1. Introduction
Consider a closed Riemannian manifold
$(M,g)$
with negative sectional curvatures. Let
$\Gamma =\pi _1(M)$
be its fundamental group, and let
$C_\Gamma $
denote the set of conjugacy classes of non-trivial elements of
$\Gamma $
. Then, the classes in
$C_\Gamma $
correspond to freely non-trivial homotopy classes of loops in M, and each class contains a unique loop of minimal g-length, which is a closed geodesic. The marked length spectrum is a function
$$ \begin{align*} \ell_g:C_\Gamma{\buildrel{}\over\longrightarrow} [0,\infty) \end{align*} $$
that assigns the g-length
$\ell _g([\gamma ])$
of the closed geodesic corresponding to the conjugacy class
$[\gamma ]\in C_\Gamma $
to
$\gamma \in \Gamma \setminus \{1\}$
.
Marked length spectrum rigidity conjecture (cf. Burns and Katok [Reference Burns and Katok7]) states that function
$\ell _g$
determines
$(M,g)$
, up to isometry.
This conjecture has been proven for surfaces by Otal [Reference Otal27] and, independently but in greater generality, by Croke [Reference Croke9] slightly later. For higher dimensions, Katok [Reference Katok20] provided a short proof for metrics in a fixed conformal class in dimension 2, which can be easily extended to all dimensions. Hamenstädt [Reference Hamenstädt16] proved the conjecture for locally symmetric space using the entropy rigidity of Besson, Courtois, and Gallot [Reference Besson, Courtois and Gallot4]. However, for general negatively curved metrics, the problem remains largely open. Guillarmou and Lefeuvre [Reference Guillarmou and Lefeuvre14] showed that the marked length spectrum of a Riemannian manifold
$(M,g)$
with Anosov geodesic flow and non-positive curvature locally determines the metric g.
In this paper, we investigate a related problem concerning the rigidity of what we refer to as the marked length pattern. We consider a closed, negatively curved Riemannian manifold
$(M,g)$
with fundamental group
$\Gamma =\pi _1(M)$
. The set of non-trivial conjugacy classes in
$\Gamma $
is denoted by
$C_\Gamma $
. The length pattern is defined as the equivalence relation
$R_g$
on
$C_\Gamma $
, which is determined by the equality of g-length:
$$ \begin{align*} R_g= \{ {(c_1,c_2)\in C_\Gamma\times C_\Gamma}\mid{\ell_g(c_1)=\ell_g(c_2)} \}. \end{align*} $$
It is important to note that the relation
$R_g$
describes the equality of g-lengths without specifying their specific values.
Theorem A. Let
$(M,g_0)$
be a closed arithmetic locally symmetric manifold of rank 1, and let g be an arbitrary negatively curved Riemannian metric on M. Then
$R_{g_0}\subset R_g$
if and only if
$(M,g)$
is isometric to
$(M,\unicode{x3bb} g_0)$
for some
$\unicode{x3bb}>0$
by an isometry isotopic to identity.
However, there is a strengthened marked length spectrum rigidity. Katok [Reference Katok20] proved that the marked length spectrum, restricted to all curves homologous to a fixed non-trivial curve
$\gamma _0$
, determines the negatively curved metric on the surface in a fixed conformal class. In [Reference Gogolev and Rodriguez Hertz12], Gogolev and Rodriguez Hertz showed that the marked length spectrum restricted to the set of conjugacy classes represented by homologically trivial geodesics is sufficient to uniquely determine the marked length spectrum. Noelle [Reference Noelle26] demonstrated the same result for the complement of a ‘small’ set when the manifold is a closed surface. The following theorem extends some of these results, including [Reference Gogolev and Rodriguez Hertz12].
Theorem B. Let
$(M,g_1)$
and
$(M,g_2)$
be two arbitrary closed negatively curved Riemannian metrics on a manifold M with fundamental group
$\Gamma $
. Let H be a subgroup of the group
$\Gamma $
such that the limit set of H is all of
$\partial \widetilde {M}$
.
Then,
$\ell _{g_1}=\ell _{g_2}$
on classes from H if and only if
$\ell _{g_1}=\ell _{g_2}$
on all of
$\Gamma $
. Moreover, if M is a surface or
$(M, g_1)$
is a locally symmetric rank one space, then
$(M,g_1)$
is isometric to
$(M,g_2)$
.
For example, this applies to any non-trivial H, which is a normal subgroup
$\Gamma $
or a normal subgroup of normal subgroup, etc.
In a forthcoming paper, in collaboration with Alexander Furman, we will provide a generalization of Noelle’s work [Reference Noelle26] using a different method.
Theorems A and B deal with different perspectives. However, the key to their proofs relies on the same framework: marked length spectrum rigidity for B-cocycles, as stated in Theorem C below. To state it, let us start with the following.
Let X be a compact space. An element
$\phi \neq \mathrm {Id}_X$
in
$\operatorname {Homeo}(X)$
is called hyperbolic if
$\phi $
has uniform north-south dynamics, as defined in Definition 2.8. It is important to note that the points
$\phi ^{\pm }\in X$
are uniquely determined by a hyperbolic
$\phi $
. Additionally, if
$\phi $
is hyperbolic, then so is
$\phi ^{-1}$
and its attracting/repelling points are
$(\phi ^{-1})^\pm =\phi ^\mp $
.
Definition 1.1. Let
$\Gamma $
be a topological group. A non-trivial compact Hausdorff
$\Gamma $
-space X is called a geometric boundary if:
-
(1)
$\Gamma $
acts on X minimally; -
(2) every
$\gamma \in \Gamma \setminus \{e\}$
is hyperbolic; -
(3) there exist
$\Gamma $
-quasi-invariant measures
$\mu $
,
$\mu '$
on X such that the
$\Gamma $
-action on
$(X\times X,\mu \times \mu ')$
is ergodic.
If, in addition,
$\mu =\mu '$
, we call X a symmetric geometric boundary.
There are many examples of symmetric geometric boundaries. A typical example is a hyperbolic group G and its boundary
$\partial G$
, or any non-elementary subgroup of a hyperbolic group with its limit set (using [Reference Kaimanovich19] and [Reference Bader and Furman3]), or certain subgroups of acylindrically hyperbolic groups and their limit sets (using [Reference Maher and Tiozzo24]).
In this paper, we mainly focus on the applications of the marked length spectrum related problems.
Let
$\Gamma =\pi _1(M)$
be the fundamental group of a closed negatively curved Riemannian manifold M. The fundamental group
$\Gamma $
acts continuously on the universal cover
$\partial \widetilde {M}$
. In fact, the
$\Gamma $
-space
$\partial \widetilde {M}$
is a geometric boundary. Fixing a base point, we have a Busemann cocycle
$\alpha :\Gamma \times \partial \widetilde {M}\rightarrow {\mathbb R}$
. The cocycle
$\alpha $
captures many geometric properties of the metric. In particular, for any non-trivial
$\gamma \in \Gamma $
,
$\alpha (\gamma ,\gamma ^+)=\ell _g([\gamma ])$
.
This inspires us to define the marked length spectrum function for a general cocycle
$\beta $
on a geometric boundary X by setting
$$ \begin{align} \ell_\beta([\gamma])=\beta(\gamma,\gamma^+) \end{align} $$
for all
$\gamma \in \Gamma \setminus \{1\}$
.
It may seem more natural to define the marked length spectrum of
$\beta $
as a pair of numbers,
$\ell _\beta ([\gamma ])=(\beta (\gamma ,\gamma ^+),\beta (\gamma ,\gamma ^-))$
. However, our definition is sufficient for a special type of cocycles called B-cocycles, which we define below. In B-cocycles, we will observe that
$\beta (\gamma ,\gamma ^-)=-\beta (\gamma ,\gamma ^+)$
and
$\ell _\beta ([\gamma ^{-1}])=\ell _\beta ([\gamma ])$
(see §4 below).
Definition 1.2. Let X be a
$\Gamma $
-space. A cocycle
$c:\Gamma \times X\rightarrow {\mathbb R}$
is a B-cocycle if:
-
(1) the map
$x\mapsto c(\gamma ,x)$
is continuous for all
$\gamma \in \Gamma $
; -
(2) there exists a continuous function
$C:X\times X\setminus \Delta \rightarrow {\mathbb R}$
such that for all
$$ \begin{align*}C(\gamma x,\gamma y)-C(x,y)=c(\gamma,x)+c(\gamma,y)\end{align*} $$
$\gamma \in \Gamma $
,
$x\neq y\in X$
, where
$\Delta $
is the diagonal
$ \{ {(x,x)}\mid {x\in X} \} $
.
When necessary to specify the function C, we refer to
$(c, C)$
as a B-cocycle.
Denote by
$Z^1_c(\Gamma ,X,{\mathbb R})$
the vector space of all continuous cocycles
$c:\Gamma \times X\to {\mathbb R}$
, and by
$B^1_c(\Gamma ,X,{\mathbb R})$
the subspace consisting of cocycles of the form
$c(\gamma ,x)=\varphi (\gamma .x)-\varphi (x)$
for some continuous function
$\varphi :X\to {\mathbb R}$
. Denote by
$H^1_c(\Gamma ,X,{\mathbb R})=Z^1_c/B^1_c$
the associated cohomology. Alternatively, we can view this as cohomology
$H^1(\Gamma ,C(X,{\mathbb R}))$
with coefficients in
$C(X,{\mathbb R})$
viewed as a
$\Gamma $
-module.
Note that the collection of all B-cocycles over a
$\Gamma $
-space X forms a vectors subspace of
$Z_c^1(\Gamma ,X,{\mathbb R})$
that contains
$B_c^1(\Gamma ,X,{\mathbb R})$
. Therefore, one can refer to cohomological classes of B-cocycles, or simply B-classes. B-classes constitute the kernel of the map
$$ \begin{align*} [i]:H^1_c(\Gamma,X,{\mathbb R})\to H^1_c(\Gamma,X\times X\setminus\Delta,{\mathbb R}) \end{align*} $$
induced by
$i\alpha (g,(x,y))=\alpha (g,x)+\alpha (g,y)$
. The following rigidity results are obtained.
Theorem C. Let
$\Gamma $
-space X be a geometric boundary and
$\alpha , \beta : \Gamma \times X\rightarrow {\mathbb R}$
be two B-cocycles. Then,
$\ell _\alpha =\ell _\beta $
if and only if we have
$[\alpha ]=[\beta ]$
in
$H_c^1(\Gamma ,X,{\mathbb R})$
.
Remark 1.3. Even though lattices in semisimple Lie groups G and their Furstenberg boundaries
$X=G/P$
are not necessarily geometric boundaries, a similar definition of B-cocycles can be applied in this case. In this scenario, the diagonal
$\Delta $
in condition (2) is replaced by the set of pairs of points that are not in general position. B-cocycle can also be defined for general acylindrical groups and their limit sets.
Theorem C remains valid in both cases. However, the proof is not covered within our current framework. We will provide the proof for general acylindrical groups in the Appendix. The proof for higher rank lattices requires a bit more effort but follows the same construction. The detailed proof for higher rank lattices will be presented elsewhere.
In the context of a negatively curved space
$(M,g)$
with
$\Gamma =\pi _1(M)$
acting on the universal cover of M,
$X=\partial \widetilde M$
, the theorem implies that the cross-ratios on the boundaries are determined by the Busemann cocycles when restricted to Busemann cocycles derived from various Riemannian metrics. This statement is weaker than the marked length spectrum rigidity and is well known in this scenario. In fact, for the Busemann cocycle of a closed negatively curved manifold, there exists a corresponding length cocycle for its geodesic flow. By fixing a visual metric on the boundary associated with a specified negatively curved metric, all pullbacks of Busemann cocycles for different metrics are Hölder continuous. In this case, Theorem C can be deduced from a theorem by Livšic [Reference Livšic22].
