all
$(x,u)$
in
$\mathcal U(R)$
$$ \begin{align} \text{for all } s\geqslant s_u,\ \ g_{\Phi_s(x,u)}\circ {\mathsf T}_{(x,u)} \Phi_{-s}\leqslant \frac{1}{2} g_{(x,u)}. \end{align} $$
Let now s be the real valued function on
$\mathsf U{\mathbf H}^2$
defined by
$$ \begin{align*}s(u)=\inf\{s_u\mid s_u \text{ satisfies assertion~(23)} \text{ on } \mathcal U(R)\}.\end{align*} $$
The function
$u\mapsto s(u)$
is upper semicontinuous and invariant under the action of
$\pi _1(S)$
. Thus by compactness of
$\mathsf U{\mathbf H}^2/\pi _1(S)$
the function has an upper bound
$s_0$
. Then for all s greater than
$s_0$
$$ \begin{align*} \Phi_s^*g \leqslant \frac{1}{2} g , \end{align*} $$
on
$\mathcal U(R)$
. In other words, the action of
$\{\Phi _{-s}\}_{s\in \mathbb R}$
is contracting on
$\mathcal {V} =\mathcal {U}(R)$
and
$\sigma _0$
is an invariant section.
Thus
$\rho $
is
$\Theta $
-Anosov according to the definition given in the beginning of the section and
$ \xi $
is its limit curve.
Now Corollary B in the introduction follows directly from the openness of the set of
$\Theta $
-Anosov representations. More precisely
Proof of Corollary B.
A positive representation
$\rho _0$
with limit map
$\xi _0$
, is
$\Theta $
-Anosov. Thus there is an open neighborhood U of
$\rho _0$
containing only
$\Theta $
-Anosov representations. For any
$\rho $
in this neighborhood, let
$\xi _\rho $
be the limit map. Note that this map is equivariant and transverse. The map
$\rho \mapsto \xi _\rho $
which sends an Anosov representation to its limit curve is continuous. Since
$\xi _{\rho _0}$
is positive it sends pairwise distinct triples in
$\partial _\infty \pi _1(S)$
to positive triples. Moreover
$\pi _1(S)$
acts cocompactly on the set of triples of pairwise distinct points of
$\partial _\infty \pi _1(S)$
, thus, for
$\rho $
close enough to
$\rho _0$
, the continuous curve
$\xi _\rho $
sends pairwise distinct triples to positive triples. Hence by Proposition 3.17,
$\xi _\rho $
is positive.
Remark 5.10. The definition of
$\Theta $
-positive representations can be made in more generality for nonelementary word hyperbolic group
$\Gamma $
whose boundary admits a cyclic ordering. This holds if
$\Gamma $
is a surface group, but also if
$\Gamma $
is virtually free. For example, an appropriate extension of the arguments in this section shows that a representation of
$\Gamma $
is
$\Theta $
-Anosov if it admits a
$\rho $
-equivariant positive boundary map
$\xi \colon \partial _\infty \Gamma \to \mathbf {F}_\Theta $
.
6 Closedness
In this section we consider the space
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
of homomorphisms from
$\pi _1(S)$
to
$\mathsf G$
, which contains a
$\Theta $
-loxodromic element.
We show that the set of
$\Theta $
-positive representations
$\operatorname {Hom}_{\Theta \textrm {-pos}}(\pi _1(S),\mathsf G)$
is an open and closed subset of
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
, hence a union of connected components.
We first have
Proposition 6.1. Every
$\Theta $
-positive representation is an element of the set
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
. Moreover, every
$\Theta $
-positive representation has a compact centralizer and does not factor through a proper parabolic subgroup of
$\mathsf G$
.
Proof. The first part is a consequence of Proposition 3.19. Let us first note that since the centralizer of a positive triple is compact, the centralizer of a positive representation is compact as well. Let
${\rho \colon \pi _1(S) \to \mathsf G}$
be a positive representation with
$\rho $
-equivariant positive boundary map
${\xi \colon \partial _\infty \pi _1(S) \to \mathbf {F}_\Theta }$
. Then
$\rho $
is
$\Theta $
-Anosov with boundary map
$\xi $
. This remains true when restricting the representation to a finite index subgroup. To argue by contradiction we can thus assume that without loss of generality
$\rho (\pi _1(S))$
is contained in a proper parabolic subgroup of
$\mathsf G$
. We consider the semi-simplification
$\rho ^{ss}$
of
$\rho $
, whose image is contained in a Levi factor of the parabolic subgroup.
