1 Introduction
Let G be a connected semisimple real algebraic group. Let
$(X, d)$
denote the associated Riemannian symmetric space. Let
$P=MAN$
be a minimal parabolic subgroup of G with fixed Langlands decomposition, where A is a maximal real split torus of G, M the maximal compact subgroup of P commuting with A, and N the unipotent radical of P. Note that N is a maximal horospherical subgroup of G, which is unique up to conjugations.
Fix a positive Weyl chamber
$\mathfrak a^+\subset \log A$
so that
$\log N$
consists of positive root subspaces, and we set
$A^+=\exp \mathfrak a^+$
. This means that N is a contracting horospherical subgroup in the sense that for any a in the interior of
$A^+$
,
$$ \begin{align*}N =\{g\in G: a^{-n} g a^n\to e\text{ as}\ n\to +\infty\}.\end{align*} $$
Let
$\Gamma $
be a Zariski dense discrete subgroup of G. In this paper, we are interested in the topological behavior of the action of the horospherical subgroup N on
$\Gamma \backslash G$
via the right translations. When
$\Gamma <G$
is a cocompact lattice, every N-orbit is dense in
$\Gamma \backslash G$
, that is, the N-action on
$\Gamma \backslash G$
is minimal. This is due to Hedlund [Reference Hedlund11] for
$G=\operatorname {PSL}_2(\mathbb R)$
and to Veech [Reference Veech19] in general. Dani gave a full classification of possible orbit closures of N-action for any lattice
$\Gamma <G$
[Reference Dani8].
For a general discrete subgroup
$\Gamma <G$
, the quotient space
$\Gamma \backslash G$
does not necessarily admit a dense N-orbit, even a dense
$NM$
-orbit, for instance in the case where
$\Gamma $
does not have a full limit set. Let
$\mathcal F$
denote the Furstenberg boundary
$G/P$
. We denote by
$\Lambda =\Lambda _\Gamma $
the limit set of
$\Gamma $
,
$$ \begin{align*}\Lambda=\Big\{\lim_{i\to \infty} \gamma_i(o) \in \mathcal F : \gamma_i\in \Gamma\Big\},\end{align*} $$
where
$o\in X$
and the convergence is understood as in Definition 2.2. This definition is independent of the choice of
$o\in X$
. The limit set
$\Lambda $
is known to be the unique
$\Gamma $
-minimal subset of
$\mathcal F$
(see [Reference Benoist1, Reference Guivarc’h9, Reference Lee and Oh15]). Thus, the set
$$ \begin{align*}\mathcal E=\{[g]\in \Gamma\backslash G: gP\in \Lambda\}\end{align*} $$
is the unique P-minimal subset of
$\Gamma \backslash G$
. For a given point
$[g]\in \mathcal E$
, the topological behavior of the horospherical orbit
$[g]N$
(or of
$[g]NM$
) is closely related to the ways in which the orbit
$\Gamma (o)$
approaches
$gP$
along its limit cone. The limit cone
$\mathcal L=\mathcal L_\Gamma $
of
$\Gamma $
is defined as the smallest closed cone of
$\mathfrak a^+$
containing the Jordan projection
$\unicode{x3bb} (\Gamma )$
. It is a convex cone with non-empty interior:
$\operatorname {int} \mathcal L\ne \emptyset $
[Reference Benoist1]. If
$\operatorname {rank } G=1$
, then
$\mathcal L=\mathfrak a^+$
. In higher ranks, the limit cone of
$\Gamma $
depends more subtly on
$\Gamma $
.
1.1 Horospherical limit points
Recall that in the rank one case, a horoball in X based at
$\xi \in \mathcal F$
is a subset of the form
$gN (\exp \mathfrak a^+ )( o)$
, where
$g\in G$
is such that
$\xi =gP$
[Reference Dalbo5]. Our generalization to higher rank of the notion of a horospherical limit point involves the limit cone of
$\Gamma $
. By a
$\Gamma $
-tight horoball based at
$\xi \in \mathcal F$
, we mean a subset of the form
$\mathcal H_{\xi }=g N (\exp \mathcal C) (o)$
, where
$g\in G$
is such that
$\xi =gP$
and
$\mathcal C$
is a closed cone contained in
$\operatorname {int} \mathcal L\cup \{0\}$
. For
$T>0$
, we write
$$ \begin{align*}\mathcal H_{\xi}(T)= gN (\exp (\mathcal C-\mathcal C_T))o,\end{align*} $$
where
$\mathcal C_T=\{u\in \mathcal C: \|u\|<T\}$
for a Euclidean norm
$\|\cdot \|$
on
$\mathfrak a$
.
Definition 1.1. We call a limit point
$\xi \in \Lambda $
a horospherical limit point of
$\Gamma $
if one of the following equivalent conditions holds:
-
• there exists a
$\Gamma $
-tight horoball
$\mathcal H_\xi $
based at
$\xi $
such that for any
$T>1$
,
$\mathcal H_\xi (T)$
contains some point of
$\Gamma (o)$
; -
• there exist a closed cone
$\mathcal C \subset {\operatorname {int} \mathcal L} \cup \{0\}$
and a sequence
$ \gamma _j \in \Gamma $
satisfying that
$\beta _\xi (o,\gamma _jo) \in \mathcal C$
for all
$j\ge 1$
and
$ \beta _\xi (o,\gamma _jo) \to \infty $
as
$j\to \infty $
, where
$\beta $
denotes the
$\mathfrak a$
-valued Busemann map (Definition 2.3).
See Lemma 3.3 for the equivalence of the above two conditions. We denote by
$$ \begin{align*}\Lambda_h\subset \Lambda\end{align*} $$
the set of all horospherical limit points of
$\Gamma $
. The attracting fixed point
$y_\gamma $
of a loxodromic element
$\gamma \in \Gamma $
whose Jordan projection
$\unicode{x3bb} (\gamma )$
belongs to
$\operatorname {int} \mathcal L$
is always a horospherical limit point (Lemma 3.5). Moreover, for any
$u\in \operatorname {int} \mathcal L$
, any u-directional radial limit point
$\xi $
(i.e.
$\xi =gP$
for some
$g\in G$
such that
$\limsup _{t\to \infty } \Gamma g \exp (tu)\ne \emptyset $
) is also a horospherical limit point (Lemma 5.3).
Remarks 1.2
-
(1) There exists a notion of horospherical limit points in the geometric boundary associated to a symmetric space, see [Reference Hattori10]. When
$\operatorname {rank }G \geq 2 $
, this notion and the one considered here are different. -
(2) Unlike the rank one case, a sequence
$\gamma _i(o)\in \mathcal H_{\xi }(T_i)$
, with
$T_i\to \infty $
, does not necessarily converge to
$\xi $
for a
$\Gamma $
-tight horoball
$\mathcal H_\xi $
based at
$\xi $
. It is hence plausible that a general discrete group
$ \Gamma $
would support a horospherical limit point outside of its limit set.
1.2 Denseness of horospheres
The following theorem generalizes Dal’bo’s theorem [Reference Dalbo5] to discrete subgroups in higher rank semisimple Lie groups.
Theorem 1.3. Let
$\Gamma <G$
be a Zariski dense discrete subgroup. For any
$[g]\in \mathcal E$
, the following are equivalent:
-
(1)
$gP\in \Lambda _h$
; -
(2)
$[g] NM$
is dense in
$\mathcal E$
.
Remarks 1.4. Conze and Guivarc’h considered the notion of a horospherical limit point for Zariski dense discrete subgroups
$\Gamma $
of
$\operatorname {SL}_d(\mathbb R)$
using the description of
$\operatorname {SL}_d(\mathbb R)/P$
as the full flag variety and the standard linear action of
$\Gamma $
on
$\mathbb R^d$
[Reference Conze, Guivarc’h, Burger and Iozzi4]. By duality, this notion coincides with ours and hence the special case of Theorem 1.3 for
$G=\operatorname {SL}_d(\mathbb R)$
also follows from [Reference Conze, Guivarc’h, Burger and Iozzi4, Theorem 4.2]. (However the claim in [Reference Conze, Guivarc’h, Burger and Iozzi4, Theorem 6.3] is incorrect.)
To extend Theorem 1.3 to N-orbits, we fix a
$P^\circ $
-minimal subset
$\mathcal E_0$
of
$\Gamma \backslash G$
, where
$P^\circ $
denotes the identity component of P. Clearly,
$\mathcal E_0\subset \mathcal E$
. Since
$P=P^\circ M$
, any
$P^\circ $
-minimal subset is a translate of
$\mathcal E_0$
by an element of the finite group
$M^\circ \backslash M$
, where
$ M^\circ $
is the identity component of
$ M $
. Denote by
$ \mathfrak D_\Gamma = \{\mathcal E_0,\ldots , \mathcal E_p\} $
the finite collection of all
$ P^\circ $
-minimal sets in
$ \mathcal E $
. To understand N-orbit closures, it is hence sufficient to restrict to
$\mathcal E_0$
.
The following is a refinement of Theorem 1.3.
Theorem 1.5. Let
$\Gamma <G$
be a Zariski dense discrete subgroup. For any
$[g]\in \mathcal E_0$
, the following are equivalent:
-
(1)
$gP\in \Lambda _h$
; -
(2)
$[g] N$
is dense in
$\mathcal E_0$
.
