We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.
We use bounded cohomology to define a notion of volume of an SO(n,1)-valued representation of a lattice SO(n,1) and, using this tool, we give a complete proof of the volume rigidity theorem of Francaviglia and Klaff in this setting. Our approach gives in particular a proof of Thurston's version of Gromov's proof of Mostow Rigidity (also in the non-cocompact case), which is dual to the Gromov--Thurston proof using the simplicial volume invariant.}
We consider a coarse version of the marked length spectrum rigidity: given a group with two left invariant metrics, if the marked length spectrum (the translation length function) under the two metrics are the same, then the two metrics are uniformly close. We prove the rigidity theorem for relatively hyperbolic groups. This generalizes a result of Fujiwara.
We develop further basic tools in the theory of continuous bounded cohomology of locally compact groups. We apply this tools to establish a Milnor-Wood type inequality in a very general context and to prove a global rigidity result which was originally announced by the authors with a sketch of a proof using bounded cohomology techniques and then proven by Koziarz and Maubon using harmonic map techniques. As a corollary one obtains that a lattice in SU(p,1) cannot be deformed nontrivially in SU(q,1), if either p is at least 2 or the lattice is cocompact. This generalizes to noncocompact lattice…
Suppose G is a hyperbolic group whose boundary has topological dimension k. If the boundary is quasisymmetrically homeomorphic to an Ahlfors k-regular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice in the isometry group of hyperbolic (k+1)-space.
We study rigidity properties of lattices in hyperbolic n-space with n>2 and of surface groups in the context of (integrable) measure equivalence. The results for lattices in hyperbolic n-space with n>2 are generalizations of Mostow rigidity; they include a cocycle version of strong rigidity and an integrable measure equivalence classification. Despite the lack of Mostow rigidity for n=2 we show that cocompact lattices in SL(2,R) allow a similar integrable measure equivalence classification.
We introduce the concept of boundary rigidity for Gromov hyperbolic spaces. We show that a proper geodesic Gromov hyperbolic space with a pole is boundary rigid if and only if its Gromov boundary is uniformly perfect. As an application, we show that for a non-compact Gromov hyperbolic complete Riemannian manifold or a Gromov hyperbolic uniform graph, boundary rigidity is equivalent to having positive Cheeger isoperimetric constant and also to being nonamenable. Moreover, several hyperbolic fillings of compact metric spaces are proved to be boundary rigid if and only if the metric spaces are un…
We make a few observations on the absence of geometric and topological rigidity for acylindrically hyperbolic and relatively hyperbolic groups. In particular, we demonstrate the lack of a well-defined limit set for acylindrical actions on hyperbolic spaces, even under the assumption of universality. We also prove a statement about relatively hyperbolic groups inspired by a remark by Groves, Manning, and Sisto about the quasi-isometry type of combinatorial cusps. Finally, we summarize these results in a table in order to assert a meta-statement about the decay of metric rigidity as the conditio…
For n>3 we study spaces obtained from finite volume complete real hyperbolic n-manifolds by removing a compact totally geodesic submanifold of codimension two. We prove that their fundamental groups are relative hyperbolic, co-Hopf, biautomatic, residually hyperbolic, not Kähler, not isomorphic to lattices in virtually connected real Lie groups, have no nontrivial subgroups with property (T), have finite outer automorphism groups, satisfy Mostow-type Rigidity, have finite asymptotic dimension and rapid decay property, and satisfy Baum-Connes conjecture. We also characterize those lattices in r…
We give the first examples of closed fibered hyperbolic 3-manifolds whose fundamental groups are distinguished from every other finitely generated, residually finite group by their finite quotients. One of the examples is also the first example of a non-orientable profinitely rigid hyperbolic 3-manifold.
In this survey paper, we outline the proofs of the rigidity results for simple, thick, hyperbolic P-manifolds found in our three earlier papers math.GR/0506518, math.GT/0410476, and math.GR/0409586. We discuss how the arguments change in the two, three, and higher dimensional settings. This paper was written for the 22nd Annual Workshop in Geometric Topology (Colorado Springs, June 2005).
In this paper, we introduce a particularly nice family of locally CAT(-1) spaces, which we call hyperbolic P-manifolds. For $X^3$ a simple, thick hyperbolic P-manifold of dimension 3, we show that certain subsets of the boundary at infinity of the universal cover of $X^3$ are characterized topologically. Straightforward consequences include a version of Mostow rigidity, as well as quasi-isometric rigidity for these spaces.