By [Reference Guéritaud, Guichard, Kassel and Wienhard21, Proposition 1.8] the semi-simplification
$\rho ^{ss}$
is
$\Theta $
-Anosov, denote
$\xi ^{ss}$
the
$\rho ^{ss}$
-equivariant boundary map. Since
$\rho ^{ss}$
belongs to the closure of the
$\mathsf G$
-orbit of
$\rho $
; there exists thus a sequence
$ \{g_m\}_{m\in \mathbb N}$
in
$\mathsf G$
such that
$\rho ^{ss}$
is the limit of
$g_m \rho g_{m}^{-1}$
. Since the boundary map
$\xi ^{ss}$
is transverse, Lemma 3.6 implies that the boundary map
$\xi ^{ss}$
is also positive as well. But this is a contradiction because the centralizer of
$\rho ^{ss}$
in
$\mathsf G$
contains the center of the Levi factor of the parabolic subgroup which is noncompact.
By a classical result of Borel and Tits [Reference Borel and Tits6, Corollaire 3.3] (proved also by Morozov [Reference Morozov35] in characteristic zero), the set
$\operatorname {Hom}^*(\pi _1(S),\mathsf G)$
is contained in the set of reductive homomorphisms, that is, representations
$\rho \colon \pi _1(S) \to \mathsf G$
, whose Zariski closure is reductive. Thus a direct consequence of Theorem 6.1 is
Corollary 6.2. Let
$\rho \colon \pi _1(S) \to \mathsf G$
be a
$\Theta $
-positive representation, then the Zariski closure of
$\rho (\pi _1(S))$
is reductive.
We also show the following result as a consequence of a result of Benoist–Labourie [Reference Benoist and Labourie2].
Proposition 6.3. Assume that the Zariski closure
$\mathsf H$
of the image of
$\rho $
is such that the exponential of every element in the open Weyl chamber of
$\mathsf H$
is loxodromic with respect to
$\mathbf {F}_\Theta $
. Then
$\rho $
belongs to
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
.
In particular, every representation with Zariski dense image belongs to
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
.
Proof. Indeed by [Reference Benoist and Labourie2, Theorem A.1.1], an element h of the image of
$\rho $
has a hyperbolic part which belongs to the Weyl Chamber. Hence h is loxodromic.
We expect that the list of possible Zariski closures of
$\Theta $
-positive representations is indeed restrictive. Classifications of the Zariski closures for maximal representations were given in [Reference Burger, Iozzi and Wienhard11, Reference Burger, Iozzi and Wienhard12, Reference Hamlet27, Reference Hamlet28] and for Hitchin representations in [Reference Guichard22, Reference Sambarino37].
Since the set of
$\Theta $
-positive representations is open in
$\operatorname {Hom}(\pi _1(S),\mathsf G)$
by Corollary B, it is also open in
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
.
We will now show
Theorem 6.4. The set of
$\Theta $
-positive homomorphisms is closed in the set
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
.
We will first prove the following proposition of independent interest:
Proposition 6.5. Let
$ \{\rho _m\}_{m\in \mathbb N}$
be a sequence of
$\Theta $
-positive representations converging to a representation
$\rho _\infty $
. Let
$\xi _m$
be the limit curve of
$\rho _m$
. Assume that we can find
$x_0$
and
$y_0$
in
$\partial _\infty \pi _1(S)$
such that
$\left \{(\xi _m(x_0),\xi _m(y_0))\right \}_{m\in \mathbb N}$
converges to a transverse pair, then
$\rho _\infty $
is positive.
6.1 Proof of Proposition 6.5
We fix a countable set A in
$\partial _\infty \pi _1(S)$
, invariant by
$\pi _1(S)$
and containing
$x_0$
and
$y_0$
. We may now assume, by the Cantor diagonal argument, that
$ \{\xi _m|_A\}_{m\in \mathbb N}$
converges simply to a map
$\xi _\infty $
from A to
$\mathbf {F}_\Theta $
. By hypothesis
$\xi _\infty (x_0)$
and
$\xi _\infty (y_0)$
are transverse.