Remark 1.6. We may consider horospherical limit points outside the context of
$\Lambda $
. In this case, our proofs of Theorems 1.3 and 1.5 show that if
$gP\in \mathcal F$
is a horospherical limit point, then the closures of
$[g]MN$
and
$[g]N$
contain
$\mathcal E$
and
$\mathcal E_i $
for some
$ \mathcal E_i \in \mathfrak D_\Gamma $
, respectively.
For
$G=\operatorname {SO}^\circ ({n,1})$
,
$n\ge 2$
, Theorem 1.5 was proved in [Reference Maucourant and Schapira16]. When G has rank one and
$\Gamma <G$
is convex cocompact, every limit point is horospherical and Winter’s mixing theorem [Reference Winter20] implies the N-minimality of
$\mathcal E_0$
.
1.3 Directional horospherical limit points
We also consider the following seemingly much stronger notion.
Definition 1.7. For
$u\in \mathfrak a^+$
, a point
$ \xi \in \mathcal F$
is called
$ u $
-horospherical if there exists a sequence
$ \gamma _j \in \Gamma $
such that
$\sup _j \| \beta _\xi (o,\gamma _jo)- \mathbb R_+ u\|<\infty $
and
$ \beta _\xi (o,\gamma _jo)\to \infty $
as
$j\to ~\infty $
.
Denote by
$ \Lambda _h(u) $
the set of
$ u $
-horospherical limit points. Surprisingly, it turns out that every horospherical limit point is u-horospherical for all
$u\in \operatorname {int}\mathcal L$
.
Theorem 1.8. For all
$u\in \operatorname {int} \mathcal L$
, we have
$$ \begin{align*}\Lambda_h=\Lambda_h(u).\end{align*} $$
1.4 Existence of non-dense horospheres
A finitely generated subgroup
$\Gamma <G$
is called an Anosov subgroup (with respect to P) if there exists
$C>0$
such that for all
$\gamma \in \Gamma $
,
$\alpha (\mu (\gamma ))\ge C|\gamma | -C$
for all simple roots
$\alpha $
of
$(\mathfrak g, \mathfrak a^+)$
, where
$\mu (\gamma )\in \mathfrak a^+$
denotes the Cartan projection of
$\gamma $
and
$|\gamma |$
is the word length of
$\gamma $
with respect to a fixed finite generating set of
$\Gamma $
.
For Zariski dense Anosov subgroups of G, almost all
$NM$
-orbits are dense in
$\mathcal E$
and almost all N-orbits are dense in
$\mathcal E_0$
with respect to any Patterson–Sullivan measure on
$\Lambda $
[Reference Lee and Oh15, Reference Lee and Oh14]. In particular, the set of all horospherical limit points has full Patterson–Sullivan measures.
However, as Anosov subgroups are regarded as higher rank generalizations of convex cocompact subgroups, it is a natural question whether the minimality of the
$NM$
-action persists in the higher rank setting. It turns out that it is not the case. Our example is based on Thurston’s theorem [Reference Thurston18, Theorem 10.7] together with the following observation on the implication of the existence of a Jordan projection of an element of
$\Gamma $
lying in the boundary
$\partial \mathcal L$
of the limit cone.
Proposition 1.9. Let
$\Gamma <G$
be a Zariski dense discrete subgroup. For any loxodromic element
$\gamma \in \Gamma $
, we have
$$ \begin{align*} \unicode{x3bb}(\gamma)\in \operatorname{int} \mathcal L \quad \text{if and only if} \; \{y_{\gamma}, y_{\gamma^{-1}}\} \subset \Lambda_h ,\end{align*} $$
where
$y_\gamma $
and
$y_{\gamma ^{-1}}$
denote the attracting fixed points of
$\gamma $
and
$\gamma ^{-1}$
, respectively.
In particular, if
$\unicode{x3bb} (\Gamma )\cap \partial \mathcal L\ne \emptyset $
, then
$\Lambda \ne \Lambda _h$
and hence there exists a non-dense
$NM$
-orbit in
$\mathcal E$
.
Thurston’s work [Reference Thurston18] provides many examples of Anosov subgroups satisfying that
$\unicode{x3bb} (\Gamma )\cap \partial \mathcal L\ne \emptyset $
. To describe them, let
$\Sigma $
be a a torsion-free cocompact lattice of
$\operatorname {PSL}_2(\mathbb R)$
and let
$\pi \kern1.3pt{:}\kern1.3pt\Sigma \kern1.3pt{\to}\kern1.3pt \operatorname {PSL}_2(\mathbb R)$
be a discrete faithful representation. Let
${0\kern1.3pt{<}\kern1.3pt d_-(\pi ) \kern1.3pt{\le}\kern1.3pt d_+ (\pi ) \kern1.3pt{<}\kern1.3pt\infty }$
be the minimal and maximal geodesic stretching constants:
$$ \begin{align} d_+(\pi) =\sup_{\sigma\in \Sigma-\{e\} }\frac {\ell(\pi(\sigma))}{\ell(\sigma)} \quad \text{and} \quad d_-(\pi) =\inf_{\sigma\in \Sigma-\{e\}}\frac{\ell(\pi(\sigma))}{\ell(\sigma)}, \end{align} $$
where
$\ell (\sigma )$
denotes the length of the closed geodesic in the hyperbolic manifold
$\Sigma \backslash \mathbb H^2$
corresponding to
$\sigma $
and
$\ell (\pi (\sigma ))$
is defined similarly.
Consider the following self-joining subgroup:
$$ \begin{align*}\Gamma_\pi:=(\text{id}\times \pi)(\Sigma)=\{(\sigma, \pi(\sigma)):\sigma\in \Sigma\} <\operatorname{PSL}_2(\mathbb R) \times \operatorname{PSL}_2(\mathbb R).\end{align*} $$
It is easy to see that
$\Gamma $
is an Anosov subgroup of
$G=\operatorname {PSL}_2(\mathbb R) \times \operatorname {PSL}_2(\mathbb R)$
. Moreover, when
$\pi $
is not a conjugate by a Möbius tranformation,
$\Gamma _\pi $
is Zariski dense in G (cf. [Reference Kim and Oh12, Lemma 4.1]). Identifying
$\mathfrak a=\mathbb R^2$
, the Jordan projection
$\unicode{x3bb} (\gamma _\pi )$
of
$\gamma _\pi =(\sigma , \pi (\sigma ))\in \Gamma _\pi $
is given by
$(\ell (\sigma ), \ell (\pi (\sigma )))\in \mathbb R^2$
. Hence, the limit cone
$\mathcal L$
of
$\Gamma _\pi $
is given by
$$ \begin{align*}\mathcal L:=\{(v_1,v_2)\in \mathbb R_{\ge 0}^2: d_-(\pi) v_1\le v_2\le d_+(\pi) v_1\}. \end{align*} $$
Thurston [Reference Thurston18, Theorem 10.7] showed that
$d_+(\pi )$
is realized by a simple closed geodesic of
$\Sigma \backslash \mathbb H^2$
in most of the cases, which hence provides infinitely many examples of
$\Gamma _\pi $
which satisfy
$\unicode{x3bb} (\Gamma _\pi )\cap \partial \mathcal L\ne \emptyset $
. Therefore, Proposition 1.9 implies (in this case, we have
$NM=N)$
the following corollary.
Corollary 1.10. There are infinitely many non-conjuagte Zariski dense Anosov subgroups
$\Gamma _{\pi }<\operatorname {PSL}_2(\mathbb R) \times \operatorname {PSL}_2(\mathbb R)$
with non-dense
$NM$
-orbits in
$\mathcal E$
.
We close the introduction by the following question (cf. [Reference Kim13, Reference Quint17]).
Question 1.11. For a simple real algebraic group G with
$\operatorname {rank} G\ge 2$
, is every discrete subgroup
$\Gamma <G$
with
$\Lambda =\Lambda _h=\mathcal F$
necessarily a cocompact lattice in G?
2 Preliminaries
Let G be a connected, semisimple real algebraic group. We fix, once and for all, a Cartan involution
$\theta $
of the Lie algebra
$\mathfrak {g}$
of G, and decompose
$\mathfrak {g}$
as
$\mathfrak g=\mathfrak k\oplus \mathfrak {p}$
, where
$\mathfrak {k}$
and
$\mathfrak {p}$
are the
$+ 1$
and
$-1$
eigenspaces of
$\theta $
, respectively. We denote by K the maximal compact subgroup of G with Lie algebra
$\mathfrak {k}$
.
Choose a maximal abelian subalgebra
$\mathfrak a$
of
$\mathfrak p$
. Choosing a closed positive Weyl chamber
$\mathfrak a^+$
of
$\mathfrak a$
, let
$A:=\exp \mathfrak a$
and
$A^+=\exp \mathfrak a^+$
. The centralizer of A in K is denoted by M, and we set N to be the maximal contracting horospherical subgroup: for
$a\in \operatorname {int} A^+$
,
$$ \begin{align*}N =\{g\in G: a^{-n} g a^n\to e\text{ as}\ n\to +\infty\}.\end{align*} $$
We set
$P=MAN$
, which is the unique minimal parabolic subgroup of G, up to conjugation.
For
$u\in \mathfrak a$
, we write
$a_u=\exp u\in A$
. We denote by
$\|\cdot \|$
the norm on
$\mathfrak g$
induced by the Killing form. Consider the Riemannian symmetric space
$X:=G/K$
with the metric induced from the norm
$\|\cdot \|$
on
$\mathfrak g$
and
$o=K\in X$
.