Although Theorem C may not be new in many cases, it offers a fresh perspective that reveals additional patterns underlying the marked length spectrum of arithmetic locally symmetric spaces. It highlights that it is not the lengths themselves but rather certain identities between them that determine the metric up to homothety. Theorem A can be derived from this observation. In fact, the combination of Livšic’s theorem [Reference Livšic22] regarding Hölder cocycles and our construction is sufficient to prove Theorem A.
It is natural to inquire about the case for non-arithmetic lattices or even negatively curved manifolds. The brief answer is that, in general, they do not possess marked length pattern rigidity.
The length spectrum of a generic negatively curved metric on a manifold is simple [Reference Abraham1]. Therefore, the answer to the question is false. However, it is worth noting that hyperbolic surfaces can have (unbound) multiplicity in the length spectrum [Reference Randol31]. Nonetheless, we have the following theorem.
For a closed surface S that admits a hyperbolic metric, let
$T_S$
be the set of marked hyperbolic metrics on S, which corresponds to its Teichmuller space.
Theorem D. There exists a subset
$T_{\mathrm {sing}}\subset T_S$
, which is a union of countably many algebraic subsets of positive codimension in
$T_S$
, such that for all
$g\in T_S\setminus T_{\mathrm {sing}}$
and
$g'\in T_S$
, the relation
$R_g$
is a subrelation of
$R_{g'}$
.
In this context, a hyperbolic metric refers to a Riemannian metric with a constant curvature
$-1$
. A hyperbolic surface is defined as a surface that possesses a complete hyperbolic metric with finite volume.
We do not currently have any examples of higher-dimensional locally symmetric manifolds that lack marked length pattern rigidity. According to Mostow rigidity, the locally symmetric structure is determined by the fundamental group alone. However, when discussing general metrics, we require additional tools and techniques to analyze and understand them.
Similar to Remark 1.3, Theorems A and B have weaker generalizations to finite volume manifolds and orbifolds. Due to the differences in setup and the resulting weaker results, we discuss these generalizations in the Appendix.
The remaining sections of the paper are organized as follows. Section 2 introduces some basic concepts and definitions related to Gromov-hyperbolic spaces, Patterson–Sullivan measures, and B-cocycles. Section 3 presents examples of geometric boundaries and discusses their properties. Section 4 proves Theorem C, establishing the connection between marked length spectrum rigidity and B-cocycles. Section 5 contains the proof of Theorem B. Section 6 discusses the extension of B-cocycles from arithmetic lattices to the Lie group G. In §7, we apply the results to negatively curved Riemannian manifolds, leading to the proof of Theorem A. Section 8 focuses on the marked length spectrum of hyperbolic surfaces and provides examples of hyperbolic metrics without marked length pattern rigidity. Appendix A considers the marked length spectrum rigidity for cocycles of general acylindrical hyperbolic groups.
2. Gromov-hyperbolic spaces and Patterson–Sullivan measures
2.1. Gromov-hyperbolic space
In this work, we will refer to [Reference Bridson and Haefliger6] as a general reference for Gromov hyperbolic spaces and provide a brief overview here.
Let
$(X,d)$
be a metric space and let p be a fixed base point in X. The Gromov product of two points x, y in X is defined as
$$ \begin{align*} (x,y)=\tfrac{1}{2}(d(x,p)+d(y,p)-d(x,y)). \end{align*} $$
We say that X is Gromov-hyperbolic if there exists
$\delta \geq 0$
such that for all x, y and z in X, the following inequality holds:
$$ \begin{align*} (x,y)\geq \min\{(x,z),(y,z)\}-\delta. \end{align*} $$
It can be shown that hyperbolicity of a space is independent of the choice of the base point. This fact is demonstrated in [Reference Gromov13, Section 1.1.B].
In the context of a metric space X, a sequence of points
$(x_n)$
is said to converge at infinity if
$(x_i,x_j)\to \infty $
as i,
$j\to \infty $
. Two such sequences,
$(x_n)$
and
$(y_n)$
, are considered equivalent if
$(x_i,y_j)\to \infty $
as i,
$j\to \infty .$
The set of equivalence classes of a sequence converging at infinity is referred to as the Gromov boundary
$\partial X$
of X. Other equivalent definitions can be found in [Reference Bridson and Haefliger6, Section H.3].
For a sequence
$(x_n)$
, we say
$\lim x_n=x$
if
$x\in X$
and
$x_n$
converge to x in X, or if
$(x_n)$
converges at infinity and x represents the equivalent class of
$(x_n)$
.
We extend the Gromov product to the extended space
$\bar {X}=X\cup \partial X$
as follows:
$$ \begin{align*} (x,y)=\sup\liminf_{i,j\to\infty}(x_i,y_j) \end{align*} $$
where the supremum is taken over all sequences
$(x_i)$
and
$(y_j)$
in X such that
$x=\lim x_i$
and
$y=\lim y_j$
.
It is important to note that the extended Gromov product is not continuous in general.
2.2. CAT
$(-1)$
spaces
A CAT
$(-1)$
-space, as defined in [Reference Bridson and Haefliger6, Definition II.1.1], is a metric geodesic space in which every geodesic triangle is thinner than its associated comparison triangle in the hyperbolic plane.
Let X be a proper CAT
$(-1)$
space, and let
$p\in X$
be a fixed base point. The Gromov product
$(\cdot ,\cdot )$
defined on
$X\times X$
can be extended continuously to a function
$\overline {X}\times \overline {X}\rightarrow [0,\infty ]$
by considering the limit:
$$ \begin{align*} (x,y)=\lim_{i,j\to \infty}(x_i,y_j), \end{align*} $$
where
$(x_i)$
and
$(y_j)$
are sequences in X such that
$x=\lim x_i$
and
$y=\lim y_j$
[Reference Bourdon5].
For each
$\epsilon \in (0,1]$
, the expression
$d_\epsilon (\xi ,\eta )=e^{-\epsilon (\xi ,\eta )_p}$
defines a metric on the boundary
$\partial X=\overline {X}\setminus X$
that is compatible with the topology of
$\overline {X}$
[Reference Bourdon5]. These metrics are referred to as visual metrics. Notably,
$\xi =\eta $
on
$\partial X$
if and only if
$(\xi , \eta )=\infty $
.
For a strictly negatively curved complete Riemannian manifold
$(M,g)$
, its universal cover
$\widetilde M$
is a Gromov-hyperbolic space, specifically a CAT
$(-k)$
space, for some
$k>0$
. The visual boundary of
$\widetilde M$
is equivalent to its Gromov boundary [Reference Bridson and Haefliger6, Lemma H.3.13]. By renormalizing the metric,
$\widetilde M$
can be transformed into a CAT
$(-1)$
space. As a result, the Gromov product extends continuously to the boundary, and visual metrics can be defined. Moreover, there exists
$\epsilon _o>0$
such that
$d_\epsilon $
defines a metric for all
$\epsilon \leq \epsilon _0$
. Once again, these metrics are referred to as visual metrics.
2.3. Patterson–Sullivan measures
Let
$\Gamma $
be a non-elementary discrete group of isometries acting on a connected, simply connected, complete Riemannian manifold
$(\widetilde M,g)$
with the curvature
$\kappa \leq -a^2<0$
. The compactification of
$\widetilde M$
, denoted by
$\overline {\widetilde {M}}=\widetilde M\cup \partial \widetilde M$
, is homeomorphic to a closed ball.
In the case of dimension two and constant curvature, Patterson [Reference Patterson28] introduced a family of measures on the boundary
$\partial \widetilde M$
indexed by points
$x\in \widetilde M$
. Sullivan [Reference Sullivan32] extended this construction to higher dimensions. Building upon their work, several authors, including Coornaert, Albuquerque, and Knieper, developed the so-called Patterson–Sullivan measures
$\nu _x$
on the boundary
$\partial \widetilde M$
in the case of variable curvature. For detailed information on the construction, please refer to [Reference Albuquerque2, Reference Knieper21].
Let us discuss some key properties of Patterson–Sullivan measures
$\nu _x$
. For more details, please refer to [Reference Quint30]. Measures that satisfy properties (2) and (3) below are known as
$\Gamma $
-conformal density.
-
(1) [Reference Quint30, Theorem 4.8] The support of
$\nu _x$
is the limit set of
$\Gamma $
. -
(2) [Reference Quint30, Theorem 4.8] Equivariance:
$\nu _{\gamma x}=\gamma _* \nu _x$
for all
$\gamma \in \Gamma $
. -
(3) [Reference Quint30, Theorem 4.8] Explicit Radon–Nikodym derivative: for all
$x,y\in \widetilde M$
, (2.1)where
$$ \begin{align} \frac{d\nu_x}{d\nu_y}(\xi)=e^{h(g)B_{x,y}(\xi)}, \end{align} $$
$B_{x,y}(\xi )$
is the Busemann function on
$\widetilde M$
and
$h(g)$
in the volume entropy of
${\widetilde M}$
.
For points
$x,y\in \widetilde M$
and
$\xi \in \partial \widetilde M$
, the function
$B:\widetilde M\times \widetilde M\times \partial \widetilde M\rightarrow {\mathbb R}$
is defined by where
$$ \begin{align*} B_{x,y}(\xi)=\lim_{t\rightarrow \infty}(d_X(y, q_\xi(t))-t), \end{align*} $$
$q_\xi $
is the unique geodesic ray with
$q_\xi (0)=x$
and
$q_\xi (\infty )=\xi $
.Remark 2.1. In general, the Patterson–Sullivan measures
$\nu _x$
are finite measures, not probability measures.
-
(4) The marked length spectrum is determined by the Patterson–Sullivan measures
$\nu _x$
. In the case where
$\widetilde M/\Gamma $
is a manifold and, as a result,
$\Gamma $
is torsion-free, consider a hyperbolic element
$\gamma \in \Gamma $
. There exists a unique attracting fixed point
$\gamma ^{+}$
of
$\gamma $
on the boundary
$\partial \widetilde M$
.Claim 2.3. For all
$x\in \widetilde M$
,
$B_{x,\gamma x}(\gamma ^{+})=\ell _g([\gamma ])$
.Proof. Using equation (2.1) and the definition of the Busemann function, we have
$$ \begin{align*} B_{x,\gamma x}(\gamma^+)=B_{y,\gamma y}(\gamma^+)+B_{x,y}(\gamma^+) -B_{\gamma x,\gamma y}(\gamma^+). \end{align*} $$
Since
$\gamma $
is an isometry, we can rewrite
$B_{\gamma x,\gamma y}(\gamma ^+)$
as
$B_{\gamma x,\gamma y}(\gamma \gamma ^+)=B_{x,y}(\gamma ^+)$
.Hence
$B_{x,\gamma x}(\gamma ^+)$
is independent of the choice of x. Taking x to be a point on the axis of
$\gamma $
, the claim follows.For a parabolic element
$\eta $
, there is only one fixed point
$\xi $
on
$\partial \widetilde M$
, and Therefore, we define
$$ \begin{align*} B_{x,\eta x}(\xi)=0. \end{align*} $$
$\ell _g([\eta ])=0$
.