For any pair of distinct points
$(x,y)$
in
$A^2$
, denote by
$]x,y[$
the interval in the oriented circle
$\partial _\infty \pi _1(S)$
with origin x and extremity y, let c be in
$A \cap ]x,y[$
and set
the convergence being for the Hausdorff topology, and using again the Cantor diagonal extraction, we can and will assume that all those sequences converge of all
$(x,y)$
in
$A^2$
with
$x\neq y$
. Observe that
$W_\infty (x,y)$
only depends on x, y, and the interval
$]x,y[$
. Furthermore the following equivariance property holds:
$\rho _\infty (\gamma ) W_\infty ( x,y) = W_\infty ( \gamma \cdot x, \gamma .\cdot y)$
.
Lemma 6.6. Assume that
$\xi _\infty (x)$
and
$\xi _\infty (y)$
are transverse then
$W_\infty (x,y)$
is a closure of a diamond with extremities
$\xi _\infty (x)$
and
$\xi _\infty (y)$
and is Zariski dense.
Proof. Since
$\xi _\infty (x)$
and
$\xi _\infty (y)$
are transverse,
$W_\infty (x,y)$
is the closure of a diamond (see Proposition 3.9). It thus contains an open set, and in particular is Zariski dense.
Lemma 6.7. For every pair of distinct points
$(x,y)$
and
$(z,t)$
in A, one has
$$ \begin{align*}\overline{W}{}_{\infty}^{Z}(x,y)=\overline{W}{}_{\infty}^{Z}(z,t) \end{align*} $$
where
$\overline {M}{}^Z$
denotes the Zariski closure of a set M. In particular, for all distinct x and y,
$W_\infty (x,y)$
is Zariski dense.
Observe that only the last assertion depends on the assumption that
$\xi _\infty (x_0)$
and
$\xi _\infty (y_0)$
are transverse.
Proof. We shall use freely the following fact. If
$\gamma $
is an algebraic automorphism of a variety V, if B is a Zariski closed subset such that
$\gamma (B)\subset B$
then
$\gamma (B)=B$
.
We first prove that if
$[u,v]\subset [w,s]$
, then we have
$$ \begin{align} \overline{W}{}_{\infty}^{Z}(u,v)=\overline{W}{}_{\infty}^{Z}(w,s). \end{align} $$
We can always find an element
$\gamma $
of
$\pi _1(S)$
such that
$$ \begin{align*}\gamma[w,s]\subset [u,v]. \end{align*} $$
Thus
$$ \begin{align*}\rho_\infty(\gamma)\bigl(\overline{W}{}_{\infty}^{Z}(w,s)\bigr)\subset \overline{W}{}_{\infty}^{Z}(u,v)\subset\overline{W}{}_{\infty}^{Z}(w,s). \end{align*} $$
By the initial observation we get that
$$ \begin{align*}\overline{W}{}_{\infty}^{Z}(w,s) \subset \overline{W}{}_{\infty}^{Z}(u,v) \subset\overline{W}{}_{\infty}^{Z}(w,s), \end{align*} $$
and thus the assertion (24) follows. Take now
$\gamma $
in
$\pi _1(S)$
such that
$$ \begin{align*}\gamma[x,y]\subset [x,y], \ \ \gamma[x,y]\cup [z,t] \not=\partial_\infty\pi_1(S). \end{align*} $$
We can then find distinct points u and v such that
$$ \begin{align*}\left(\gamma[x,y]\cup [z,t]\right)\subset [u,v]. \end{align*} $$
Thus, applying thrice assertion (24), we have
$$ \begin{align*}\overline{W}{}_{\infty}^{Z}(x,y) = \overline{W}{}_{\infty}^{Z}(\gamma\cdot x,\gamma\cdot y) = \overline{W}{}_{\infty}^{Z}(u,v)=\overline{W}{}_{\infty}^{Z}(z,t). \end{align*} $$
The last assertion follows from the fact that
$\xi _\infty (x_0)$
and
$\xi _\infty (y_0)$
are transverse and thus
$W_\infty (x_0,y_0)$
is Zariski dense by Lemma 6.6.
We are now in the position to show that
$\rho _\infty $
is
$\Theta $
-positive. This will be a consequence of the following proposition:
Proposition 6.8. For any pair of distinct points
$(x,y)$
, the pair
$(\xi _\infty (x),\xi _\infty (y))$
is transverse. Moreover,
$\xi _\infty $
is a positive map.