Let
$\mathcal {F}=G/P$
denote the Furstenberg boundary. Since K acts transitively on
$ \mathcal {F} $
and
$K\cap P=M$
, we may identify
$\mathcal {F}=K/M$
. We denote by
$\mathcal F^{(2)}$
the unique open G-orbit in
$\mathcal {F} \times \mathcal {F}$
.
Denote by
$w_0\in K$
the unique element in the Weyl group such that
$\operatorname {Ad}_{w_0}\mathfrak a^+= -\mathfrak a^+$
; it is the longest Weyl element. We then have
$\check {P}:=w_0 P w_0^{-1}$
is an opposite parabolic subgroup of G, with
$ \check {N} $
its unipotent radical. The map
$\operatorname {i}=-\operatorname {Ad}_{w_0}: \mathfrak a^+\to \mathfrak a^+$
is called the opposition involution.
For
$g\in G$
, we consider the following visual maps:
$$ \begin{align*}g^+:=gP\in \mathcal F\quad\text{and}\quad g^-:=gw_0P\in \mathcal F.\end{align*} $$
Then
$\mathcal F^{(2)}=\{(g^+, g^-)\in \mathcal F\times \mathcal F: g\in G\}$
.
Any element
$ g \in G $
can be uniquely decomposed as the commuting product
$ g_h,g_e,g_u $
, where
$ g_h $
,
$ g_e $
, and
$ g_u $
are hyperbolic, elliptic, and unipotent elements, respectively. The Jordan projection of g is defined as the element
$\unicode{x3bb} (g) \in \mathfrak a^+ $
satisfying
$ g_h= \varphi \exp \unicode{x3bb} (g) \varphi ^{-1} $
for some
$\varphi \in G$
.
An element
$g\in G$
is called loxodromic if
$ \unicode{x3bb} (g) \in \operatorname {int} \mathfrak a^+ $
; in this case,
$g_u$
is necessarily trivial. For a loxodromic element
$g\in G$
, the point
$\varphi ^+ \in \mathcal {F}$
is called the attracting fixed point of g, which we denote by
$y_g$
. For any loxodromic element
$g\in G$
and
$\xi \in \mathcal F$
with
$(\xi , y_{g^{-1}})\in \mathcal F^{(2)}$
, we have
$\lim _{k\to \infty } g^k \xi = y_{g}$
and the convergence is uniform on compact subsets.
Note that for any loxodromic element
$g\in G$
,
$$ \begin{align*}\unicode{x3bb}(g^{-1})= \operatorname{i}\unicode{x3bb}(g).\end{align*} $$
Let
$\Gamma <G$
be a Zariski dense discrete subgroup of G. The limit cone
$\mathcal L=\mathcal L_\Gamma $
of
$\Gamma $
is the smallest closed cone of
$\mathfrak a^+$
containing
$\unicode{x3bb} (\Gamma )$
. It is a convex cone with non-empty interior [Reference Benoist1].
We will use the following simple lemma.
Lemma 2.1. For any
$ v \in \unicode{x3bb} (\Gamma ) $
and
$ \zeta \in \mathcal {F} $
, there exists a loxodromic element
$ \gamma \in \Gamma $
with
$ \unicode{x3bb} (\gamma )=v $
and a neighborhood
$ U $
of
$ \zeta $
in
$ \mathcal {F} $
such that
$ \{y_\gamma \}\times U $
is a relatively compact subset of
$\mathcal {F}^{(2)} $
and as
$k\to \infty $
,
$$ \begin{align*} \gamma^{-k} \zeta \to y_{\gamma^{-1}} \quad \text{uniformly on}\ U. \end{align*} $$
Proof. Let
$\zeta \in \mathcal F$
. Choose
$\gamma _1\in \Gamma $
such that
$\unicode{x3bb} (\gamma _1)=v$
. Since the set of all loxodromic elements of
$\Gamma $
is Zariski dense in G [Reference Benoist, Kobayashi, Kashiwara, Matsuki, Nishiyama and Oshima2] and
$\mathcal F^{(2)}$
is Zariski open in
$\mathcal F\times \mathcal F$
, there exists
$\gamma _2\in \Gamma $
such that
$(\zeta , \gamma _2 y_{\gamma _1})\in \mathcal F^{(2)}$
. Let
$\gamma =\gamma _2 \gamma _1\gamma _2^{-1}$
, so that
$y_{\gamma }=\gamma _2y_{\gamma _1}$
. It now suffices to take any neighborhood U of
$\zeta $
such that
$U\times \{ \gamma _2y_{\gamma _1}\}$
is a relatively compact subset of
$\mathcal F^{(2)}$
.
2.1 Convergence of a sequence in X to
$\mathcal F$
By the Cartan decomposition
$G=KA^+K$
, for
$g\in G$
, we may write
$$ \begin{align*}g=\kappa_1(g)\exp (\mu(g))\kappa_2(g)\in KA^+K,\end{align*} $$
where
$\mu (g)\in \mathfrak a^+$
, called the Cartan projection of g, is uniquely determined, and
$\kappa _1(g), \kappa _2(g)\in K$
. If
$\mu (g)\in \operatorname {int} \mathfrak a^+$
, then
$[\kappa _1(g)]\in K/M=\mathcal F$
is uniquely determined.
Let
$\Pi $
be the set of simple roots for
$(\mathfrak g, \mathfrak a)$
. For a sequence
$g_i\to G$
, we say
$g_i\to \infty $
regularly if
$\alpha (\mu (g_i))\to \infty $
for all
$\alpha \in \Pi $
. Note that if
$g_i\to \infty $
regularly, then for all sufficiently large i,
$\mu (g_i)\in \operatorname {int}\mathfrak a^+$
and hence
$[\kappa _1(g_i)]$
is well defined.
Definition 2.2. A sequence
$p_i \in X$
is said to converge to
$\xi \in \mathcal F$
if there exists
$g_i\to \infty $
regularly in G with
$p_i=g_i(o)$
and
$\lim _{i\to \infty }[\kappa _1(g_i)]=\xi $
.
2.2
$P^\circ $
-minimal subsets
We denote by
$\Lambda \subset \mathcal F$
the limit set of
$\Gamma $
, which is defined as
$$ \begin{align} \Lambda=\{\lim \gamma_i(o): \gamma_i\in \Gamma\}.\end{align} $$
For a non-Zariski dense subgroup,
$\Lambda $
may be an empty set. For
$\Gamma <G$
Zariski dense, this is the unique
$\Gamma $
-minimal subset of
$\mathcal F$
[Reference Benoist1, Reference Lee and Oh15].
It follows that the following set
$\mathcal E$
is the unique P-minimal subset of
$\Gamma \backslash G$
:
$$ \begin{align*}\mathcal E=\{[g]\in \Gamma\backslash G: g^+\in \Lambda\}.\end{align*} $$
Let
$P^\circ $
denote the identity component of P. Then
$\mathcal E$
is a disjoint union of at most
$[P: P^{\circ }]$
-number of
$P^\circ $
-minimal subsets. We fix one
$P^{\circ }$
-minimal subset
$\mathcal E_0$
once and for all. Note that any
$P^{\circ }$
-minimal subset is then of the form
$\mathcal E_0 m$
for some
$m\in M$
. We set
$$ \begin{align} \Omega:=\{[g]\in \Gamma\backslash G: g^+, g^-\in \Lambda\}\quad\text{and}\quad \Omega_0:=\Omega\cap \mathcal E_0 .\end{align} $$
2.3 Busemann map
The Iwasawa cocycle
$\sigma : G\times \mathcal {F} \to \mathfrak a$
is defined as follows: for
$(g, \xi )\in G\times \mathcal {F} $
with
$\xi =[k]$
for
$k\in K$
,
$\exp \sigma (g,\xi )$
is the A-component of
$g k$
in the
$KAN$
decomposition, that is,
$$ \begin{align*}gk\in K \exp (\sigma(g, \xi)) N.\end{align*} $$
The
$\mathfrak a$
-valued Busemann function
$\beta : \mathcal {F} \times X \times X \to \mathfrak a $
is defined as follows: for
$\xi \in \mathcal {F} $
and
$g, h\in G$
,
$$ \begin{align*}\beta_\xi ( ho, go):=\sigma (h^{-1}, \xi)-\sigma(g^{-1}, \xi).\end{align*} $$
We note that for any
$g\in G$
,
$\xi \in \mathcal F$
, and
$x,y, z\in X$
,
$$ \begin{align} \beta_\xi(x,y)=\beta_{g\xi}(gx, gy)\quad\text{and}\quad \beta_\xi(x, y)=\beta_\xi(x,z)+\beta_\xi(z, y).\end{align} $$
In particular,
$ \beta _\xi (o,go) \in \mathfrak a $
is defined by
$$ \begin{align} g^{-1}k_\xi \in K \exp(-\beta_\xi(o, go)) N, \end{align} $$
and hence
$\beta _P (o, a_u o)=u$
for any
$u\in \mathfrak a$
. For
$h, g\in G$
, we set
$\beta _\xi (h, g):=\beta _{\xi }(ho, go)$
.