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set gra…
We show that if $Γ= Γ_1\times\dotsb\times Γ_n$ is a product of $n\geq 2$ non-elementary ICC hyperbolic groups then any discrete group $Λ$ which is $W^*$-equivalent to $Γ$ decomposes as a $k$-fold direct sum exactly when $k=n$. This gives a group-level strengthening of Ozawa and Popa's unique prime decomposition theorem by removing all assumptions on the group $Λ$. This result in combination with Margulis' normal subgroup theorem allows us to give examples of lattices in the same Lie group which do not generate stably equivalent II$_1$ factors.
We construct arithmetic Kleinian groups that are profinitely rigid in the absolute sense: each is distinguished from all other finitely generated, residually finite groups by its set of finite quotients. The Bianchi group $\mathrm{PSL}(2,\mathbb{Z}[ω])$ with $ω^2+ω+1=0$ is rigid in this sense. Other examples include the non-uniform lattice of minimal co-volume in $\mathrm{PSL}(2,\mathbb{C})$ and the fundamental group of the Weeks manifold (the closed hyperbolic $3$-manifold of minimal volume).
For $i= 1,2$, let $G_i$ be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension $n$, $n \geq 3$. Let $H_i \subset G_i$ be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of $H_i$ is a codimension one topological sphere. 2) limit set of $H_i$ is an even dimensional topological sphere. 3) $H_i$ is a codimension one duality group. This generalizes (1). In particular, if $n = 3$, $H_i$ could be any freely indecomposable subgroup of $G_i$. 4) $H_i$ is an odd-dimensional Poincare Duality group $PD(2k+1)$. This generalizes (2). We…
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem…
We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable. We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficien…
We introduce and study the notion of relative rigidity for pairs $(X,\JJ)$ where 1) $X$ is a hyperbolic metric space and $\JJ$ a collection of quasiconvex sets 2) $X$ is a relatively hyperbolic group and $\JJ$ the collection of parabolics 3) $X$ is a higher rank symmetric space and $\JJ$ an equivariant collection of maximal flats Relative rigidity can roughly be described as upgrading a uniformly proper map between two such $\JJ$'s to a quasi-isometry between the corresponding $X$'s. A related notion is that of a $C$-complex which is the adaptation of a Tits complex to this context. We prove t…
We prove that simple, thick hyperbolic P-manifolds of dimension >2 exhibit Mostow rigidity. We also prove a quasi-isometry rigidity result for the fundamental groups of simple, thick hyperbolic P-manifolds of dimension >2. The key tool in the proofs of these rigidity results is a strong form of the Jordan separation theorem, for maps from S^n to S^{n+1} which are not necessarily injective.
Given a convex representation $ρ:Γ\to\textrm{PGL}(d,\mathbb{R})$ of a convex co-compact group $Γ$ of $\mathbb{H}^k$ we find upper bounds for the quantity $αh_ρ,$ where $h_ρ$ is the entropy of $ρ$ and $α$ is the Hölder exponent of the equivariant map $\partialΓ\to\mathbb{P}(\mathbb{R}^d).$ We also give rigidity statements when the upper bound is attained. We then study Hitchin representations and prove that if $ρ:π_1Σ\to\textrm{PSL}(d,\mathbb{R})$ is in the Hitchin component then $αh_ρ\leq 2/(d-1)$ (where $α$ is the Hölder exponent of the map $ζ:\partial\mathbb{H}^2\to\mathscr{F}$) with equalit…
A well-known question asks whether any two non-isometric finite volume hyperbolic 3-manifolds are distinguished from each other by the finite quotients of their fundamental groups. At present, this has been proved only when one of the manifolds is a once-punctured torus bundle over the circle. We give substantial computational evidence in support of a positive answer, by showing that no two manifolds in the SnapPea census of 72 942 finite volume hyperbolic 3-manifolds have the same finite quotients.
We define a complete Riemannian manifold X to be large-scale conformally rigid if all groups that are quasi-isometric to some complete Riemannian manifold of bounded geometry conformal to X are quasi-isometric to X. We prove that many 3-manifolds, including Euclidean 3-space, hyperbolic 3-space and the product of the hyperbolic plane with the real line are large-scale conformally rigid. This implies new characterizations of groups that can act properly, cocompactly by isometries on those manifolds.
This paper overviews recent developments in the classification up to quasi-isometry of finitely generated groups, and more specifically of relatively hyperbolic groups.