Proof. Let
$(x, y, z)$
be a triple of pairwise distinct points in
$\partial _\infty \pi _1(S)$
. Let us denote for simplicity
$x_n=\xi _n(x)$
,
$y_n=\xi _n(y)$
and
$z_n=\xi _n(z)$
for n in
$\mathbb N\cup \{\infty \}$
. We choose diamonds by letting
$$ \begin{align*} V^0_n=V^*_{z_n}(x_n,y_n),\ \ V^1_n=V^*_{y_n}(x_n,z_n),\ \ V^2_n=V^*_{y_n}(z_n,y_n). \end{align*} $$
Since
$W_\infty (x,y)$
,
$W_\infty (y,z)$
, and
$W_\infty (z,x)$
are Zariski dense, we can pick three points a, b, and c so that
-
1.
$a\in W_\infty (x,y)$
,
$b\in W_\infty (y,z)$
,
$c\in W_\infty (z,x)$
, -
2.
$a,b,c$
are pairwise transverse, -
3. any point in
$\{a,b,c\}$
is transverse to any point in
$\{x_\infty ,y_\infty ,z_\infty \}$
.
Let us now pick sequences
$ \{a_m\}_{m\in \mathbb N}$
,
$ \{b_m\}_{m\in \mathbb N}$
, and
$ \{c_m\}_{m\in \mathbb N}$
, with
$a_m\in V^0_m$
,
$b_m\in V^1_m$
, and
$c_m\in V^2_m$
, and converging to a, b, and c respectively.
We will now apply the necklace property several times. By Proposition 3.2,
$(a_m, b_m ,c_m)$
is a positive triple and since
$a,b,c$
are pairwise transverse it follows that
$(a,b,c)$
is a positive triple.
Then, since
$x_m$
belongs to
$V^*_{b_m}(a_m,c_m)$
, it follows that
$x_\infty $
belongs to
$\overline {V}{}^{*}_b(a,c)$
. Since
$x_\infty $
is transverse to both a and c,
$x_\infty $
belongs to
$V^*_b(a,c)$
. Symmetrically
$y_\infty $
belongs to
$V^*_c(a,b)$
,
$z_\infty $
belongs to
$V^*_a(c,b)$
. Applying Proposition 3.2 again,
$(x_\infty ,y_\infty ,z_\infty )$
is a positive triple.
The fact that, for any cyclically oriented quadruple
$(x,y,z,w)$
, the quadruple
$(\xi _\infty (x), \xi _\infty (y), \xi _\infty (z), \xi _\infty (w))$
is positive, now follows from Proposition 3.1.(3). Hence the positivity of
$\xi _\infty $
by definition.
6.2 Proof of Theorem 6.4
We consider a sequence
$ \{\rho _m\}_{m\in \mathbb N}$
of
$\Theta $
-positive representations converging to a representation
$\rho _\infty $
. Let
$ \{\xi _m\}_{m\in \mathbb N}$
be the corresponding sequence of positive limit maps. Our assumption is that image of
$\rho _\infty $
contains a
$\Theta $
-loxodromic element
$\rho _\infty (\gamma _0)$
.
We fix a countable set A in
$\partial _\infty \pi _1(S)$
, invariant by
$\pi _1(S)$
, and containing
$\gamma _0^+$
and
$\gamma _0^-$
. We may now assume, by applying the Cantor diagonal argument, that
$ \{\xi |_A\}_{m\in \mathbb N}$
converges simply to a map
$\xi _\infty $
from A to
$\mathbf {F}_\Theta $
.
Observe now that if
$y^+$
is the attracting fixed point of
$\rho _\infty (\gamma _0)=\lim _{n\to \infty } \rho _\infty (\gamma _0)$
, then
$y^+$
is the limit of the attracting fixed points of
$\{\rho _m(\gamma _0)\}_{m\in \mathbb N}$
. It follows that
$y^+=\xi _\infty (\gamma ^+)$
. The same holds for the repelling fixed point
$y^-$
of
$\rho _\infty (\gamma _0)$
, and we have
$y^-=\xi _\infty (\gamma ^-)$
. Since
$y^+$
and
$y^-$
are transverse, we can apply Proposition 6.5 to obtain that
$\rho _\infty $
is positive.