2.4 Shadows
For
$ q\in X$
and
$r>0$
, we set
$B(q,r)=\{x\in X: d(x, q)\le r\}$
. For
$p=g(o)\in X$
, the shadow of the ball
$B(q,r)$
viewed from p is defined as
$$ \begin{align*}O_r(p,q):=\{(gk)^+\in \mathcal F: k\in K,\; gk\operatorname{int} A^+o\cap B(q,r)\ne \emptyset\}.\end{align*} $$
Similarly, for
$\xi \in \mathcal F$
, the shadow of the ball
$B(q,r)$
as viewed from
$\xi $
is
$$ \begin{align*}O_r(\xi,q):=\{h^+\in \mathcal F: h\in G\text{ satisfies } h^-=\xi,\, ho\in B(q,r) \}. \end{align*} $$
Lemma 2.3. [Reference Lee and Oh15, Lemmas 5.6 and 5.7]
-
(1) There exists
$\kappa>0$
such that for any
$g\in G$
and
$r>0$
,
$$ \begin{align*}\sup_{\xi \in O_r(g(o),o)}\|\beta_{\xi}(g(o),o)-\mu(g^{-1})\|\le \kappa r .\end{align*} $$
-
(2) If a sequence
$p_i\in X$
converges to
$\xi \in \mathcal F$
, then for any
$0<\varepsilon <r$
, we have for all sufficiently large i.
$$ \begin{align*}O_{r-\varepsilon} (p_i, o)\subset O_r(\xi, o)\subset O_{r+\varepsilon} (p_i, o)\end{align*} $$
3 Horospherical limit points
Let
$\Gamma <G$
be a Zariski dense discrete subgroup. A
$\Gamma $
-tight horoball based at
$\xi \in \mathcal F$
is a subset of the form
$\mathcal H_{\xi }=g N (\exp \mathcal C) (o)$
, where
$g\in G$
is such that
$\xi =gP$
and
$\mathcal C$
is a closed cone contained in
$\operatorname {int} \mathcal L\cup \{0\}$
. For
$T>0$
, we write
$\mathcal H_{\xi }(T)= gN (\exp (\mathcal C-\mathcal C_T))o$
. We recall the definition from the introduction.
Definition 3.1. We say that
$\xi \in \mathcal F$
is a horospherical limit point of
$\Gamma $
if there exists a
$\Gamma $
-tight horoball
$\mathcal H_\xi $
based at
$\xi $
such that
$\mathcal H_\xi (T)\cap \Gamma (o)\ne \emptyset $
for all
$T>1$
.
In this section, we provide a mostly self-contained proof of the following theorem.
Theorem 3.2. Let
$[g]\in \mathcal E$
. The following are equivalent:
-
(1)
$g^+=gP\in \Lambda $
is a horospherical limit point; -
(2)
$[g]NM$
is dense in
$\mathcal E$
.
The main external ingredient in our proof is the density of the group generated by the Jordan projection
$ \unicode{x3bb} (\Gamma )$
, due to Benoist [Reference Benoist, Kobayashi, Kashiwara, Matsuki, Nishiyama and Oshima2], that is,
$$ \begin{align*} \mathfrak a = \overline{\langle \unicode{x3bb}(\Gamma) \rangle} \end{align*} $$
for every Zariski dense discrete subgroup
$ \Gamma < G $
. In fact, for every cone
$ \mathcal C \subset \mathcal L $
with non-empty interior, there exists a Zariski dense subgroup
$ \Gamma ' < \Gamma $
with
$ \mathcal L_{\Gamma '} \subset \mathcal C $
(see [Reference Benoist1]); therefore, we have
$$ \begin{align*} \mathfrak a = \overline{\langle \unicode{x3bb}(\Gamma) \cap \operatorname{int} \mathcal L \rangle}. \end{align*} $$
It is convenient to use a characterization of horospherical limit points in terms of the Busemann function.
Lemma 3.3. For
$ \xi \in \Lambda $
, we have
$\xi \in \Lambda _h$
if and only if there exists a closed cone
${\mathcal C \subset {\operatorname {int} \mathcal L} \cup \{0\}}$
and a sequence
$ \gamma _j \in \Gamma $
satisfying
$$ \begin{align} \beta_\xi(o,\gamma_jo) \to \infty \quad \text{and} \quad \beta_\xi(o,\gamma_jo) \in \mathcal C \quad\text{for all large } j\ge 1. \end{align} $$
Proof. Let
$\xi =gP\in \Lambda _h$
be as defined in Definition 3.1. Then there exists
${\gamma _j= gp n_j a_{u_j}k_j\in \Gamma }$
for some
$p\in P$
,
$n_j\in N$
,
$k_j\in K$
, and
$u_j\to \infty $
in some closed cone
$\mathcal C$
contained in
$\operatorname {int}\mathcal L\cup \{0\}$
. Fix some closed cone
$\mathcal C'\subset \operatorname {int} \mathcal L\cup \{0\}$
whose interior contains
$\mathcal C$
. Note that
$$ \begin{align*} \beta_{\xi}(o, \gamma_j o) &= \beta_{gP}(e, g)+ \beta_{gP}(g, gpn_j a_{u_j})\\ &= \beta_P(g^{-1}, e)+ \beta_P(e,p)+ \beta_P(e, n_j)+\beta_P(e, a_{u_j})\\ & =\beta_{P}(g^{-1},p)+u_j. \end{align*} $$
Therefore, the sequence
$\beta _{\xi }(o, \gamma _j)-u_j$
is uniformly bounded. Since
$u_j\in \mathcal C$
,
$\beta _{\xi }(o,\gamma _j o)\in \mathcal C'$
for all large j. Therefore, equation (3.1) holds. For the other direction, let
$\gamma _j$
and
$\mathcal C$
satisfy equation (3.1) for
$\xi =gP$
for
$g\in G$
. Since
$G=gNAK$
, we may write
$\gamma _j= gn_ja_{u_j} k_j $
for some
$n_j\in N, u_j\in \mathfrak a$
and
$k_j\in K$
. By a similar computation as above, the sequence
$\beta _\xi (o, \gamma _j o)-u_j$
is uniformly bounded. It follows that
$u_j\in \mathcal C'$
for all large j and
$u_j\to \infty $
. Therefore, for any
$T>1$
, there exists
$j>1$
such that
$\gamma _j (o)\in g N \exp (\mathcal C'-\mathcal C^{\prime }_T)(o)$
. This proves
$\xi \in \Lambda _h$
.
We note that the condition in equation (3.1) is independent of the choice of basepoint
$ o $
. Indeed, for any
$g\in G$
and
$\xi \in \mathcal F$
and for all
$ \gamma \in \Gamma $
, we have
$$ \begin{align*} \beta_\xi(o, \gamma o) = \beta_\xi(o, go)+\beta_\xi (go, \gamma go)+\beta_\xi (\gamma go, \gamma o) ,\end{align*} $$
and hence
$$ \begin{align*} \|\beta_\xi(o, \gamma o)-\beta_\xi (go, \gamma go)\|&=\|\beta_\xi(o, go)+\beta_\xi (\gamma go, \gamma o)\|\\ &=\|\beta_\xi(o, go)-\beta_{\gamma^{-1}\xi} (o,go)\| \\ & \le 2 \cdot \max_{\eta \in \mathcal{F}} \|\beta_\eta (o, go)\|. \end{align*} $$
Since this bound is independent of
$\gamma \in \Gamma $
, the condition in equation (3.1) implies that for any
$p=go \in X$
,
$$ \begin{align} \beta_\xi(p,\gamma_j p) \to \infty \quad \text{and} \quad \beta_\xi(p,\gamma_jp) \in \mathcal C \quad\text{for all large }j. \end{align} $$
Let us now consider the following seemingly stronger condition for a limit point being horospherical.
Definition 3.4. For
$u\in \mathfrak a^+$
, a point
$ \xi \in \mathcal F$
is called a
$ u $
-horospherical limit point if for some
$p\in X$
(and hence for any
$p\in X$
), there exists a constant
$ R>0 $
and a sequence
$ \gamma _j \in \Gamma $
satisfying
$$ \begin{align*} \beta_\xi(p,\gamma_jp)\to \infty \quad \text{and}\quad \|\beta_\xi(p,\gamma_jp)-\mathbb{R}_+u\| < R \quad \text{for all }j. \end{align*} $$
We denote the set of
$ u $
-horospherical limit points by
$ \Lambda _h(u) $
.
By
$ G $
-invariance of the Busemann map, the set of horospherical (respectively u-horospherical) limit points is
$\Gamma $
-invariant. Therefore, for
$x=[g]\in \Gamma \backslash G$
, we may say
$x^+:=\Gamma gP$
horospherical (respectively u-horospherical) if
$g^+$
is.
For
$u\in \mathfrak a$
, we call
$x\in \Gamma \backslash G$
a u-periodic point if
$x a_u = xm_0$
for some
$m_0\in M$
; note that
$xa_{\mathbb R u} M_0$
is then compact. Note that for
$u\in \operatorname {int}\mathfrak a^+$
, the existence of a u-periodic point is equivalent to the condition that
$u\in \unicode{x3bb} (\Gamma )$
.
Lemma 3.5. Let
$u\in \mathfrak a^+$
. If
$x\in \Gamma \backslash G$
is u-periodic, then
$x^+\in \mathcal F$
is a u-horospherical limit point.
Proof. Since x is u-periodic, there exist
$g\in G$
with
$x=[g]$
and
$\gamma \in \Gamma $
such that
$\gamma =g a_u m g^{-1}$
for some
$m\in M$
, and
$y_\gamma =g^+ \in \Lambda $
. Moreover, for any
$k\ge 1$
,
$$ \begin{align*}\beta_{gP} (go, \gamma^k go)=\beta_{P}(o, a^k_u o)=k u.\end{align*} $$
This implies
$gP$
is u-horospherical.
Proposition 3.6. Let
$x\in \Gamma \backslash G$
. If
$ x^+$
is u-horospherical for some
$u\in \unicode{x3bb} (\Gamma )$
, then the closure
$\overline {xN}$
contains a
$ u $
-periodic point.