-
(5) According to [Reference Furman11, Proposition 1], the Bowen–Margulis–Sullivan geodesic current on
$\partial \widetilde M\times \partial \widetilde M$
can be defined as follows. Let
$p\in \widetilde M$
be a base point. Choose a small enough
$\epsilon>0$
so that
$d_\epsilon $
is a metric. Through a direct computation, it can be shown that defines a
$$ \begin{align*} d\mu(\xi,\eta)=d_\epsilon(\xi,\eta)^{-2h(g)/\epsilon}\cdot dv_p(\xi)\,dv_p(\eta) \end{align*} $$
$\Gamma $
-invariant measure on
$\partial \widetilde M\times \partial \widetilde M$
. This measure is referred to as the Bowen–Margulis–Sullivan geodesic current. It is important to note that while the Gromov product and the visual metric depend on the choice of the base point p, the Bowen–Margulis–Sullivan geodesic current is independent of both the choice of p and the chosen value of
$\epsilon $
.
2.4. Cocycles
This part of the paper is largely based on [Reference Zimmer33, §4.2]. We provide a summary of definitions and results, and refer to the proofs in that section and the references cited therein.
In the context of a
$\Gamma $
-space X, a function
$\alpha :\Gamma \times X\rightarrow {\mathbb R}$
is considered a cocycle if it satisfies the condition:
$$ \begin{align} \alpha(\gamma\eta,x)=\alpha(\gamma,\eta x)+\alpha(\eta,x) \end{align} $$
for all
$\gamma $
,
$\eta \in \Gamma $
, and
$x\in X$
. When
$\alpha $
is a continuous function, it is referred to as a continuous cocycle. This terminology is analogous to Borel cocycles.
For any mapping
$k: X\rightarrow {\mathbb R}$
, the coboundary of k is defined as
$$ \begin{align*} d k(\gamma, x)=k(\gamma x)-k(x)\quad(\gamma\in \Gamma,\ x\in X). \end{align*} $$
Similar to before, we classify
$dk$
as continuous or Borel depending on the regularity of k. It is important to note that all coboundaries are also cocycles.
Definition 2.4. Two cocycles
$\alpha $
,
$\beta $
are called equivalent, if there exist a map
$k:X\rightarrow {\mathbb R}$
such that
$$ \begin{align*}\alpha-\beta=d k.\end{align*} $$
We refer to
$\alpha $
,
$\beta $
continuous equivalent if the map k is continuous.
It is indeed possible to define the (continuous) cohomology group of a
$\Gamma $
-space by considering (continuous) equivalent cocycles as cohomological classes. We denote the set of (continuous) cocycles as Z, and the set of coboundaries of continuous maps as B. Here, Z forms an abelian group, and B is a subgroup of Z. The first (continuous) cohomological group, denoted as
$H^1_c(\Gamma , X,{\mathbb R})$
, is then defined as the quotient group
$Z/B$
. Notably, this cohomological group is equivalent to the usual group cohomology of
$\Gamma $
with coefficient in
$C(X,{\mathbb R})$
, which represents the set of continuous maps from X to
${\mathbb R}$
viewed as a
$\Gamma $
-module.
Indeed, there is a more general definition of cocycles that allows for the target to be any second countable group, and in this setting, the resulting cohomology may not necessarily be a group.
Recall the following definition in [Reference Zimmer33].
Definition 2.5. Two Borel cocycles
$\alpha $
,
$\beta $
are called strictly equivalent if there is a Borel function
$\phi $
such that
$$ \begin{align*}\alpha-\beta=d \phi.\end{align*} $$
Let
$G_0$
be a closed subgroup of G. Then G acts on
$G/G_0$
via left translation. Let H be a second countable group. The following proposition was proven in [Reference Zimmer33].
Proposition 2.6. [Reference Zimmer33, Proposition 4.2.13]
There is a bijection
$$ \begin{align*} \{\mathrm{Borel\ cocycles} \ G\times G/G_0\rightarrow H\}\rightarrow \mathrm{{Hom}}(G_0, H) \end{align*} $$
between strict equivalence classes of Borel cocycles and conjugacy classes of homomorphisms.
2.5. B-cocycles
The Busemann-cocycles are closely related to the Patterson–Sullivan measures. By fixing a base point
$p\in M$
, the Busemann cocycle
$B(\gamma , \xi )$
is defined as
$$ \begin{align*}B(\gamma, \xi)=B_{\gamma^{-1}p,p}(\xi)=\frac{1}{h(g)}\ln \frac{d(\gamma^{-1}_\ast \nu_p)}{d\nu_p}(\xi)\end{align*} $$
for all
$\gamma \in \Gamma $
and
$\xi \in \partial M$
.
Based on property (4) in §2.3, the marked length spectrum can be determined by the Busemann cocycle. Property (5) in §2.3 is equivalent to the following expression:
$$ \begin{align*}-2(\gamma\xi,\gamma\eta)_p+2(\xi,\eta)_p=B(\gamma,\xi)+B(\gamma,\eta)\end{align*} $$
for all
$\gamma \in \Gamma $
,
$\xi ,\eta \in \partial M$
and
$\xi \neq \eta $
.
In Definition 1.2, we generalized the Busemann cocycle to define B-cocycles.
Given any
$\Gamma $
-space X, there is a diagonal action of
$\Gamma $
on
$X\times X$
. We can consider the map
$$ \begin{align*} [i]: H^1_c(\Gamma, X,{\mathbb R})\rightarrow H^1_c(\Gamma, X\times X\setminus\Delta,{\mathbb R}), \end{align*} $$
defined as
$i(\alpha )(\gamma ,(x,y))=\alpha (\gamma ,x)+\alpha (\gamma ,y)$
. Then B-cocycles are precisely the cocycles that represent the classes in
$\text {Ker}([i])$
.
Lemma 2.7. In a B-cocycle
$(c,C)$
, the function
$C:X\times X\setminus \Delta \to {\mathbb R}$
determines the cocycle
$c:\Gamma \times X{\buildrel {}\over \longrightarrow } {\mathbb R}$
.
Proof. Let
$(C, c)$
be a B-cocycle, and consider three pairwise different points x, y,
$z\in X$
. Define
$h(x,y,z)=C(x,y)+C(x,z)-C(y,z)$
. Then by definition of B-cocycle, we have
$$ \begin{align*} 2c(\gamma, x)=h(\gamma x,\gamma y,\gamma z)-h(x,y,z).\\[-35pt] \end{align*} $$
2.6. North-South dynamics
In this paper, all north-south dynamics are uniform north-south dynamics in the sense of [Reference Lustig and Uyanik23]. More precisely, let us recall the following definition.
Definition 2.8. [Reference Lustig and Uyanik23, Definition 3.1] Let
$f: X \to X$
be a homeomorphism of a topological space X.
-
(1) The map
$\phi $
is said to have (pointwise) north-south dynamics if it has two distinct fixed points
$\phi ^{+}$
and
$\phi ^-$
, called attractor and repeller, respectively, such that for every
$x\in X\setminus \{\phi ^+,\phi ^-\}$
, one has and
$$ \begin{align*} \lim_{n\to \infty}\phi^n(x)=\phi^+ \end{align*} $$
$$ \begin{align*} \lim_{n\to -\infty}\phi^{n}(x)=\phi^-. \end{align*} $$
-
(2) The map
$\phi $
is said to have uniform north-south dynamics if the following conditions are satisfied: there exist two distinct fixed points
$\phi ^{+}$
and
$\phi ^-$
, such that for every compact set
$K\subset X\setminus \{\phi ^-\}$
and every neighborhood
$U^+$
of
$\phi ^+$
, there exists an integer
$N^+\geq 0$
such that for all
$n\geq N^+$
, one has Similarly, for every compact set
$$ \begin{align*} \phi^n(K)\subset U^+. \end{align*} $$
$K\subset X\setminus \{\phi ^+\}$
and every neighborhood
$U^-$
of
$\phi ^-$
, there exists an integer
$N^-\leq 0$
such that for every
$n\leq N^-$
, one has
$$ \begin{align*} \phi^{n}(K)\subset U^-. \end{align*} $$
Note that when X is compact, the two conditions in part (2) of Definition 2.8 are dual to each other, as the complement of a compact subset is open, and vice versa.
3. Geometric boundaries
In this section, we present some basic properties of geometric boundaries that will be useful in the next section.
First, note that in Definition 1.1, the support of
$\mu $
is a
$\Gamma $
-invariant closed subset of X. Therefore, since the
$\Gamma $
-action is minimal,
$\mu $
has full support. The same is true for
$\mu '$
.
In Lemma 2.7, it was shown that for a B-cocycle
$(c, C)$
, the function C determines the cocycle c. Conversely, we have the following lemma.
Lemma 3.1. Let
$\Gamma $
-space X be a geometric boundary, and
$(c,C)$
be a B-cocycle on X. Then the cocycle c determines C up to a constant.
Proof. Let
$(c,C)$
and
$(c,C')$
be two B-cocycles.
By definition of B-cocycles, the difference of the two coboundaries
$dC-dC'=0$
. Therefore,
$C-C'$
is a
$\Gamma $
-invariant function.
Since
$\Gamma $
is
$\nu \times \nu '$
-ergodic,
$\nu \times \nu '$
has full support, and
$C-C'$
is continuous,
$C-C'$
is a constant function.
We give some examples of geometric boundaries.
Example 3.2. Let
$\Gamma $
be the fundamental group of a closed negatively curved Riemannian manifold
$(M,g)$
, and let
$\partial \widetilde {M}$
be the boundary of the Riemannian universal cover
$\widetilde {M}$
. The
$\Gamma $
-space
$\partial \widetilde {M}$
is a symmetric geometric boundary.
Let us verify conditions (1)–(3) in Definition 1.1 for the
$\Gamma $
-space
$\partial \widetilde {M}$
.
-
(1) Since the limit set of
$\Gamma $
coincides with
$\partial \widetilde {M}$
, the action of
$\Gamma $
on
$\partial \widetilde {M}$
is minimal. -
(2) It is well known and can be deduced by the classification of isometries of hyperbolic spaces.
-
(3) Consider the Patterson–Sullivan measures
$\nu _x$
on
$\partial \widetilde {M}$
. There is the Bowen–Margulis– Sullivan geodesic current on
$\partial ^2 \widetilde {M}$
which is in the measure class
$[\nu _x\times \nu _x]$
. It is well known that the Bowen–Margulis–Sullivan geodesic current is
$\Gamma $
-ergodic. Therefore, the action of
$\Gamma $
on
$\partial \widetilde {M}$
satisfies condition (3) in Definition 1.1.
Example 3.3. Consider the same setting as in Example 3.2. Let H be a normal subgroup of
$\Gamma $
such that
$\Gamma /H$
is a finite extension of
${\mathbb Z}$
or
${\mathbb Z}^2$
. Then
$\partial \widetilde {M}$
is a symmetric geometric H-boundary.
Let us verify conditions (1)–(3) in Definition 1.1 for the H-space
$\partial \widetilde {M}$
.
-
(1) Since H is normal, the limit set of H is equal to the limit set of
$\Gamma $
. Therefore, the action is minimal. -
(2) H is a non-trivial group.
-
(3) The Bowen–Margulis–Sullivan geodesic current is an H-invariant ergodic measure (cf. [Reference Guivarc’h15]).
Now we will present some basic properties of geometric boundaries. All of these statements are well known. We provide short proofs for the sake of completeness.
Lemma 3.4. Geometric boundaries have infinitely many points.
Proof. Let X be a geometric boundary of
$\Gamma $
. According to condition (2) in Definition 1.1, and since X is a non-trivial
$\Gamma $
-space, there exists a hyperbolic element
$\gamma $
. Since
$\gamma $
fixes
$\gamma ^+$
and
$\gamma ^-$
, and
$\gamma $
acts non-trivially on X, X must contains at least three points. For any
$x\neq \gamma ^+$
and
$x\neq \gamma ^-$
, we have
$$ \begin{align*}\lim_{n\rightarrow +\infty}\gamma^n x=\gamma^+.\end{align*} $$
It is clear that
$\gamma ^n x\neq \gamma ^+$
for all n. This completes the proof.