6.3 Proof of Theorem C
By Theorem 6.4 the set of
$\Theta $
-positive representations is closed in
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
. Using furthermore Corollary B this set is open. Thus the set of
$\Theta $
-positive representations is a union of connected components of
$\operatorname {Hom}^\Theta (\pi _1(S),\mathsf G)$
. Since finally, we can obtain positive representation by factoring in a positive
$\mathsf {PSL}_2(\mathbb R)$
we deduce the Theorem.
7 Positive representations and Cayley components
Let us recall that for a real split Lie group
$\mathsf G$
, the Hitchin component was originally defined by Hitchin as the image of the Hitchin section
$\Phi $
which assigns to a tuple of holomorphic differentials on a Riemann surface
$\Sigma $
a
$\mathsf G$
-Higgs bundle on
$\Sigma $
. Let us denote the image of
$\Phi $
by
$\mathcal {P}(\Sigma , \mathsf G)$
. Through the non-Abelian Hodge correspondence the set
$\mathcal {P}(\Sigma , \mathsf G)$
corresponds to a subset of the representation variety
$\operatorname {Rep}^+(\pi _1(S),\mathsf G)$
, which we denote by the same symbol. Hitchin showed that
$\mathcal {P}(\Sigma , \mathsf G)$
is open and closed (hence a union of connected components) and the map
$\Phi $
gives a parametrization of
$\mathcal {P}(\Sigma , \mathsf G)$
. In the case of maximal representations a similar but more complicated parametrization of the space of maximal representations was obtained in [Reference Bradlow, García-Prada and Gothen8], [Reference García-Prada, Gothen and Riera18], and [Reference Biquard, García-Prada and Rubio5]. For any simple Lie groups admitting a positive structure relative to
$\Theta $
, the authors of [Reference Bradlow, Collier, García-Prada, Gothen and Oliveira7] define in a similar way subsets
$\mathcal {P}_e(\Sigma ,\mathsf G)$
of the moduli space of
$\mathsf G$
-Higgs bundles by giving explicit parametrizations, see also [Reference Collier14] and [Reference Aparicio-Arroyo, Bradlow, Collier, García-Prada, Gothen and Oliveira1] for indefinite orthogonal groups. They prove that
$\mathcal {P}_e(\Sigma ,\mathsf G)$
is open and closed in
$\operatorname {Rep}^+(\pi _1(S),\mathsf G)$
. They further prove that all representations in
$\mathcal {P}_e(\Sigma ,\mathsf G)$
have compact centralizer and thus do not factor through a proper parabolic subgroup. They further show that the set
$\mathcal {P}_e(\Sigma ,\mathsf G)$
contains an open subset of
$\Theta $
-positive representations.
Theorem C implies that any connected component of
$\mathcal {P}^\Theta _e(\Sigma ,\mathsf G) = \mathcal {P}_e(\Sigma ,\mathsf G)\cap \operatorname {Rep}^\Theta (\pi _1(S),\mathsf G)/\mathsf G$
that contains a
$\Theta $
-positive representation consists entirely of
$\Theta $
-positive representations. Due to the extension of our main result in [Reference Beyrer, Guichard, Labourie, Pozzetti and Wienhard3] we further have that any connected component of
$\mathcal {P}_e(\Sigma ,\mathsf G)$
that contains at least one
$\Theta $
-positive representation consists entirely of
$\Theta $
-positive representations. For many G this implies that
$\mathcal {P}_e(\Sigma ,\mathsf G) \subset \operatorname {Hom}_{\Theta \textrm {-pos}}(\Gamma ,\mathsf G)$
.
For this let us introduce the standard components of
$\mathcal {P}_e(\Sigma ,\mathsf G)$
. Consider an embedding of
$\mathsf {SL}_2(\mathbb R)$
, such that the induced map from
${\mathbf P}^1(\mathbb {R})$
to
$\mathbf {F}_\Theta $
is a positive circle; then the corresponding Fuchsian representation is positive. These Fuchsian representations can now in addition be twisted by a representation of
$\pi _1(S)$
into the centralizer of this
$\mathsf {SL}_2(\mathbb R)$
in
$\mathsf G$
. This is called a twisted positive Fuchsian representation. We call a component of
$\mathcal {P}_e(\Sigma ,\mathsf G)$
standard if it contains a twisted positive Fuchsian representation.