Proof. Choose
$g\in G$
so that
$x=[g]$
. We may assume without loss of generality that
$g=k\in K$
, since
$kan N= k N a$
, and a translate of a u-periodic point by an element of A is again a u-periodic point. Since
$u\in \unicode{x3bb} (\Gamma )$
, there exists a u-periodic point, say,
$x_0\in \Gamma \backslash G$
. It suffices to show that
$$ \begin{align} \overline{[k]N}\cap x_0 AM\ne \emptyset\end{align} $$
as every point in
$x_0AM$
is u-periodic.
Since
$k^+$
is
$ u $
-horospherical and using equation (2.4), there exists
$ R>0 $
and sequences
$ \gamma _j \in \Gamma $
,
$ u_j \to \infty $
in
$ \mathfrak a^+ $
and
$ k_j \in K $
and
$ n_j \in N $
satisfying
$\gamma _j^{-1} k = k_j a_{-u_j} n_j $
or
$$ \begin{align} k_j = \gamma_j^{-1} k n_j^{-1} a_{u_j}, \end{align} $$
with
$ \|\mathbb {R}_+u - u_j \| < R $
for all
$ j $
. Let
$ \ell _j \to \infty $
be a sequence of integers satisfying
$$ \begin{align} \|\ell_ju - u_j \| < R+\|u\| \;\quad\text{for all} j \ge 1. \end{align} $$
By passing to a subsequence, we may assume without loss of generality that
$ \gamma _j^{-1}kP$
converges to some
$\xi _0 \in \mathcal {F} $
. Since
$\check {N}P$
is Zariski open and
$ \Gamma $
is Zariski dense, we may choose
$g_0\in G$
such that
$x_0=[g_0]$
and
$g_0^{-1}\xi _0\in \check {N}P$
. Let
$h_0\in \check {N}$
be such that
$ \xi _0=g_0 h_0 P $
. Since
$ g_0\check {N}P $
is open and
$\gamma _j^{-1}kP\to g_0h_0P$
, we may assume that for all j, there exists
$ h_j \in \check {N} $
satisfying
$ g_0 h_j P = \gamma _j^{-1}k P = k_j P $
with
$ h_j \to h_0 $
. Let
$p_j= a_{v_j}m_j \tilde n _j \in P=AMN$
be such that
$g_0 h_j p_j= k_j$
; since
$h_j\to h_0$
and the product map
$\check {N}\times P\to \check {N}P$
is a diffeomorphism, the sequence
$p_j$
, as well as
${v_j}\in \mathfrak a$
, are bounded.
Therefore, by equation (3.4), we get for all j,
$$ \begin{align*} g_0 &= k_j p_j^{-1}h_j^{-1} \\& =\gamma_j^{-1} kn^{-1}_j a_{u_j} ( \tilde n^{-1}_j m_j^{-1} a_{-v_j}) h_j^{-1} \\ &= \gamma_j^{-1} k n^{-1}_j (a_{u_j} \tilde n^{-1}_j a_{-u_j}) a_{u_j} m_j^{-1} a_{-v_j}h_j^{-1} \\ &= \gamma_j^{-1} k n^{-1}_j (a_{u_j} \tilde n^{-1}_j a_{-u_j}) m_j^{-1} (a_{u_j-v_j}h_j^{-1} a_{-u_j+v_j}) a_{u_j-v_j}. \end{align*} $$
Since
$h_j^{-1} \in \check {N}$
and
$ v_j \in \mathfrak a $
are uniformly bounded and since
$u_j\to \infty $
within a bounded neighborhood of the ray
$\mathbb R_+ u\in \operatorname {int}\mathfrak a^+$
, we have
$$ \begin{align*} {\tilde h}_j = a_{u_j-v_j}h_j^{-1} a_{-u_j+v_j} \to e \quad \text{in }\check{N}. \end{align*} $$
By setting
$ n^{\prime }_j = n^{-1}_j (a_{u_j} \tilde n ^{-1}_j a_{-u_j}) \in N $
, we may now write
$$ \begin{align*} g_0 = \gamma_j^{-1} k n^{\prime}_j m_j^{-1} {\tilde h}_j a_{u_j-v_j}. \end{align*} $$
Since
$x_0$
is u-periodic, there exists
$ \gamma _0 \in \Gamma $
such that
$ \gamma _0 = g_0 a_u m_0 g_0^{-1} $
for some
$ m_0 \in M $
. Hence, for all
$j\ge 1$
,
$$ \begin{align*} \gamma_0^{-\ell_j} &= g_0 a_{-\ell_j u} m_0^{-\ell_j} g_0^{-1} = (\gamma_j^{-1} kn^{\prime}_j m_j^{-1} {\tilde h}_j a_{u_j-v_j} ) ( a_{-\ell_j u} m_0^{-\ell_j }) g_0^{-1}. \end{align*} $$
In other words,
$$ \begin{align*} \gamma_j^{-1} k n^{\prime}_j = \gamma_0^{-\ell_j}g_0 m_0^{\ell_j} a_{-u_j+\ell_j u+v_j} {\tilde h}_j^{-1}m_j. \end{align*} $$
Since the sequence
$ -u_j+\ell _j u+ v_j \in \mathfrak a $
is uniformly bounded by equation (3.5) and
$ {\tilde h}_j \to e $
in
$ \check {N} $
, we conclude that the sequence
$ \Gamma k n^{\prime }_j $
has an accumulation point in
$ \Gamma g_0 AM $
. This proves equation (3.3).
It turns out that a horospherical limit point is also
$ u $
-horospherical for any
$u\in \operatorname {int} \mathcal L$
.
Proposition 3.7. For each
$ u \in {\operatorname {int} \mathcal L} $
, we have
$ \Lambda _h=\Lambda _h(u) $
.
Proof. Let
$ \xi \in \Lambda _h $
. By definition, there is a sequence
$ \gamma _j \in \Gamma $
satisfying
$v_j:=\beta _\xi (e,\gamma _j)\to \infty $
with the sequence
$\|v_j\|^{-1}v_j$
converging to some point
$v_0\in \operatorname {int} \mathcal L$
. By passing to a subsequence, we may assume that
$\gamma _j^{-1} \xi $
converges to some
$\xi _0\in \mathcal F$
.
Let
$u\in \operatorname {int}\mathcal L$
. We claim that
$\xi \in \Lambda _h(u)$
. We first consider the case
$u\not \in \mathbb R_+ v_0$
. Let
$r:=\operatorname {rank} G-1\ge 0$
. Since
$\bigcup _{\gamma \in \Gamma }\mathbb R_+\unicode{x3bb} (\gamma )$
is dense in
$\mathcal L$
, there exist
$w_1, \ldots , w_r\in \unicode{x3bb} (\Gamma )$
such that
$v_0$
belongs to the interior of the convex cone spanned by
$u, w_1, \ldots , w_r$
, so that
$$ \begin{align*}v_0= c_0 u+ \sum_{\ell=1}^r c_\ell w_\ell \end{align*} $$
for some positive constants
$c_0,\ldots , c_\ell $
.
Since
$\|v_j\|^{-1}v_j \to v_0$
, we may assume, by passing to a subsequence, that for each
$j\ge 1$
, we have
$$ \begin{align} \|v_j\|^{-1} v_j = c_{0,j} u +\sum_{\ell =1}^r c_{\ell,j} w_\ell \end{align} $$
for some positive
$c_{\ell ,j}$
,
$\ell =0, \ldots , r$
. Note that for each
$0\le \ell \le r$
,
$c_{\ell ,j}\to c_\ell $
as
$j\to \infty $
.
By Lemma 2.1, we can find a loxodromic element
$g_1\in \Gamma $
and a neighborhood
$U_1$
of
$\xi _0$
such that
$\unicode{x3bb} (g_1^{-1})=w_1$
,
$\{y_{g_1}\}\times U_1\subset \mathcal F^{(2)}$
and
$g_1^{-k}U_1\to y_{g_1^{-1}}$
uniformly. Applying Lemma 2.1 once more, we can find
$ g_2 \in \Gamma $
satisfying
$ \unicode{x3bb} (g_2^{-1}) = w_2 $
and a neighborhood
$ U_2 \subset \mathcal {F} $
of
$ y_{g_1^{-1}} $
satisfying
$ \{y_{g_2} \}\times U_2 \subset \mathcal {F}^{(2)} $
and that
$ g_2^{-k}U_2 \to y_{g_2^{-1}} $
uniformly.
Continuing inductively, we get elements
$ g_1,\ldots ,g_r \in \Gamma $
and open sets
$ U_1,\ldots ,U_r \subset \mathcal {F} $
satisfying that for all
$ \ell = 1,\ldots ,r $
:
-
(1)
$ {w}_\ell =\unicode{x3bb} (g_\ell ^{-1}) $
; -
(2)
$ y_{g_{\ell -1}^{-1}} \in U_{\ell } $
; -
(3)
$ g_\ell ^{-k}U_\ell \to y_{g_\ell ^{-1}} $
uniformly; and -
(4)
$ \{ y_{g_\ell } \}\times U_\ell $
is a relatively compact subset of
$ \mathcal {F}^{(2)} $
.
We set
$\xi _\ell :=y_{g_\ell ^{-1}}$
for each
$1\le \ell \le r$
; so
$U_{\ell }$
is a neighborhood of
$\xi _{\ell -1}$
for each
$1\le \ell \le r$
.