It can be shown that geometric boundaries are perfect spaces.
Lemma 3.5. Let X be a geometric boundary of the group
$\Gamma $
. Let
$\gamma $
be a hyperbolic element. There exists an element
$\theta \in \Gamma $
such that
$\theta \gamma ^+\neq \gamma ^-$
,
$\theta \gamma ^-\neq \gamma ^-$
.
Proof. Assume the lemma is false. Then for any element
$\theta \in \Gamma $
, either
$\theta \gamma ^+=\gamma ^-$
or
$\theta \gamma ^-=\gamma ^-$
holds.
If
$\theta \gamma ^+=\gamma ^-$
, then
$\theta ^2 \gamma ^+=\theta \gamma ^-\neq \gamma ^-$
. Hence, we have
$\theta ^2\gamma ^-=\gamma ^-$
. Therefore,
$\theta \gamma ^-=\theta ^{-1}\gamma ^-=\gamma ^+$
.
Otherwise,
$\theta \gamma ^-=\gamma ^-$
.
As a result, we see that the
$\Gamma $
-orbit of
$\gamma ^-$
consists only of the two points
$\gamma ^+$
and
$\gamma ^-$
. The action of
$\Gamma $
is not minimal by Lemma 3.4. This is a contradiction.
Therefore, the lemma holds.
A direct computation shows the following lemmas.
Lemma 3.6. Conjugations of a hyperbolic element are hyperbolic. Moreover, for a hyperbolic element
$\gamma $
, we have
$(\theta \gamma \theta ^{-1})^\pm =\theta \gamma ^\pm .$
Lemma 3.7. Let
$\gamma , \eta \in \Gamma $
with
$\gamma $
hyperbolic. Then for all but possibly one n,
$\eta \gamma ^n$
is hyperbolic. Furthermore, if
$\eta \gamma ^+\neq \gamma ^-$
, then we have
$$ \begin{align*}\lim_{n\rightarrow +\infty}(\eta\gamma^n)^+=\eta\gamma^+,\end{align*} $$
$$ \begin{align*}\lim_{n\rightarrow +\infty}(\eta\gamma^n)^-=\gamma^-.\end{align*} $$
Proof. The first claim follows from the fact that
$\gamma $
acts non-trivially on X and has infinite order. Therefore,
$\eta \gamma ^n$
acts non-trivially on X for all but at most one n. Using condition (2) in Definition 1.1, this implies that
$\eta \gamma ^n$
is hyperbolic for all but at most one n.
Now assume
$\eta \gamma ^+\neq \gamma ^-$
. Let U and V be two open sets such that
$U\cap V=\emptyset $
,
$\eta \gamma ^+\in U$
,
$\gamma ^-\in V$
. According to Definition 2.8, there exists an integer
$N_{U,V}$
such that for all
$n\geq N_{U,V}$
,
$\gamma ^n(X-V)\subset \eta ^{-1}U$
. This implies
$\eta \gamma ^n(X-V)\subset U$
for
$n\geq N_{U,V}$
. Since
$U\subset X-V$
, we have
$\eta \gamma ^n(U)\subset U$
for all
$n\geq N_{U,V}$
.
For the hyperbolic element
$\eta \gamma ^n$
, again using Definition 2.8, when
$x\neq (\eta \gamma ^n)^-$
, we have
$$ \begin{align*} \lim_{k\rightarrow +\infty}(\eta\gamma^n)^k x=(\eta\gamma^n)^+. \end{align*} $$
Hence,
$(\eta \gamma ^n)^+\in U$
for all
$n\geq N_{U,V}$
. Since U can be arbitrarily small, we conclude that
$$ \begin{align*} \lim_{n\rightarrow \infty}(\eta\gamma^n)^+=\eta\gamma^+. \end{align*} $$
This proves the first equation.
For the second equation, note that
$\eta ^{-1}((\gamma ^{-1})^+)=\eta ^{-1}\gamma ^-\neq \gamma ^+=(\gamma ^{-1})^-$
. Therefore,
$$ \begin{align*} \lim_{n\to \infty}(\eta^{-1}\gamma^{-n})^+)=\eta^{-1}(\gamma^{-1})^+=\eta^{-1}\gamma^-. \end{align*} $$
Finally, using Definition 2.8 and Lemma 3.6,
$$ \begin{align*}\lim_{n\rightarrow \infty}(\eta\gamma^n)^-=\lim_{n\rightarrow \infty}(\gamma^{-n}\eta^{-1}))^+=\lim_{n\rightarrow \infty}\eta(\eta^{-1}\gamma^{-n}))^+=\gamma^-,\end{align*} $$
since
$\eta ^{-1}\gamma ^{-n}=\eta ^{-1}(\gamma ^{-n}\eta ^{-1})\eta $
.
4. Marked length spectrum rigidity for B-cocycles
We will now proceed with the proof of Theorem C in this section.
Before presenting the proof, we first make the following observation. Let
$(c,C)$
be a B-cocycle over a geometric
$\Gamma $
-boundary X. Since
$$ \begin{align*} c(\gamma,\gamma^+)+c(\gamma,\gamma^-)=C(\gamma \gamma^+,\gamma \gamma^{-})-C(\gamma^+,\gamma^-)=0 \end{align*} $$
for all
$\gamma \in \Gamma \setminus \{1\}$
, we have
$$ \begin{align*} c(\gamma,\gamma^-)=-c(\gamma,\gamma^+). \end{align*} $$
Furthermore, using the fact that
$$ \begin{align*} 0=c(1,\gamma^-)=c(\gamma^{-1}\gamma,\gamma^-)=c(\gamma^{-1},\gamma\gamma^-)+c(\gamma,\gamma^-)=c(\gamma^{-1},\gamma^-)+c(\gamma,\gamma^-), \end{align*} $$
we obtain
$$ \begin{align*}\ell_c([\gamma^{-1}])=c(\gamma^{-1},\gamma^-)=c(\gamma,\gamma^+)=\ell_c([\gamma]). \end{align*} $$
Now we are ready to prove Theorem C.
Proof of Theorem C
The ‘if’ part is straightforward.
For the ‘only if’ part, let
$\delta =\alpha -\beta $
. Since
$\delta $
is a B-cocycle and
$\ell _\delta =0$
, we aim to demonstrate that
$\delta $
can be expressed as
$\delta =d\varphi $
for some continuous function
$\varphi $
. Let us assume that the pair
$(\delta ,f)$
is a B-cocycle.
The proof is divided into three steps.
Step 1: Show that the cocycle
$\delta $
is bounded. Choose any hyperbolic element
$\gamma \in \Gamma $
with fixed points
$\gamma ^+$
and
$\gamma ^{-}$
. Let
$\eta \in \Gamma $
be any element such that
$\eta \gamma ^+\neq \gamma ^-$
. We have
$$ \begin{align} \delta(\eta \gamma^n,\gamma^+)=\delta(\eta,\gamma^n\gamma^+)+\delta(\gamma^n,\gamma^+)=\delta(\eta,\gamma^+) \end{align} $$
for all n, based on the assumption and the cocycle identity.
Now, for sufficiently large n,
$\eta \gamma ^n$
is hyperbolic according to Lemma 3.7. Hence, we have
$$ \begin{align} \delta(\eta,\gamma^+)=\delta(\eta \gamma^n, \gamma^+)=\delta(\eta \gamma^n, \gamma^+)+\delta(\eta \gamma^n, (\eta\gamma^n)^-). \end{align} $$
Here, the last equality follows from the fact that
$\delta (\eta \gamma ^n,{\kern-1pt} (\eta \gamma ^n)^-){\kern-1pt}={\kern-2pt}-\delta (\eta \gamma ^n,{\kern-1pt} (\eta \gamma ^n)^+){\kern-1pt}={\kern-1pt}0$
.
Since
$(\delta ,f)$
is a B-cocycle, according to Definition 1.2, we have
$$ \begin{align} \delta(\eta \gamma^n, \gamma^+)+\delta(\eta \gamma^n, (\eta\gamma^n)^-)=f(\eta \gamma^n \gamma^+, \eta \gamma^n (\eta\gamma^n)^-)-f(\gamma^+,(\eta\gamma^n)^-). \end{align} $$
The right-hand side of equation (4.3) can be simplified as
$f(\eta \gamma ^+, (\eta \gamma ^n)^-))-f(\gamma ^+,(\eta \gamma ^n)^-))$
.
Taking the limit as
$n\rightarrow +\infty $
, using Lemma 3.7, and equations (4.2) and (4.3), we obtain
$$ \begin{align} \delta(\eta,\gamma^+)=f(\eta\gamma^+,\gamma^-)-f(\gamma^+,\gamma^-). \end{align} $$
Next, choose
$\theta \in \Gamma $
such that
$\theta (\gamma ^+)\neq \gamma ^-$
and
$\theta (\gamma ^-)\neq \gamma ^-$
. This choice is possible according to Lemma 3.5. Applying the same argument as before, but with
$\theta \gamma \theta ^{-1}$
instead of
$\gamma $
, we have the following. When
$\eta \theta \gamma ^+\neq \theta \gamma ^-$
,
$$ \begin{align} \delta(\eta,\theta\gamma^+)=f(\eta\theta\gamma^+,\theta\gamma^-)-f(\theta\gamma^+,\theta\gamma^-). \end{align} $$
Fix two open neighborhoods
$U_1$
and
$U_2$
of
$\gamma ^-$
and
$\theta \gamma ^-$
, respectively, such that
$U_1\cap U_2=\emptyset $
,
$\gamma ^+\notin U_1$
,
$\theta \gamma ^+\notin U_1$
, and
$\theta \gamma ^+\notin U_2$
. The existence of
$U_1$
and
$U_2$
is guaranteed since X is Hausdorff.
Recall that f is continuous, particularly
$f(\cdot ,\gamma ^-):X-\{\gamma ^-\}\rightarrow {\mathbb R}$
is continuous. Since X is compact and
$U_1$
is open, the compliment
$X-U_1$
is compact. By the extreme value theorem, there exists
$N_1$
such that
$|f(x,\gamma ^-)|\leq N_1$
for all
$x\notin U_1$
. Similarly, there exists
$N_2$
such that
$|f(y,\theta \gamma ^-)|\leq N_2$
for all
$y\notin U_2$
.
Set
$N=\max \{N_1, N_2\}$
. Now we are ready to show that
$\delta $
is bounded.
First, we show that
$\delta $
has a uniform bound on one point
$\gamma ^+$
. Consider any element
$\zeta \in \Gamma $
.
Case 1:
$\zeta \gamma ^+\notin U_1$
. Note that
$\zeta \gamma ^+\neq \gamma ^-$
since
$\gamma ^-\in U_1$
. Then by equation (4.4), we have
$$ \begin{align*} |\delta(\zeta,\gamma^+)|=|f(\zeta\gamma^+,\gamma^-)-f(\gamma^+,\gamma^-)|\leq 2N. \end{align*} $$
Case 2:
$\zeta \gamma ^+\in U_1$
. Then,
$\zeta \gamma ^+\notin U_2$
. Since
$\theta \gamma ^-\in U_2$
, we have
$$ \begin{align*} \zeta\theta^{-1}((\theta\gamma\theta^{-1})^+)=\zeta\gamma^+\neq \theta\gamma^-=(\theta\gamma\theta^{-1})^-. \end{align*} $$
By equation (4.5), we have
$$ \begin{align} |\delta(\zeta\theta^{-1},\theta\gamma^+)|=|f(\zeta\gamma^+,\theta\gamma^-)-f(\theta\gamma^+,\theta\gamma^-)|\leq 2N. \end{align} $$
However,
$\theta \gamma ^+\notin U_1$
. Applying case 1, we have
$$ \begin{align} |\delta(\theta,\gamma^+)|=|f(\theta\gamma^+,\gamma^-)-f(\gamma^+,\gamma^-)|\leq 2N. \end{align} $$
Hence, combining equations (4.6) and (4.7), we obtain
$$ \begin{align*} |\delta(\zeta,\gamma^+)|=|\delta(\zeta\theta^{-1}\theta,\gamma^+)|=|\delta(\zeta\theta^{-1},\theta \gamma^+)+\delta(\theta,\gamma^+)|\leq 4N. \end{align*} $$
In both cases, there is a uniform bound:
$|\delta (\zeta ,\gamma ^+)|\leq 4N$
for all
$\zeta \in \Gamma $
.