When
$\mathsf G$
is a classical group and not locally isomorphic to
$\mathsf {Sp}_4(\mathbb {R})$
, or
$ \mathsf {SO}(p, p+1)$
, every component of
$\mathcal {P}_e(\Sigma ,\mathsf G)$
is standard [Reference Bradlow, Collier, García-Prada, Gothen and Oliveira7].
For
$\mathsf {Sp}_4(\mathbb {R})$
with the positive structure relative to
$\Theta \neq \Delta $
, positive representations correspond precisely to maximal representations [Reference Burger, Iozzi, Labourie and Wienhard10, Reference Burger, Iozzi and Wienhard12]. In particular, the exceptional connected components of maximal representations in
$\mathcal {P}_e(\Sigma ,\mathsf G)$
which do not contain any twisted positive Fuchsian representation [Reference Gothen20], are positive. Similarly for the exceptional Hermitian Lie group of tube type,
$\mathcal {P}_e(\Sigma ,\mathsf G) = \operatorname {Hom}_{\Theta \textrm {-pos}}(\Gamma ,\mathsf G)$
is the set of maximal representations.
For
$ \mathsf {SO}(p, p+1)$
with the positive structure relative to
$\Theta \neq \Delta $
, there also exist exceptional connected components in
$\mathcal {P}_e(\Sigma ,\mathsf G)$
which do not contain any twisted positive Fuchsian representation. To deduce that they are positive we can use an embedding argument. Embedding
$ \mathsf {SO}(p, p+1) \to \mathsf {SO}(p, p+2) $
, these components are sent to standard components for
$\mathsf {SO}(p, p+2)$
[Reference Bradlow, Collier, García-Prada, Gothen and Oliveira7], and thus they consist also entirely of
$\Theta $
-positive representations seen in
$\mathsf {SO}(p, p+2) $
. Since any
$\Theta $
-positive representation in
$\mathsf {SO}(p, p+2) $
, whose image is contained in
$ \mathsf {SO}(p, p+1)$
is also
$\Theta $
-positive as a representation into
$ \mathsf {SO}(p, p+1)$
, we conclude that these exceptional components consist entirely of
$\Theta $
-positive representations.
However, for exceptional groups G whose restricted root system have a Dynkin diagram of type
$F_4$
, we do not know if all connected components of
$\mathcal {P}_e(\Sigma ,\mathsf G)$
contain positive representations. If this were the case, this would imply
$\mathcal {P}_e(\Sigma ,\mathsf G) \subset \operatorname {Hom}_{\Theta \textrm {-pos}}(\Gamma ,\mathsf G)$
. We further expect to have
$\mathcal {P}_e(\Sigma ,\mathsf G) = \operatorname {Hom}_{\Theta \textrm {-pos}}(\Gamma ,\mathsf G)$
. For this to hold one would need to show that any positive representation lies in
$\mathcal {P}_e(\Sigma ,\mathsf G)$
, or what might be the better approach, that none of the connected components of
$ \operatorname {Hom}^+(\Gamma ,\mathsf G)\backslash \mathcal {P}_e(\Sigma ,\mathsf G)$
contain a
$\Theta $
-positive representation.
Acknowledgments
We thank Michel Brion, Steve Bradlow, Brian Collier, Beatrice Pozzetti, and Jérémy Toulisse for interesting discussions regarding the topics of this paper. We also thank Nicolas Tholozan and Tengren Zhang for pointing out mistakes in previous versions.
Competing interests
The authors have no competing interest to declare.
Financial support
O.G. and F.L. acknowledge the funding of the ANR grant DynGeo ANR-11-BS01-013 and of the Institut Universitaire de France. A.W. acknowledges funding by the DFG, Germany’s Excellence Strategy EXC-2181/1 - 390900948 (the Heidelberg STRUCTURES Cluster of Excellence), by the ERC under ERC-Consolidator grant 614733 and ERC-Advanced Grant 101018839. She thanks the Klaus Tschira Foundation, the Hector Fellow Academy, and the Clay Foundation for support. F.L., O.G., and A.W. acknowledge support from N.S.F grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties”. F.L. and A.W. acknowledge support from N.S.F Grant No. 1440140 while in residence in Fall 2019, at the MSRI in Berkeley, F.L. acknowledges funding by the University of California at Berkeley and by the ERC under ERC-Advanced grant 101095722.
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