Since
$\mathcal Q_{\eta _0}:=\{\eta \in \mathcal F: (\eta _0, \eta )\in \mathcal F^{(2)}\}=\bigcup _{R>0} O_R (\eta _0, o)$
for any
$\eta _0\in \mathcal F$
and
$U_\ell \subset \mathcal Q_{y_{g_\ell }}$
is a relatively compact subset of
$\mathcal {F}^{(2)} $
, there exists
$ R_\ell>0 $
such that
$ U_\ell \subset O_{R_\ell }(y_{g_\ell },o) $
. Since
$g_\ell ^ko $
converges to
$y_{g_\ell }$
as
$k\to +\infty $
, by Lemma 2.3(2),
$$ \begin{align} O_{R_\ell}(y_{g_\ell}o,o) \subset O_{R_\ell+1}(g_\ell^k o,o) \end{align} $$
for all sufficiently large
$k>1$
.
For each
$1\le \ell \le r$
and
$j\ge 1$
, let
$k_{\ell ,j}$
be the largest integer smaller than
$c_{\ell , j}\|v_j\|$
. As
$\|v_j\|\to \infty $
and
$c_{\ell ,j}\to c_\ell $
, we have
$k_{\ell , j}\to \infty $
as
$j\to \infty $
. By the uniform contraction
$ g_\ell ^{-k}U_i \to \xi _\ell $
, there exists
$j_0>1$
such that for all
$j\ge j_0$
,
$$ \begin{align} \gamma_{j}^{-1}\xi \in U_1, \quad g_\ell^{-k_{\ell,j}}U_\ell \subseteq U_{\ell+1} ,\quad \text{and} \quad U_\ell \subset O_{R_\ell+1}(g_\ell^{k_\ell, j}o,o) \end{align} $$
for all
$ \ell =1,\ldots ,r $
.
For each
$j\ge j_0$
, we now set
$$ \begin{align*} \tilde{\gamma}_j:=\gamma_j g_1^{k_{1,j}} g_2^{k_{2,j}} \ldots g_r^{k_{r,j}} \in \Gamma. \end{align*} $$
We claim that
$\beta _\xi (e,\tilde {\gamma }_j)\to \infty $
as
$j\to \infty $
and that
$$ \begin{align} \sup_{j\ge j_0} \|\beta_\xi(e,\tilde{\gamma}_j)-\mathbb R_+ u \|<\infty ;\end{align} $$
this proves that
$\xi $
is u-horospherical.
Fix
$ j\ge j_0 $
and for each
$1\le \ell \le r$
, let
$ k_\ell :=k_{\ell ,j} $
,
$b_\ell :=c_{\ell ,j}\|v_j\|$
, and set
$$ \begin{align*} h_\ell = g_1^{k_1} g_2^{k_2} \ldots g_\ell^{k_\ell}, \end{align*} $$
and
$ g_0=e $
. The cocycle property of the Busemann function gives that
$$ \begin{align} \beta_\xi(e,\tilde{\gamma}_j) =\beta_\xi(e,\gamma_j) -\sum_{\ell=1}^{r} \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e). \end{align} $$
By equation (3.8),
$\gamma _j^{-1}\xi \in U_1$
and for each
$1\le \ell \le r$
,
$$ \begin{align*}h_{\ell -1}^{-1}\gamma_j^{-1} \xi \in g_\ell^{-k_\ell}\ldots g_1^{-k_1}U_1\subset U_{\ell+1}\subset O_{R_\ell +1}(g_\ell^{k_\ell}o, o ).\end{align*} $$
Hence, by Lemma 2.3(1), there exists
$\kappa \ge 1$
such that for each
$1\le \ell \le r$
,
$$ \begin{align*} \| \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e) -\mu(g^{-k_\ell}_\ell)\|\le \kappa (R_{\ell}+1).\end{align*} $$
Note that for some
$C_\ell>0$
,
$\|\mu (g_\ell ^{-k}) -k \unicode{x3bb} (g_\ell ^{-1})\|\le C_\ell $
for all
$k\ge 1$
. Since
$\unicode{x3bb} (g_\ell ^{-1})=w_\ell $
, we get
$$ \begin{align*} \| \beta_{h_{\ell-1}^{-1}\gamma_j^{-1}\xi}(g^{k_\ell}_\ell, e) -k_\ell w_\ell \|\le \kappa (R_{\ell}+1)+ C_\ell .\end{align*} $$
Therefore, by equation (3.10), we obtain
$$ \begin{align*}\bigg\|\beta_\xi(e, \tilde \gamma_j) -\bigg(v_j -\sum_{\ell=1}^r k_\ell w_\ell \bigg) \bigg\| \le \kappa \sum_{\ell=1}^r (R_\ell + C_\ell+1) .\end{align*} $$
By equation (3.6), we have
$$ \begin{align*}c_{0,j} \|v_j\| u = v_j -\sum_{\ell=1}^{r} b_\ell w_\ell .\end{align*} $$
Since
$|b_\ell -k_\ell |\le 1$
and
$c_{0,j}>0$
, we deduce that for all
$j\ge j_0$
,
$$ \begin{align*} &\|\beta_\xi(e, \tilde \gamma_j)-\mathbb R_+ u\| \le \| \beta_\xi(e, \tilde \gamma_j)- c_{0,j} \|v_j\|\cdot u \| \\ &\quad\le \bigg\| \beta_\xi(e, \tilde \gamma_j)- \bigg(v_j -\sum_{\ell=1}^r k_\ell w_\ell \bigg) \bigg\| + \sum_{\ell=1}^r \| k_\ell w_\ell -b_\ell w_\ell \| \\&\quad\le \kappa \sum_{\ell=1}^r (R_\ell + C_\ell +\|w_\ell\| +1). \end{align*} $$
This proves equation (3.9) and, consequently,
$\xi $
is u-horospherical for any
$u\notin \mathbb R_+ v_0.$
To show that
$\xi $
is
$v_0$
-horospherical, fix any
$u\notin \mathbb R_+v_0$
and
$\tilde \gamma _j\in \Gamma $
be a sequence as in equation (3.9) associated to u. If we set
$\tilde v_j=\beta _\xi (e, \tilde \gamma _j)$
, then
$\|\tilde v_j\|^{-1} \tilde v_j$
converges to a unit vector in
$\operatorname {int} \mathcal L$
proportional to u. Therefore, by repeating the same argument only now switching the roles of
$v_0$
and u, we prove that
$\xi $
is
$v_0$
-horospherical as well. This completes the proof.
We may now prove Theorem 3.2.
Proof of Theorem 3.2
Let
$g\in G$
be such that
$\xi =g^+\in \Lambda $
is a horospherical limit point. Set
$Y:=\overline {[g]NM}$
. We claim that
$Y=\mathcal E$
. By Benoist [Reference Benoist1], the group generated by
$\unicode{x3bb} (\Gamma )\cap \operatorname {int} \mathcal L$
is dense in
$\mathfrak a$
. Hence, for every
$ \varepsilon> 0 $
, there exist loxodromic elements
$ \gamma _1,\ldots ,\gamma _{q} \in \Gamma $
such that
$$ \begin{align*} \unicode{x3bb}(\gamma_1),\ldots,\unicode{x3bb}(\gamma_{q}) \in \mathrm{Int}\mathcal L \end{align*} $$
and the group
$\mathbb Z\unicode{x3bb} (\gamma _1)+\cdots +\mathbb Z \unicode{x3bb} (\gamma _{q})$
is an
$\varepsilon $
-net in
$\mathfrak a$
, that is, its
$\varepsilon $
-neighborhood covers all
$\mathfrak a$
. Denote
$u_i = \unicode{x3bb} (\gamma _i) $
for
$ i=1,\ldots ,q $
. By Proposition 3.7, the point
$ \xi $
is
$u_1 $
-horospherical. By Proposition 3.6, there exists a
$ u_1 $
-periodic point
$x_1 \in \mathcal E $
contained in
$ Y $
, set
$$ \begin{align*} Y_1: = \overline{x_1 NM} \subset Y. \end{align*} $$
By Lemma 3.5,
$x_1^+$
is
$u_1$
-horospherical; in particular, it is a horospherical limit point. Therefore, we can inductively find a
$u_i$
-periodic point
$ x_i $
in
$ Y_{i-1}=\overline {x_{i-1} NM} $
for each
$2\le i\le q$
. By periodicity,
$x_i (\exp {u_i}) M=x_i M$
, and hence
$Y_i \exp {\mathbb Z u_i} = Y_i$
for each
$1\le i\le q$
. Therefore, we obtain
$$ \begin{align*}Y\supset Y_1\, {\exp {\mathbb Z u_1}}\supset Y_2\, {\exp ({\mathbb Z u_1+\mathbb Z u_2})}\supset \cdots \supset Y_q\, {\exp {\bigg(\sum_{i=1}^q \mathbb Z u_i}\bigg)}.\end{align*} $$
Recalling the dependence of
$Y_q$
and
$\sum _{i=1}^q \mathbb Z u_i$
on
$\varepsilon $
, set
$$ \begin{align*}Z_\varepsilon:= Y_q MN \exp \bigg(\sum_{i=1}^q \mathbb Z u_i\bigg) \subset Y.\end{align*} $$
Since
$MN \exp (\sum _{i=1}^q \mathbb Z u_i)$
is an
$\varepsilon $
-net of P and
$\mathcal E$
is P-minimal,
$Z_\varepsilon $
is a
$2\varepsilon $
-net of
$\mathcal E$
for all
$\varepsilon>0$
. Since Y contains a
$2\varepsilon $
-net of
$\mathcal E$
for all
$\varepsilon>0$
and Y is closed, it follows that
$Y=\mathcal E$
.