Second, we show
$\delta $
is bounded on a
$\Gamma $
-orbit. By cocycle identity,
$$ \begin{align*} \delta(\rho,\varrho \gamma^+)=\delta(\rho\varrho,\gamma^+)-\delta(\varrho,\gamma^+), \end{align*} $$
which is bounded by
$8N$
for all
$\rho ,\varrho \in \Gamma $
.
Since the orbit of
$\gamma ^+$
is dense and
$\delta $
is a continuous cocycle,
$\delta $
is globally bounded.
Step 2: Find
$\phi $
as a Borel function. It is well known that a bounded cocycle is a Borel coboundary, that is, it can be written as
$d\varphi '$
for some Borel function
$\varphi '$
. This result can be found, for example, in [Reference Furman11, Proof of Proposition 1]. In our case, since we have
$d(f(x,y)-\varphi '(x)-\varphi '(y))=0$
, it follows that
$f(x,y)-\varphi '(x)-\varphi '(y)$
is a
$\Gamma $
-invariant Borel function.
Step 3:
$\phi $
is essentially continuous. By Definition 1.1,
$\Gamma $
acts ergodicly on
$(X\times X, \mu \times \mu ')$
. We observe that the function
$f(x,y)-\varphi '(x)-\varphi '(y)$
is constant
$\mu \times \mu '$
-almost everywhere (a.e.). Using a Fubini-type argument, we can find a subset A of X such that
$\mu '(A)=1$
and for every
$y_0\in A$
, the function
$f(x,y_0)-\varphi '(x)-\varphi '(y_0)$
is constant
$\mu $
-a.e. Consequently,
$\varphi '(x)$
is essentially equal, up to a constant, to the continuous function
$\varphi _1(x):=f(x,y_0)-\varphi '(y_0)$
. Technically,
$f(y_0,y_0)$
is undefined, so
$\varphi _1$
is only continuous on
$X\setminus \{y_0\}$
. However, by applying the same argument for
$y_1\neq y_0$
in A, we obtain a second function
$\varphi _2(x)$
. Note that
$\varphi _1-\varphi _2$
is a
$\Gamma $
-invariant function (on
$X\setminus (\Gamma y_0\cup \Gamma y_1$
)) that is continuous on
$X-\{y_0,y_1\}$
. By adjusting
$\varphi _2$
by a constant if necessary, we can glue the two functions together to form a new function
$\varphi $
that is continuous on X. It is worth mentioning that shifting a constant does not alter the coboundary of a function.
Since
$d\varphi $
and
$\delta $
are two continuous cocycles that are identical
$\mu $
-a.e., they are the same cocycle, and we conclude that
$\delta =d\varphi $
.
The marked length spectrum
$\ell $
defines a map from
$H^1(\Gamma ,X,{\mathbb R})$
to
$C(\Gamma )^{\mathbb R}$
. We have shown that this map is injective when restricted to
$\text {Ker}([i])$
. However, the image of this map is still unknown to us and remains mysterious.
5. Proof of Theorem B
We will now proceed with the proof of Theorem B.
Proof of Theorem B
The ‘if’ part is clear.
For the ‘only if’ part, let
$(M,g)$
be a closed Riemannian manifold of negative curvature,
$\Gamma =\pi _1(M)$
be its fundamental group,
$\partial \widetilde {M}$
be the boundary of the universal covering, and
$H<\Gamma $
be a subgroup that acts minimally on
$\partial \widetilde {M}$
.
Let
$\mu $
be a symmetric generating measure on H that has a finite first moment with respect to the distance
$d_{\widetilde {g}}$
on
$\widetilde {M}$
, that is,
$\sum _{h\in H}\mu (h)\cdot d_{\widetilde {g}}(e,h)<\infty $
. According to the work of Kaimanovich [Reference Kaimanovich19, §7.3], the Poisson boundary of
$(H,\mu )$
is realized on
$\partial \widetilde {M}$
, specifically, there exists a unique
$\mu $
-stationary probability measure
$\nu $
on
$\partial \widetilde {M}$
, such that
$(\partial \widetilde {M},\nu )$
serves as the Poisson boundary for
$(H,\mu )$
.
Furthermore, by the result of Bader and Furman [Reference Bader and Furman3, Theorem 2.7], we know that the product measure
$\nu \times \nu $
on
$\partial \widetilde {M}\times \partial \widetilde {M}$
is H-ergodic. This implies that
$\partial \widetilde {M}$
is a symmetric H-boundary.
Let
$g_1$
and
$g_2$
be two negatively curved Riemannian metrics on M, and let
$\beta _1,\beta _2\in Z^1_c(\Gamma ,\partial \widetilde {M},{\mathbb R})$
be the Busemann cocycles associated with the lifted metrics
$\widetilde {g}_1$
and
$\widetilde {g}_2$
on
$\widetilde {M}$
. These cocycles are B-cocycles, meaning that there exist (geometrically defined) continuous functions
$f_1,f_2:\partial \widetilde {M}\times \partial \widetilde {M}\setminus \Delta \to {\mathbb R}$
so that
$$ \begin{align*} \beta_i(\gamma,x)+\beta_i(\gamma,y)=f_i(\gamma x,\gamma y)-f_i(x,y)\quad (\gamma\in\Gamma,\ x\ne y\in \partial\widetilde{M},\ i=1,2). \end{align*} $$
The restrictions
$\bar \beta _i: H\times \partial \widetilde {M}\to {\mathbb R}$
are still B-cocycles, denoted by
$(\bar \beta _i,f_i)$
. According to the assumption,
$\ell _{\bar \beta _1}=\ell _{\bar \beta _2}$
. By Theorem C, there exists a continuous map
$\varphi $
such that
$\bar \beta _1-\bar \beta _2=d \varphi $
.
Direct calculation shows that the continuous function
$\partial \widetilde {M}\times \partial \widetilde {M}\setminus \Delta \to {\mathbb R}$
given by
$$ \begin{align*} f_1(x,y)-f_2(x,y)-\varphi(x)-\varphi(y) \end{align*} $$
is H-invariant. Since H is
$\nu \times \nu $
-ergodic and
$\nu $
has full support, the function is
$\nu \times \nu $
-a.e. constant. Being continuous and
$\nu \times \nu $
having full support on
$\partial \widetilde {M}\times \partial \widetilde {M}$
, the functions are actually constant. This implies that
$\beta _1-\beta _2=d \varphi $
. Hence,
$\ell _{\beta _1}=\ell _{\beta _2}$
on
$\Gamma $
. In other words,
$(M,g_1)$
and
$(M,g_2)$
have the same marked length spectrum.
The results for surface and locally symmetric spaces can be deduced from the marked length spectrum rigidity.
6. Extension of B-cocycles
Now we will begin the preparation for the proof of Theorem C.
In this section,
$\Gamma $
will denote a torsion-free uniform arithmetic lattice in a rank-one simple center-free Lie group G. Let K be the maximal compact subgroup of G, and X be the Gromov boundary of
$G/K$
. All cocycles in this section will be defined over X. For a subgroup
$G'<G$
, we define a cocycle
$\omega : G'\times X\rightarrow {\mathbb R}$
as a cocycle of
$G'$
.
Denoting
$\Gamma \backslash G/K$
as the locally symmetric space of
$\Gamma $
, and the double coset
$\Gamma e K\in \Gamma \backslash G/K$
as the base point, we consider
$\alpha $
as the Busemann cocycle of this space and
$(\beta ,f)$
as another B-cocycle of
$\Gamma $
. We define the equivalence relation
$R_\alpha $
on
$C_\Gamma $
as follows:
$$ \begin{align*} ([\gamma_1], [\gamma_2])\in R_\alpha \Longleftrightarrow\, \ell_\alpha ([\gamma_1])=\ell_\alpha([\gamma_2]). \end{align*} $$
Similarly, we define the equivalence relation
$R_\beta $
.
The main goal of this section is to prove the following proposition.
Proposition 6.1. If
$R_\alpha $
is a sub-relation of
$R_\beta $
, then there exists a cocycle
$\bar \beta : G\times X\to {\mathbb R}$
that extends
$\beta $
to the Lie group G.
Before explaining the strategy of the proof of this proposition, let us recall the following definition.
Definition 6.2. [Reference Druţu and Kapovich10, Definition 5.17] Let
$\Gamma <G$
be a subgroup, the commensurator of the group
$\Gamma $
in G is given by
$$ \begin{align*} \text{Comm}_G(\Gamma)= \{ {g\in G}\mid{g\Gamma g^{-1}\cap\Gamma \text{ has finite index in both } \Gamma \text{ and } g\Gamma g^{-1}.} \} \end{align*} $$
An important criterion of Margulis for arithmeticity is the following.
Theorem 6.3. [Reference Margulis25, Theorem 9]
Let G be a connected simple Lie group with a trivial center. A lattice
$\Gamma $
in G is arithmetic if and only if
$\text {Comm}_G(\Gamma )$
is dense in G.
There are two steps involved in this extension. First, we extend
$\beta $
to the commensurator
$\text {Comm}_G(\Gamma )$
of
$\Gamma $
in G. Then, we further extend
$\beta $
from
$\text {Comm}_G(\Gamma )$
to the entire group G using a density-type argument. In this process, we make use of the following observation. Since
$\alpha $
is a restriction of a G-cocycle
$G\times X\to {\mathbb R}$
, if
$\gamma _1$
,
$\gamma _2\in \Gamma $
are conjugate in G, then
$([\gamma _1], [\gamma _2]))\in R_\alpha $
and consequently
$([\gamma _1], [\gamma _2])\in R_\beta $
.
6.1. Extension to the commensurability subgroup
Let s be an element of
$\text {Comm}_G(\Gamma )$
. By definition, there exist finite index subgroups
$\Gamma '$
,
$\Gamma "$
of
$\Gamma $
such that
$s\Gamma 's^{-1}=\Gamma "$
. We denote the restrictions of
$\beta :\Gamma \times X\to {\mathbb R}$
to
$\Gamma '$
and
$\Gamma "$
as
$\beta '$
and
$\beta "$
, respectively. We can define another cocycle
$s_\ast \beta "$
of
$\Gamma '$
as follows:
$$ \begin{align*} s_\ast\beta"(\gamma,\xi)=\beta"(s\gamma s^{-1}, s\xi)\quad (\gamma\in\Gamma'). \end{align*} $$
Similarly, we can construct
$\alpha '$
and
$s_\ast \alpha "$
.
Since
$\gamma $
and
$s\gamma s^{-1}$
are conjugate in G, it follows that
$([\gamma ]_\Gamma , [s\gamma s^{-1}]_\Gamma )\in R_{\alpha }\subset R_{\beta }$
for
$\gamma \in \Gamma '$
, and therefore,
$\ell _{\beta '}=\ell _{s_\ast \beta "}$
on
$\Gamma '$
. Note that
$(s_\ast \beta ", f_s)$
is a B-cocycle, where
$f_s(x,y)=f(s x,s y)$
. Applying Theorem C to
$\Gamma '$
, we obtain a continuous function
$\varphi _s$
such that
$s_\ast \beta "-\beta '=\delta \varphi _s$
.
Now, consider the function
$f(s\xi ,s\eta )-f(\xi ,\eta )-\varphi _s(\xi )-\varphi _s(\eta )$
, which is a
$\Gamma '$
-invariant continuous function. Let
$\mu $
be a G-invariant measure on
$X\times X\setminus \Delta $
. Recall that
$\Gamma '$
acts ergodically on
$(X\times X\setminus \Delta ,\mu )$
by Howe–Moore’s ergodicity theorem. Therefore,
$f(s\xi ,s\eta )-f(\xi ,\eta )-\varphi _s(\xi )-\varphi _s(\eta )$
is
$\mu $
-a.e. a constant. Since it is a continuous function and
$\mu $
has full support, the function is constant. For any
$\eta \neq \xi $
, we define the cocycle
$\hat \beta :\text {Comm}_G(\Gamma )\times X\to {\mathbb R}$
as follows:
$$ \begin{align*} \hat\beta(s,\xi)=\varphi_s(\xi)+\frac{f(s\xi,s\eta)-f(\xi,\eta)-\varphi_s(\xi)-\varphi_s(\eta)}{2}. \end{align*} $$
Note that the right-hand side is independent of
$\eta $
.
Indeed, it is clear that
$$ \begin{align*} f(s\xi,s\eta)-f(\xi,\eta)=\hat\beta(s,\xi)+\hat\beta(s,\eta). \end{align*} $$
We now show that
$\hat \beta $
defines a cocycle of
$\text {Comm}_G(\Gamma )$
. Let
$X^{(3)}$
be the set of pairwise distinct triple of points in X. Define a new function
$h:X^{(3)}\to {\mathbb R}$
by
${h(\xi ,\eta ,\omega )=f(\xi ,\eta )+f(\xi ,\omega )-f(\eta ,\omega )}$
. It is straightforward to see that for all
$s\in \text {Comm}_G(\Gamma )$
, and
$(\xi ,\eta ,\omega )\in X^{(3)}$
, we have
$$ \begin{align*} h(s \xi,s \eta,s \omega)-h(\xi,\eta,\omega)=2\hat\beta(s,\xi). \end{align*} $$
The left-hand side is a cocycle from
$\text {Comm}_G(\Gamma )\times X^{(3)}$
to
${\mathbb R}$
. Hence, the right-hand side is also a cocycle in the same form. However, the right-hand side depends only on the first factor and induces a cocycle of
$\text {Comm}_G(\Gamma )$
.
It implies that
$\hat \beta (s,\xi )$
is a B-cocycle of
$\text {Comm}_G(\Gamma )$
.
6.2. Extension to G
By the fact that
$\Gamma $
is arithmetic and Margulis’s Theorem 6.3, we know that
$\text {Comm}_G(\Gamma )$
is dense in G with respect to the Hausdorff topology.
We will now conclude the argument by using the following lemma.
Lemma 6.4. Let
$L<G$
be a dense subgroup with a B-cocycle
$(\hat \beta , f)$
of L. Then
$\hat \beta $
extends to a B-cocycle
$(\bar \beta ,f)$
of G.
Proof. As before, define
$h(\xi ,\eta ,\omega )=f(\xi ,\eta )+f(\xi ,\omega )-f(\eta ,\omega )$
on
$X^{(3)}$
. It is clear that for all
$l\in L$
,
$(\xi ,\eta ,\omega )\in X^{(3)}$
, we have
$$ \begin{align*} h(l \xi,l \eta,l \omega)-h(\xi,\eta,\omega)=2\hat\beta(l,\xi). \end{align*} $$
Now, let us fix an arbitrary element
$g\in G$
, and a sequence
$\{l_n\}_{n=1}^{\infty }$
in L such that
$\lim _{n\rightarrow \infty } l_n=g.$
For any element
$l\in L$
, we can identify
$\hat \beta (l,\cdot )$
as a function on
$X^{(3)}$
. With this in mind, we have
$$ \begin{align*} 2\hat\beta(l_n,\xi)-2\hat\beta(l_m,\xi)=h(l_n \xi,l_n \eta, l_n \omega)-h(l_m \xi,l_m \eta, l_m \omega). \end{align*} $$
The right-hand side can be written as
$$ \begin{align*} \Omega(n,m):=[h(l_n \xi,l_n \eta, l_n \omega)-h(g\xi,g\eta,g\omega)]-[h(l_m \xi,l_m \eta, l_m \omega)-h(g\xi,g\eta,g\omega)], \end{align*} $$
where
$\Omega (m,n): X^{(3)}\to {\mathbb R}$
is a family of functions indexed by
${\mathbb Z}^2$
.
Fixing
$\xi $
, we can choose
$\eta $
,
$\omega $
such that the three points are pairwise different. Since G acts continuously on X, we have
$\lim _{n,m\rightarrow \infty }\Omega (m,n)=0$
uniformly on compact subsets of pairwise different triples. In other words,
$\hat \beta (l_n,x)$
converges uniformly on compact sets. This implies that
$\hat \beta (l_n,\xi )$
is a Cauchy sequence that uniformly converges to some continuous function, denoted by
$\bar \beta : G\times X\to {\mathbb R}$
.
We then have
$$ \begin{align*} h(g\xi,g\eta,g\omega)-h(\xi,\eta,\omega)=2\bar\beta(g,\xi). \end{align*} $$
Hence, as before,
$\bar \beta $
is a cocycle of G, and
$\bar \beta (g,\xi )+\bar \beta (g,\eta )=f(g\xi ,g\eta )-f(\xi ,\eta )$
by the definition of h.
It implies that
$\bar \beta $
is a B-cocycle of G.
7. Marked length pattern rigidity
We will now prove the marked length pattern rigidity, which is stated in Theorem A.
Proof of Theorem A
One direction is trivial. We will show the hard direction: the ‘only if’ part of the proof.
Let
$\Gamma $
be the fundamental group of a closed arithmetic locally symmetric rank-one manifold
$(M,g_0)$
, and let
$(M,g)$
be another closed Riemannian manifold with negative sectional curvature on M.
We consider the Riemannian universal covers
$(\widetilde {M},\widetilde {g_0})$
and
$(\widetilde {M},\widetilde {g})$
of
$(M,g_0)$
and
$(M,g)$
, respectively. We first establish an identification between their Gromov boundaries.
The lifting map of the identity map of M is a quasi-isometry of
$(\widetilde {M},\widetilde {g_0})$
and
$(\widetilde {M},\widetilde {g})$
by the Švarc–Milnor lemma [Reference Bridson and Haefliger6, Proposition I.8.19]. This quasi-isometry extends to a
$\Gamma $
-equivalent homeomorphism
$\Phi $
of their boundaries [Reference Bridson and Haefliger6, Theorem III.H.3.9].
We can pull back the Busemann cocycle
$\beta '$
defined on the boundary of
$(\widetilde {M},\widetilde {g})$
to a cocycle on the boundary of
$(\widetilde {M},\widetilde {g_0})$
, denoted by
$\beta $
. More specifically,
$\beta (\gamma , \xi )=\beta '(\gamma , \Phi \xi )$
, where
$\Phi $
is the homeomorphism between the boundaries induced by the lifting map. It is important to note that the pullback operation does not change the marked length spectrum of cocycles, as
$\Phi $
maps
$\gamma ^+\in \partial (\widetilde {M}, g_0)$
to
$\gamma ^+\in \partial (\widetilde {M},g)$
for all
$\gamma \in \Gamma \setminus \{1\}$
. Moreover, the pullback of a B-cocycle is still a B-cocycle.
Let
$\alpha $
be the Busemann cocycle for the locally symmetric manifold
$(M,g)$
with base-point
$[e]$
as described in §6. Here,
$\alpha $
is a restriction of a cocycle
$\bar {\alpha }$
defined on G. According to the assumption,
$R_\alpha $
is a sub-relation of
$R_\beta $
. Proposition 6.1 implies that
$\beta $
can be extended to a cocycle
$\bar {\beta }$
for G.
Let P be the minimal parabolic subgroup of G and A be the maximal
${\mathbb R}$
-split torus (Cantan subgroup) contained in P. It is known that the Borel G-cocycle on the Furstenberg boundary
$X=G/P$
, up to strictly equivalence, are classified by
$\text {Hom}(P,{\mathbb R})=\text {Hom}(A,{\mathbb R})$
up to equivalence (Proposition 2.6). In the case of rank-one Lie group G,
$A={\mathbb R}$
. Since
$\text {Hom}({\mathbb R},{\mathbb R})={\mathbb R}$
, and
$\bar {\alpha }$
is nontrivial, each class in
$\text {Hom}(A,{\mathbb R})$
has a B-cocycle representative
$\unicode{x3bb} \bar {\alpha }$
for some
$\unicode{x3bb} \in {\mathbb R}$
.
It follows that the continuous B-cocycle
$\bar {\beta }$
is strictly equivalent to
$\unicode{x3bb} \bar {\alpha }$
for some
$\unicode{x3bb} \in {\mathbb R}$
. Since the marked length spectrum of a cocycle is an invariant under strict equivalence, we conclude that there exists
$\unicode{x3bb} $
such that
$\unicode{x3bb} \ell _g=\ell _{g'}$
. Furthermore, by Theorem C, we have
$\beta -\unicode{x3bb} \alpha =d \varphi $
for some continuous map
$\varphi $
.
Then the Riemannian manifold
$(M,({1}/{\unicode{x3bb} })g)$
has the same marked length spectrum as
$(M,g_0)$
. According to marked length spectrum rigidity for locally symmetric manifolds [Reference Hamenstädt16],
$(M,g_0)$
and
$(M,({1}/{\unicode{x3bb} })g)$
are isometric.
8. Hyperbolic surfaces without marked length pattern rigidity
In this section, we will demonstrate that most finite-volume complete hyperbolic surfaces do not possess marked length pattern rigidity.
8.1. Fricke moduli
We begin by introducing a suitable coordinate system in the Teichmuller space, known as the Fricke space [Reference Imayoshi and Taniguchi18], which will be useful for our purposes.
Let S be a surface with genus g and n punctures where
$2-2g-n<0$
. We denote
$\Gamma $
as the fundamental group of S and
$T_{g,n}$
as the Teichmuller space of S. When a hyperbolic structure is imposed on S, it induces a lattice embedding of
$\Gamma $
into
$G_1:=\operatorname {PSL}_2({\mathbb R})$
. We lift this representation to a representation
$\bar {\pi }$
of a possibly index-two covering of
$\Gamma $
, to
$G=\operatorname {SL}_2({\mathbb R})$
.
We consider a canonical system of generators {
$\alpha _i$
,
$\beta _i$
,
$\gamma _j$
}, (
$1\leq i\leq g$
,
$1\leq j\leq n$
) for the fundamental group
$\Gamma $
, where the generators satisfy the fundamental relation:
$$ \begin{align*} \prod_{i=1}^g[\alpha_i,\beta_i]\prod_{j=1}^n\gamma_j=e. \end{align*} $$
Assuming that
$\pi $
is a lattice embedding of
$\Gamma $
into
$G_1$
, up to conjugacy, we impose the following normalization conditions for the lifting representation:
-
(1)
$\pi (\alpha _1)$
has its repelling and attractive fixed points at 0 and
$\infty $
, respectively, -
(2)
$\pi (\beta _1)$
has a fixed point at 1.