For the other direction, suppose that
$[g]NM$
is dense in
$\mathcal E$
for
$g\in G$
. Choose any
$u\in \operatorname {int} \mathcal L$
and a closed cone
$\mathcal C \subset \operatorname {int} \mathcal L\cup \{0\}$
which contains u. Then
$\mathcal H_\xi =gN(\exp \mathcal C )(o)$
is a
$\Gamma $
-tight horoball. Let
$t>1$
. Since
$ga_{-2t u}\in \mathcal E$
, there exist
$\gamma _i\in \Gamma $
,
$n_i\in N$
,
$m_i\in M$
, and
$q_i\to e$
in G such that for all
$i\ge 1$
,
$\gamma _i g n_i m_i q_i= g a_{-2t u}$
. Since
$d(\gamma _i^{-1}g, gn_i m_ia_{2t u})\le d(q_i a_{2t u},a_{2t u})\to 0 $
as
$i\to \infty $
, it follows that for all sufficiently large
$i\ge 1$
,
$\gamma _i^{-1} go \in \mathcal H_\xi (t)$
. Hence,
$g^+$
is a horospherical limit point by Definition 3.1.
4 Topological mixing and directional limit points
There is a close connection between denseness of
$ N $
-orbits and the topological mixing of one-parameter diagonal flows with direction in
$ \operatorname {int} \mathcal L $
. This connection allows us to make use of recent topological mixing results by Chow and Sarkar [Reference Chow and Sarkar3]: recall the notation
$\Omega _0$
from equation (2.2).
Theorem 4.1. [Reference Chow and Sarkar3]
For any
$u\in \operatorname {int} \mathcal L$
,
$\{a_{tu} : t\in \mathbb R\}$
is topologically mixing on
$\Omega _0$
, that is, for any open subsets
$\mathcal O_1, \mathcal O_2$
of
$\Gamma \backslash G$
intersecting
$\Omega _0$
,
$$ \begin{align*}\mathcal O_1\exp tu \cap \mathcal O_2\ne \emptyset \quad\text{ for all large}\ |t|\gg 1.\end{align*} $$
The above theorem was predated by a result of Dang [Reference Dang6] in the case where
$ M $
is abelian.
4.1
N-orbits based at directional limit points along
$\operatorname {int} \mathcal L$
Definition 4.2. For
$u\in \operatorname {int} \mathfrak a^+$
, denote by
$\Lambda _u$
the set of all u-directional limit points, that is,
$\xi \in \Lambda _u$
if and only if
$\limsup _{t\to +\infty } \Gamma g \exp (tu)\ne \emptyset $
for some (and hence any)
$g\in G$
with
$gP=\xi $
.
It is easy to see that
$\Lambda _u\subset \Lambda $
for
$u\in \operatorname {int} \mathfrak a^+$
.
Proposition 4.3. If
$[g]\in \mathcal E_0$
satisfies
$g^+\in \Lambda _u$
for some
$u\in \operatorname {int}\mathcal L$
, then
$$ \begin{align*}\overline{[g]N}=\mathcal E_0.\end{align*} $$
Proof. Since
$\Omega _0 N=\mathcal E_0$
, we may assume without loss of generality that
$x=[g]\in \Omega _0$
. There exist
$\gamma _i\in \Gamma $
and
$t_i\to +\infty $
such that
$\gamma _i g a_{t_iu} $
converges to some
$h\in G$
. In particular,
$x \exp (t_i u)\to [h]$
. Since
$x a_{t_iu}\in \Omega _0$
and
$\Omega _0$
is
$ A $
-invariant and closed, we have
$[h]\in \Omega _0$
. We write
$\gamma _i g a_{t_iu}=hq_i$
, where
$q_i\to e$
in G. Therefore,
$xN=[h] q_i N a_{-t_iu}$
for all
$i\ge 1$
. Let
$\mathcal O\subset \Gamma \backslash G$
be any open subset intersecting
$\Omega _0$
. It suffices to show that
$xN\cap \mathcal O\ne \emptyset $
. Let
$\mathcal O_1$
be an open subset intersecting
$\Omega _0$
and
$V\subset \check {P}$
be an open symmetric neighborhood of e such that
$\mathcal O_1 V\subset \mathcal O$
.
Since
$q_i\to e$
and
$NV$
is an open neighborhood of e in G, there exists an open neighborhood, say, U of e in G and
$i_0$
such that
$U\subset q_i NV$
for all
$i\ge i_0$
. By Theorem 4.1, we can choose
$i>i_0$
such that
$[h] U\kern1.3pt{\cap}\kern1.3pt \mathcal O_1 a_{t_iu}\kern1.3pt{\ne}\kern1.3pt \emptyset $
. It follows that
${[h] q_i NV a_{-t_iu} \kern1.3pt{\cap}\kern1.3pt \mathcal O_1\kern1.3pt{\ne}\kern1.3pt \emptyset }$
. Since
$V\subset a_{-t_iu} V a_{t_iu}$
as
$u\in \mathfrak a^+$
, we have
$$ \begin{align*}[h] q_i NV a_{-t_iu} \cap \mathcal O_1 \subset [h] q_i N a_{-t_iu} V \cap \mathcal O_1 .\end{align*} $$
Since
$V=V^{-1}$
, we get
$[h] q_i N a_{-t_iu} \cap \mathcal O_1 V \ne \emptyset $
. Therefore,
$xN\cap \mathcal O\ne \emptyset $
, as desired.
This immediately implies the following corollary.
Corollary 4.4. If
$[g]\in \Omega _0$
is u-periodic for some
$u\in \operatorname {int}\mathcal L$
, then
$$ \begin{align*}\overline{[g]N}=\mathcal E_0.\end{align*} $$
Proof. Since
$ [g] (\exp ku) =[g]m_0^k$
for any integer k and M is compact, we have
$g^+\in \Lambda _u$
. Therefore, the claim follows from Proposition 4.3.
We may now conclude our main theorem in its fullest form.
Theorem 4.5. Let
$[g]\in \mathcal E_0$
. The following are equivalent:
-
(1)
$g^+\in \Lambda $
is a horospherical limit point; -
(2)
$[g]N$
is dense in
$\mathcal E_0$
; -
(3)
$[g]NM$
is dense in
$\mathcal E$
.
Proof. The implication
$(2)\Rightarrow (3)$
is trivial and
$ (3) \Rightarrow (1) $
was shown in Theorem 3.2. Hence, let us prove
$ (1)\Rightarrow (2) $
.
Let
$x=[g]\in \mathcal E_0$
. Suppose that
$g^+\in \Lambda _h$
. Fix any
$u\in \unicode{x3bb} (\Gamma )\cap \operatorname {int} \mathcal L_\Gamma $
. By Propositions 3.7 and 3.6,
$xN$
contains a u-periodic point, say,
$x_0$
. Hence, by Corollary 4.4,
$\overline {xN}\supset \overline {x_0N}\supset \Omega _0 N=\mathcal E_0$
. This proves
$(1)\Rightarrow (2)$
.
5 Conical limit points, minimality, and Jordan projection
A point
$\xi \in \mathcal F$
is called a conical limit point of
$\Gamma $
if there exists a sequence
$ u_j \to \infty $
in
$ \mathfrak a^+ $
such that for some (and hence every)
$ g \in G $
with
$ \xi = gP $
,
$$ \begin{align*} \limsup_{j\to \infty} \Gamma g a_{u_j} \neq \emptyset. \end{align*} $$
A conical limit point of
$\Gamma $
is indeed contained in
$\Lambda $
. We consider the following restricted notion.
Definition 5.1. We call
$\xi \in \mathcal F$
a strongly conical limit point of
$\Gamma $
if there exists a closed cone
$ \mathcal C \subset \operatorname {int} \mathcal L \cup \{0\} $
and a sequence
$ u_j \to \infty $
in
$ \mathcal C $
such that for some (and hence every)
$ g \in G $
with
$ \xi = gP $
,
$$ \begin{align*} \limsup_{j\to \infty} \Gamma g a_{u_j} \neq \emptyset. \end{align*} $$
Remarks 5.2. We mention that a conical limit point defined in [Reference Conze, Guivarc’h, Burger and Iozzi4] for
$\Gamma <\operatorname {SL}_d(\mathbb R)$
coincides with our strongly conical limit point.
Lemma 5.3. Any strongly conical limit point of
$\Gamma $
is horospherical.
Proof. Suppose that
$ \xi = gP $
is strongly conical, that is, there exist
$\gamma _j\in \Gamma $
and
$ u_j \to \infty $
in some closed cone
$\mathcal C\subset \operatorname {int} \mathcal L \cup \{0\}$
such that
$\gamma _j g a_{u_j}$
converges to some
$h\in G$
. Write
$\gamma _j g a_{u_j}=hq_j$
, where
$q_j\to e$
in G. Let
$\mathcal C'$
be a closed cone contained in
$\operatorname {int} \mathcal L \cup \{0\}$
whose interior contains
$\mathcal C \smallsetminus \{0\}$
.