For the lifting representation, let the matrix representation of
$\bar \pi (\alpha _1)$
and
$\bar \pi (\beta _1)$
be given by
$$ \begin{align*} \begin{array}{ll} \begin{pmatrix} \unicode{x3bb}&0\\ 0&\unicode{x3bb}^{-1} \end{pmatrix},& \unicode{x3bb}>1,\\ \begin{pmatrix} a_1&b_1\\ c_1&d_1 \end{pmatrix}, &a_1d_1-b_1c_1=1, \quad a_1+b_1=c_1+d_1>0, \end{array} \end{align*} $$
respectively. Also, for
$2\leq i\leq g$
,
$\bar \pi (\alpha _i)$
and
$\bar \pi (\beta _i)$
are represented uniquely by the matrices:
$$ \begin{align*} \begin{array}{lll} \begin{pmatrix} a_i&b_i\\ c_i&d_i \end{pmatrix},&a_i d_i-b_i c_i=1,&c_i>0,\\ \begin{pmatrix} a^{\prime}_i&b^{\prime}_i\\ c^{\prime}_i&d^{\prime}_i \end{pmatrix},&a^{\prime}_i d^{\prime}_i-b^{\prime}_i c^{\prime}_i=1,&c^{\prime}_i>0. \end{array} \end{align*} $$
Similarly, for
$1\leq j\leq n$
,
$\bar \pi (\gamma _j)$
is written uniquely in the form:
$$ \begin{align*}\begin{pmatrix} e_j&f_j\\ g_j&h_j \end{pmatrix}\end{align*} $$
with
$e_j h_j-f_j g_j=1$
,
$e_j+h_j=2$
.
We define the Fricke coordinates by assigning a lattice embedding to the sequence
$(a_i, c_i, d_i, a^{\prime }_i, c^{\prime }_i, d^{\prime }_i, e_j, g_j)$
, where
$2\leq i\leq g$
,
$1\leq j\leq n$
. In [Reference Imayoshi and Taniguchi18, Theorem 2.25], it is shown that the Fricke coordinates defines an embedding of the Teichmuller space
$T_{g,n}$
into
${\mathbb R}^{6g-6+2n}$
.
We recall the algorithm to recover
$\bar \pi $
, and hence
$\pi $
, from its Fricke coordinates. For more details, please refer to [Reference Imayoshi and Taniguchi18, p. 49].
It is clear that
$\bar \pi (\alpha _i)$
,
$\bar \pi (\beta _i)$
, and
$\bar \pi (\gamma _j)$
, where
$2\leq i\leq g$
and
$1\leq j\leq n$
are uniquely determined by the Fricke coordinates. It is worth noting that all the
$b_i$
,
$b^{\prime }_i$
(
$2\leq i\leq g$
),
$f_j$
, and
$h_j\ (1 \leq j \leq n)$
can be expressed as rational functions of the Fricke coordinates.
To show that
$\bar \pi (\alpha _1)$
and
$\bar \pi (\beta _1)$
are determined by the Fricke coordinate, consider the element
$$ \begin{align*} \bigg(\bar\pi\bigg(\prod_{i=2}^g[\alpha_i.\beta_i]\prod_{1=1}^n\gamma_j\bigg)\bigg)^{-1}=\begin{pmatrix} a&b\\ c&d \end{pmatrix}. \end{align*} $$
Using the fundamental relation of
$\Gamma $
:
$$ \begin{align*}\begin{pmatrix} \unicode{x3bb}&0\\ 0&\unicode{x3bb}^{-1} \end{pmatrix} \begin{pmatrix} a_1&b_1\\ c_1&d_1 \end{pmatrix} \begin{pmatrix} \unicode{x3bb}^{-1}&0\\ 0&\unicode{x3bb} \end{pmatrix}=\pm\begin{pmatrix} a&b\\ c&d \end{pmatrix}\begin{pmatrix} a_1&b_1\\ c_1&d_1 \end{pmatrix}.\end{align*} $$
Note that the negative sign is possible because not every representation into
$\operatorname {PSL}_2({\mathbb R})$
lifts to a representation (of the same group) into
$\operatorname {SL}_2({\mathbb R})$
.
By replacing
$(\begin {smallmatrix} a&b\\ c&d \end {smallmatrix})$
by
$(\begin {smallmatrix} -a&-b\\ -c&-d \end {smallmatrix})$
, if
$$ \begin{align*}\begin{pmatrix} \unicode{x3bb}&0\\ 0&\unicode{x3bb}^{-1} \end{pmatrix} \begin{pmatrix} a_1&b_1\\ c_1&d_1 \end{pmatrix} \begin{pmatrix} \unicode{x3bb}^{-1}&0\\ 0&\unicode{x3bb} \end{pmatrix}=-\begin{pmatrix} a&b\\ c&d \end{pmatrix}\begin{pmatrix} a_1&b_1\\ c_1&d_1 \end{pmatrix},\end{align*} $$
we have
$$ \begin{align} \unicode{x3bb}^2=\frac{a-1}{1-d}. \end{align} $$
Additionally,
$$ \begin{align*}a_1=\frac{b}{1-a}c_1,\end{align*} $$
$$ \begin{align*}d_1=\frac{c}{1-d}b_1.\end{align*} $$
Using
$a_1+b_1=c_1+d_1$
, we get
$$ \begin{align*}\frac{a+b-1}{1-a}c_1=\frac{c+d-1}{1-d}b_1.\end{align*} $$
Recalling that
$a_1d_1-b_1c_1=1$
, it follows that
$$ \begin{align} c^2_1\bigg[\frac{bc(a+b-1)}{(1-a)^2(c+d-1)}-\frac{c+d-1}{1-d}\bigg]=1. \end{align} $$
Equations (8.1) and (8.2) demonstrate that
$\unicode{x3bb} ^2$
and
$c_1^2$
are rational functions of the Fricke coordinates. Therefore, we can view the Teichmuller space as a subset of
${\mathbb R}^{6g-6+2n}$
via Fricke coordinates.
8.2. Length of geodesic and Horowitz’s theorem
For any hyperbolic element A in
$\operatorname {PSL}_2({\mathbb R})$
, the translation length
$\ell _A$
determines its trace
$\mathrm {tr}(A)$
as
$|\mathrm {tr}(A)| = 2\cosh (\ell _A)$
. Therefore, the marked length pattern of a Fuchsian group can be determined by the traces of its elements.
In the case of a lattice representation
$\pi $
of the surface group
$\Gamma $
into
$\operatorname {PSL}_2({\mathbb R})$
, the Fricke coordinates X determine the values of
$\unicode{x3bb} $
and
$c_1$
through equations (8.1), (8.2), as well as the normalization conditions.
Let s and t be two real numbers with
$st\neq 0$
. Consider the family of representations
$\pi ^{\prime }_{s,t}: F_{2g+n}\rightarrow \operatorname {GL}_2({\mathbb R})$
defined as follows:
$$ \begin{align*}\pi_{s,t}'(\alpha_1)=\begin{pmatrix} s&0\\ 0&s^{-1} \end{pmatrix},\end{align*} $$
$$ \begin{align*}\pi_{s,t}'(\beta_1)=\begin{pmatrix} \dfrac{bt}{1-a}&\dfrac{(1-d)(a+b-1)t}{(1-a)(c+d-1)}\\[8pt] t&\dfrac{(a+b-1)ct}{(1-a)(c+d-1)} \end{pmatrix},\end{align*} $$
$$ \begin{align*}\pi_{s,t}'(\alpha_i)=\begin{pmatrix} a_i&\dfrac{a_i d_i-1}{c_i}\\ c_i&d_i \end{pmatrix},\end{align*} $$
$$ \begin{align*}\pi_{s,t}'(\beta_i)=\begin{pmatrix} a^{\prime}_i&\dfrac{a^{\prime}_id^{\prime}_i-1}{c^{\prime}_i}\\ c^{\prime}_i&d^{\prime}_i \end{pmatrix},\end{align*} $$
$$ \begin{align*}\pi_{s,t}'(\gamma_j)=\begin{pmatrix} e_j&\dfrac{2e_j-e^2_j-1}{g_j}\\ g_j&2-e_j \end{pmatrix},\end{align*} $$
for
$2\leq i\leq g$
and
$1\leq j\leq n$
, where a, b, c, d are determined by the equation
$$ \begin{align*}\begin{pmatrix} a&b\\ c&d \end{pmatrix}\prod_{i=2}^g[\pi_{s,t}'(\alpha_i),\pi_{s,t}'(\beta_i)]\prod_{j=1}^n\pi^{\prime}_{s,t}(\gamma_j)=\pm\begin{pmatrix} 1&0\\ 0&1 \end{pmatrix},\end{align*} $$
as before. It is worth noting that when
$s=\unicode{x3bb} $
and
$t=c_1$
, the representation
$\pi ^{\prime }_{\unicode{x3bb} ,c_1}$
corresponds to the lifting representation
$\bar \pi $
.
It is evident that all the traces of
$\pi _{s,t}'(w)$
, where
$w\in F_{2g+n}$
, are rational functions of X, s, and t. By induction, it can be shown that for any word
$w\in F_{2g+n}$
,
$\text {tr}(\pi ^{\prime }_{s,t}(w))$
has the form:
$$ \begin{align*}\text{tr}(\pi^{\prime}_{s,t}(w))=t^k\sum_{l=-\infty}^\infty \omega_{i}s^l, \end{align*} $$
where
$\omega _{i}$
are rational functions on X,
$k\in {\mathbb Z}$
, and all but a finite number of
$\omega _{i}$
are 0.
When s and t are fixed, there are four different representations
$\pi ^{\prime }_{\pm s,\pm t}$
, all with the same traces up to sign. Specifically, we just replace
$\pi _{s,t}'(\alpha _1)$
and
$\pi ^{\prime }_{s,t}(\beta _1)$
by
$\pm \pi ^{\prime }_{s,t}(\alpha _1)$
and
$\pm \pi ^{\prime }_{s,t}(\beta _1)$
, respectively. Consequently,
$(\text {tr}(\pi ^{\prime }_{\pm s,\pm t}(w)))^2$
is independent of the choice of these four representations. Therefore,
$(\text {tr}(\pi ^{\prime }_{s,t}(w)))^2$
has the form:
$$ \begin{align*}(\text{tr}(\pi_{s,t}'(w)))^2=t^{2k}\sum_{l=-\infty}^\infty \Omega_{i}s^{2l}, \end{align*} $$
where
$\Omega _{i}$
are rational functions on X, and all but a finite number of
$\Omega _{i}$
are 0.
Indeed, when we set
$s=\unicode{x3bb} $
,
$t=c_1$
, the representation
$\pi ^{\prime }_{\unicode{x3bb} ,c_1}$
is a valid representation of
$F_{2g+n}$
to
$\operatorname {SL}_2({\mathbb R})$
. By equations (8.1) and (8.2), we can see that
$(\text {tr}(\pi _{\unicode{x3bb} , c_1}'(w)))^2$
is a rational function on X for all
$w\in F_{2g+n}$
. Furthermore, we have
$$ \begin{align*} \pi_{\unicode{x3bb}, c_1}'\bigg(\prod_{i=1}^g[\alpha_i,\beta_i]\prod_{j=1}^n\gamma_j\bigg)=\pm \begin{pmatrix} 1&0\\ 0&1 \end{pmatrix}. \end{align*} $$
Therefore, the representation
$\pi _{\unicode{x3bb} , c_1}'$
is a lifting of the representation
$\pi $
. The natural projection p from
$F_{2g+n}$
to
$\Gamma $
maps to the fundamental group. It can be observed that the traces of
$\pi $
and
$\pi ^{\prime }_{\unicode{x3bb} ,c_1}$
are the same up to sign. In other words, for any
$w\in F_{2g+n},$
we have
$$ \begin{align} \mathrm{{tr}(\pi_{\unicode{x3bb},c_1}'(w))}=\pm \mathrm{{tr}}(\pi(p(w))). \end{align} $$
We can deduce that
$(\mathrm {{tr}}(\pi (\gamma )))^2$
are rational functions on X for all
$\gamma \in \Gamma $
. Since Teichmuller space is an open subset of
${\mathbb R}^{6g-6+2n}$
in the Hausdorff topology, it is Zariski dense. As a result, there exist unique extensions of
$(\mathrm {{tr}}(\pi (\gamma )))^2$
as rational functions on
${\mathbb R}^{6g-6+2n}$
. We refer to the function
$(\mathrm {{tr}}(\pi (\gamma )))^2$
as the rational function of
$\gamma $
and denote it by
$Q_\gamma $
.
A stronger version of this result can be obtained from a theorem by Horowitz [Reference Horowitz17].
Theorem 8.1. Let
$F=\langle s_1,s_2,\ldots , s_m\rangle $
be a free group on m generators. For any word
$w\in F$
, there exists a polynomial
$P_w$
depending only on w, with integer coefficients in
$2^m-1$
variables such that for any representation
$\phi : F\rightarrow \operatorname {SL}_2({\mathbb R})$
, we have
$$ \begin{align*}\mathrm{{tr}}(\phi(w))=P_w(t_1,t_2,\ldots, t_{12},\ldots,t_{12\ldots m}),\end{align*} $$
where
$t_{i_1i_2\cdots i_v}=\mathrm {{tr}}(\phi (s_{i_1}s_{i_2}\cdots s_{i_v}))$
for all
$1\leq i_1<i_2<\cdots <i_v\leq m$
.
We define a relation
$R_{\mathrm {\min }}$
on
$\Gamma $
as follows:
$\gamma R_{\mathrm {\min }}\eta $
when
$Q_\gamma =Q_\eta $
. It is clear that for any hyperbolic metric g on S,
$R_{\mathrm {\min }}$
is a sub-relation of
$R_g$
. It has been known for a long time that there are elements in different conjugacy classes with the same rational function [Reference Horowitz17, Reference Randol31]. Hence,
$R_{\mathrm {\min }}$
is not a trivial relation.
For any
$\gamma $
and
$\eta \in \Gamma $
such that
$(\gamma ,\eta )\notin R_{\mathrm {\min }}$
, the rational equation
$Q_\gamma =Q_\eta $
defines an algebraic subset of
${\mathbb R}^{6g-6+2n}$
. The intersection of this set and the Teichmuller space is a subset of positive codimension. Since there are only countably many pairs of elements, we obtain countably many algebraic subsets of positive codimension. Based on dimension considerations or by referring to the Lebesgue measure, we conclude that the union of all these countable subsets of positive codimension, denoted as
$T_{\mathrm {singular}}$
, is a proper subset of the Teichmuller space.
In particular, by choosing two points in the complementary of
$T_{\mathrm {singular}}$
, we obtain two hyperbolic metrics
$g_1$
and
$g_2$
such that
$R_{\mathrm {\min }}=R_{g_1}=R_{g_2}$
. This implies that
$g_1$
does not share marked length pattern rigidity.
We have shown Theorem D.
Acknowledgements
This work was conducted at the University of Illinois at Chicago and forms part of the author’s thesis. I am immensely grateful to my supervisor, Alexander Furman, for his invaluable guidance, support, and expertise throughout this research. His mentorship played a crucial role in overcoming challenges. I also extend my gratitude to Andrey Gogolyev and the referees for their valuable insights and constructive feedback, which enhanced the quality of this work.
A Appendix. General geometric boundaries
In this Appendix, we extend our results to general acylindrical groups. The proof follows a similar approach as in the paper, with some necessary modifications. We will outline the key changes and restate the theorems in this context.
In the context of general acylindrical groups, the Poisson–Furstenberg boundaries may not necessarily correspond to geometric boundaries due to the presence of elliptic and parabolic elements. To address this issue, we introduce a notation to distinguish the behavior of hyperbolic elements.
Let X be a
$\Gamma $
-space and
$\gamma {\kern-1pt}\in{\kern-1pt} \Gamma $
acts hyperbolicly on X. For any
$\eta {\kern-1pt}\in{\kern-1pt} \Gamma $
such that
$\eta \gamma ^+{\kern-1pt}\neq{\kern-1pt} \gamma ^-$
, let
$$ \begin{align*} A_{\eta,\gamma}= \{ {n}\mid{\eta\gamma^n \ \mathrm{is\ hyperbolic}} \} = \{ {n}\mid{\gamma^n\eta \ \mathrm{ is\ hyperbolic}} \}. \end{align*} $$
The equality in the last equation holds because
$\eta \gamma ^n=\eta (\gamma ^n\eta )\eta ^{-1}.$
Notice that
$A_{\eta ,\gamma }=A_{\eta ^{-1},\gamma ^{-1}}$
.
Definition A.1. Let
$\Gamma $
be a topological group. A non-trivial compact Hausdorff
$\Gamma $
-space X is called a general geometric boundary if the following conditions hold:
-
(1)
$\Gamma $
acts on X minimally; -
(ii) there exist hyperbolic elements in
$\Gamma $
and for every hyperbolic element
$\gamma $
, let
$\eta \in \Gamma $
such that
$\eta \gamma ^+\neq \gamma ^-$
, we have
$\sup A_{\eta ,\gamma }=+\infty $
; -
(3) there exist
$\Gamma $
-quasi invariant measures
$\mu $
,
$\mu '$
on X such that the product measure
$\mu \times \mu '$
is ergodic under the action of
$\Gamma $
.
If
$\mu =\mu '$
, we call X a symmetric general geometric boundary.
We just replace condition (2) in the definition of geometric boundary by condition (ii).
First, we show that general acylindrical groups and their Possion–Furstenberg boundaries are general geometric boundaries.
Recall that a group G is called acylindrically hyperbolic if the group G admits a non-elementary acylindrical isometric action on a geodesic (Gromov-)hyperbolic space M. Maher and Tiozzo showed in [Reference Maher and Tiozzo24] that the Furstenberg–Poisson boundary of a spread-out generating measure, which has finite entropy and finite logarithmic moment on G, is the same as the limit set of G in
$\partial M$
with the hitting measure. By the work of Bader and Furman [Reference Bader and Furman3], condition (3) follows. Condition (1) is true since G acts on its limit set minimally.
For condition (ii), recall that there is visual metric d on the limit set X. For a hyperbolic element
$\gamma $
and any compact subset
$K\subset X-\{\gamma ^-\}$
, there exist L and
$\kappa>0$
such that
$d(\gamma ^n x,\gamma ^+)\leq L\exp (-n\kappa )$
for all
$x\in K$
.
Now let
$\eta \in \Gamma $
. Here,
$\eta $
acts on
$(X,d)$
by Lipschitz homemorphism. Take an open neighborhood U of
$\eta \gamma ^+$
with
$\gamma ^-\notin U$
. There exists an integer N such that for all
$n\geq N$
,
$\gamma ^n U\subset \eta ^{-1}U$
. If necessary, increase n until
$\eta \gamma ^n|_U$
becomes a contraction. Hence, there is a contracting fixed point of
$\eta \gamma ^n$
in U. By the classification of isometries of hyperbolic spaces,
$\eta \gamma ^n$
is hyperbolic for all sufficiently large n.
Second, all lemmas in §3 still hold. The proofs follow a similar approach as in the paper. The statement of Lemma 3.7 in this case is slightly different, and hence we reproduce it here.
Lemma A.2. Let
$\gamma $
,
$\eta \in \Gamma $
with
$\gamma $
hyperbolic. If
$\eta \gamma ^+\neq \gamma ^-$
, then we have
$$ \begin{align*}\lim_{n\in A_{\eta,\gamma},n\rightarrow \infty}(\eta\gamma^n)^+=\eta\gamma^+,\end{align*} $$
$$ \begin{align*}\lim_{n\in A_{\eta,\gamma},n\rightarrow \infty}(\eta\gamma^n)^-=\gamma^-.\end{align*} $$
Define the marked length spectrum function for a general cocycle
$\beta $
on a general geometric boundary X by setting
$$ \begin{align} \ell_\beta([\gamma])=\beta(\gamma,\gamma^+) \end{align} $$
for all hyperbolic elements
$\gamma \in \Gamma \setminus \{1\}$
.
The same proof of Theorem C using these lemmas gives the following theorem.
Theorem A.3. Let
$\Gamma $
-space X be a general geometric boundary and
$\alpha , \beta : \Gamma \times X\rightarrow {\mathbb R}$
two B-cocycles. If
$\ell _\alpha =\ell _\beta $
, then
$\alpha -\beta =d \varphi $
for some continuous function
$\varphi $
. In other words,
$[\alpha ]=[\beta ]$
in
$H^1_c(\Gamma ,X,{\mathbb R})$
.
We generalize the definition of the marked length spectrum function to finite volume negatively curved manifolds.
Let
$(M,g)$
be a complete finite volume manifold with negative curvature, and let
$\Gamma =\pi _1(M)$
be its fundamental group, which may contain parabolic elements. For a parabolic element
$\gamma \in \Gamma $
, there are arbitrarily short (non-closed) geodesics that represent
$\gamma $
. We adopt the convention that
$\ell _g([\gamma ])=0$
for parabolic classes. This function is sometimes referred to as the minimal marked length spectrum. It represents the infimum of the lengths of all closed geodesics in the class
$[\gamma ]$
.
We define the marked length pattern in the same way as before:
$$ \begin{align*} R_g= \{ {(c_1,c_2)\in C_\Gamma\times C_\Gamma}\mid{\ell_g(c_1)=\ell_g(c_2)} \}. \end{align*} $$
As an application of Theorem A.3, using the same construction as in this paper, we are able to prove the following two theorems.
Theorem A.4. Let
$(M,g_1)$
and
$(M,g_2)$
be two arbitrary finite volume complete closed strictly negatively curved Riemannian metrics on a manifold M with fundamental group
$\Gamma $
. Let H be a subgroup of
$\Gamma $
such that the limit set of H is all of
$\partial \widetilde {M}$
.
Then,
$\ell _{g_1}=\ell _{g_2}$
on classes from H if and only if
$\ell _{g_1}=\ell _{g_2}$
on all of
$\Gamma $
.
Theorem A.5. Let
$(M,g_0)$
be a finite volume arithmetic locally symmetric manifold of rank 1, and let g be an arbitrary strictly negatively curved complete finite volume Riemannian metric on M. Then
$R_{g_0}\subset R_g$
if and only if
$\ell _{g_0}=\unicode{x3bb} \ell _{g}$
for some
$\unicode{x3bb}>0$
.
Remark A.6. In [Reference Cao8], Cao showed that if two orientable, uniform visibility surfaces of finite area and bounded non-positive curvature have the same marked length spectrum, then they must be isometric. Hence, we can strengthen Theorems A.4 and A.5 in dimension 2.
In higher dimensions, the marked length spectrum rigidity is not known. Peigné and Sambusetti [Reference Peigné and Sambusetti29] showed the following:
Let M be a finite volume n-manifold with pinched negative curvature, specifically satisfying
$-b^2\leq \kappa \leq -1$
. Assume that M is homotopy equivalent to a locally symmetric manifold
$M_0$
with curvature normalized between
$-$
4 and
$-$
1. If M and
$M_0$
have the same marked length spectrum, then they are isometric.
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