Then
$\gamma _j^{-1}=g a_{u_j}q_j^{-1} h^{-1}$
and
$$ \begin{align*}\beta_{gP} (e, \gamma_j^{-1} ) =\beta_P (g^{-1}, a_{u_j}q_j^{-1} h^{-1})= \beta_{P} (g^{-1}, q_j^{-1}h^{-1}) +\beta_P(e, a_{u_j}) .\end{align*} $$
Since
$\beta _P(e, a_{u_j})=u_j$
and
$q_j^{-1}h^{-1}$
are uniformly bounded, the sequence
$$ \begin{align*}\beta_{gP}(e, \gamma_j^{-1}) - u_j\end{align*} $$
is uniformly bounded. Since
$u_j\in \mathcal C$
and
$\mathcal C\subset \operatorname {int}\mathcal C'\cup \{0\}$
, it follows that
$$ \begin{align*}\beta_{gP}(e, \gamma_j^{-1})\in \mathcal C'\end{align*} $$
for all sufficiently large j. This proves that
$\xi \in \Lambda _h$
.
Corollary 5.4. For any
$g\in G$
with strongly conical
$g^+\in \mathcal F$
, we have
$$ \begin{align*}\overline{[g] NM}=\mathcal E .\end{align*} $$
5.1 Directionally conical limit points
If
$v\in {\operatorname {int} \mathcal L}$
, then clearly
$\Lambda _v$
is contained in the horospherical limit set of
$\Gamma $
, and hence any
$NM$
-orbit based at a point of
$\Lambda _v$
is dense in
$\mathcal E$
. However, we would like to show in this section that the existence of a point in
$\Lambda _v$
for
$v\in \partial \mathcal L_\Gamma $
implies the existence of a non-dense
$NM$
-orbit in
$\mathcal E$
.
The flow
$\exp (\mathbb R u)$
is said to be topologically transitive on
$\Omega /M=\{\Gamma g M : g^{\pm }\in \Lambda \}$
if for any open subsets
$\mathcal O_1, \mathcal O_2$
intersecting
$\Omega /M$
, there exists a sequence
$t_n \to +\infty $
such that
$\mathcal O_1\cap \mathcal O_2 a_{t_n u}\ne \emptyset $
.
We make the following simple observation.
Lemma 5.5. For
$g\in \Omega $
, we have
$$ \begin{align*}\overline{gNM}\supset \Omega\quad \text{if and only if}\, \ \overline {gw_0\check{N}M}\supset \Omega.\end{align*} $$
Proof. We have
$\check {N}=w_0 N w_0^{-1}$
. Note that
$[g]\in \Omega $
if and only if
$[gw_0]\in \Omega $
, since
$(g w_0)^{\pm }=g^{\mp }$
. So
$\Omega w_0=\Omega $
. Hence,
$gNM$
is dense in
$\Omega $
if and only if
$gw_0 \check {N}M w_0^{-1} $
is dense in
$\Omega $
if and only if
$[g]w_0\check {N}M$
is dense in
$\Omega w_0=\Omega $
.
Since the opposition involution preserves
$\mathcal L$
and
$\unicode{x3bb} (g^{-1})=\operatorname {i}\unicode{x3bb} (g)$
for any loxodromic element, it follows that
$\unicode{x3bb} (\gamma )\in \partial \mathcal L$
if and only if
$\unicode{x3bb} (\gamma ^{-1})\in \partial \mathcal L$
.
Proposition 5.6
-
(1) If
$\Lambda =\Lambda _h$
, then
$\exp (\mathbb R v)$
is topologically transitive on
$\Omega /M$
for any
$v\in \operatorname {int}\mathfrak a^+$
such that
$\Lambda _v\ne \emptyset $
. -
(2) For any loxodromic element
$\gamma \in \Gamma $
with
$\{y_{\gamma }, y_{\gamma ^{-1}}\}\subset \Lambda _h$
, the flow
$\exp (\mathbb R \unicode{x3bb} (\gamma ))$
is topologically transitive on
$\Omega /M$
.
Proof. Assume that
$\Lambda =\Lambda _h$
; so the
$NM$
-action on
$\mathcal E$
is minimal. Suppose that
$ \Lambda _v\ne \emptyset $
for some
$v\in \operatorname {int}\mathfrak a^+.$
We claim that for any
${\mathcal O}_1, {\mathcal O}_2$
be two right M-invariant open subsets intersecting
$\Omega $
,
${\mathcal O}_1 \exp (t_iv) \cap {\mathcal O}_2\ne \emptyset $
for some sequence
$t_i \to +\infty $
. Choose
$x=[g]\in \Omega $
so that
$g^+\in \Lambda _v$
. Then there exists
$\gamma _i\in \Gamma $
and
$t_i\to +\infty $
such that
$\gamma _i g a_{t_i v}$
converges to some
$g_0$
. Note that
$x_0:=[g_0]\in \Omega $
. So write
$\gamma _i g a_{t_i v}= g_0 h_i$
with
$h_i\to e$
. By the
$NM$
-minimality assumption,
$x NM $
intersects every open subset of
$\Omega $
. Since
$v\in \operatorname {int} \mathfrak a^+$
and hence
$a_{-tv} n a_{tv}\to e$
as
$t\to +\infty $
, we may assume without loss of generality that
$x \in {\mathcal O}_1$
. Choose an open neighborhood U of e in G so that
${\mathcal O}_1\supset x U M$
. Note that there exists a sequence
$T_i\to \infty $
as
$i\to \infty $
such that for all i,
$$ \begin{align*}x U M a_{t_i v} \supset x a_{t_i v} a_{-t_i v} \check{N}_\varepsilon M a_{t_i v} \supset x_0 h_i \check{N}_{T_i}, \end{align*} $$
where
$ \check {N}_{R}=\check {N} \cap B_R^G $
is the set of elements of
$ \check {N} $
of norm
$ \leq R $
. So
${\mathcal O}_1 a_{t_i v}\supset x_0h_i \check {N}_{T_i}$
.
Choose an open neighborhood V of e in G and some open subset
${\mathcal O}_2'$
intersecting
$\Omega $
so that
${\mathcal O}_2\supset {\mathcal O}_2' V$
. Since
$x_0\check {N}M$
is dense in
$\Omega $
,
$x_0 n\in {\mathcal O}_2'$
for some
$n\in \check {N}$
. Hence,
$x_0 h_i n = x_0 n (n^{-1} h_i n)\in {\mathcal O}_2' V\subset {\mathcal O}_2 $
for all i large enough so that
$n^{-1} h_i n\in V$
. Therefore, for all i such that
$n\in \check {N}_{T_i}$
, we get
$$ \begin{align*}x_0h_in\in {\mathcal O}_1 a_{t_i v}\cap {\mathcal O}_2\ne \emptyset.\end{align*} $$
This proves the first claim.
Now suppose that
$\gamma \in \Gamma $
is a loxodromic element with
$y_{\gamma }, y_{\gamma ^{-1}}\in \Lambda _h$
. Write
$\gamma =g m a_{ v} g^{-1} $
for some
$g\in G$
and
$m\in M$
. Since
$y_\gamma =g^+$
and
$y_{\gamma ^{-1}}=gw_0^+,$
we have each
$[g]NM$
and
$[g]w_0 NM$
contains
$\Omega $
in its closure. Now in the notation of the proof of the first claim, note that
$x_0=[g_0]\in [g]M$
since
$[g]\exp (\mathbb R v) M$
is closed. Therefore, each
$\overline {x_0 NM}$
and
$\overline {x_0 \check {N}M}$
contains
$ \Omega $
. Based on this, the same argument as above shows the topological transitivity of
$\exp \mathbb R v$
, which finishes the proof since
$v=\unicode{x3bb} (\gamma )$
.
Since
$\mathcal L$
is invariant under the opposition involution
$\operatorname {i}$
and
$\unicode{x3bb} (\gamma )=\operatorname {i} \unicode{x3bb} (\gamma ^{-1})$
for any loxodromic element
$\gamma \in \Gamma $
, the Jordan projection
$\unicode{x3bb} (\gamma )$
belongs to
$ \partial \mathcal L$
if and only if the Jordan projection
$\unicode{x3bb} (\gamma ^{-1})$
belongs to
$ \partial \mathcal L$
. Together with the result of Dang and Gloriuex [Reference Dang and Glorieux7, Proposition 4.7], which says that
$\exp (\mathbb R u)$
is not topologically transitive on
$\Omega /M$
for any
$u\in \partial \mathcal L\cap \operatorname {int} \mathfrak a^+$
, Proposition 5.6 implies the following corollary.
Corollary 5.7
-
(1) If
$\Lambda _v\ne \emptyset $
for some
$v\in \partial \mathcal L\cap \operatorname {int} \mathfrak a^+$
, then
$$ \begin{align*}\Lambda\ne \Lambda_h.\end{align*} $$
-
(2) For any loxodromic element
$\gamma \in \Gamma $
, we have
$\unicode{x3bb} (\gamma )\in \partial \mathcal L$
if and only if Hence, if
$$ \begin{align*}\{y_\gamma, y_{\gamma^{-1}}\} \not\subset \Lambda_h .\end{align*} $$
$\Lambda =\Lambda _h$
, then
$\unicode{x3bb} (\Gamma )\subset \operatorname {int} \mathcal L$
.
Acknowledgements
We would like to thank Richard Canary and Pratyush Sarkar for helpful conversations regarding Corollary 1.10. O. Landesberg would also like to thank Subhadip Dey and Ido Grayevsky for helpful and enjoyable discussions. We thank the anonymous referee for pointing out to us the paper [Reference Conze, Guivarc’h, Burger and Iozzi4]. H. Oh is partially supported by NSF grant DMS-1900101
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