Let G be a closed subgroup of the isometry group of a proper CAT(0)-space X. We show that if G is non-elementary and contains a rank-one element then its second bounded cohomology group with coefficients in the regular representation is non-trivial. As a consequence, up to passing to an open subgroup of finite index, either G is a compact extension of a totally disconnected group or G is a compact extension of a simple Lie group of rank one.
Given a CAT(0) cube complex X, we show that if Aut(X) $\neq$ Isom(X) then there exists a full subcomplex of X which decomposes as a product with $\mathbb{R}^n$. As applications, we prove that if X is $δ$-hyperbolic, cocompact and 1-ended, then Aut(X) $=$ Isom(X) unless X is quasi-isometric to $\mathbb{H}^2$, and extend the rank-rigidity result of Caprace-Sageev to any lattice $ Γ\leq$ Isom(X).
The purpose of this note is twofold. In the first part we observe that two finitely generated non-amenable groups are quasi-isometric if and only if they admit topologically orbit equivalent Cantor minimal actions. In particular, free groups if different rank admit topologically orbit equivalent Cantor minimal actions unlike in the measurable setting. In the second part we introduce the measured orbit equivalence category of a Cantor minimal system and construct (in certain cases) a representation of this category on the category of finite-dimensional vector spaces. This gives rise to novel fu…
Let $X$ be a proper, geodesically complete Hadamard space, and $\ Γ<\mbox{Is}(X)$ a discrete subgroup of isometries of $X$ with the fixed point of a rank one isometry of $X$ in its infinite limit set. In this paper we prove that if $Γ$ has non-arithmetic length spectrum, then the Ricks' Bowen-Margulis measure -- which generalizes the well-known Bowen-Margulis measure in the CAT$(-1)$ setting -- is mixing. If in addition the Ricks' Bowen-Margulis measure is finite, then we also have equidistribution of $Γ$-orbit points in $X$, which in particular yields an asymptotic estimate for the orbit coun…
The Morse boundary of a proper geodesic metric space is designed to encode hypberbolic-like behavior in the space. A key property of this boundary is that a quasi-isometry between two such spaces induces a homeomorphism on their Morse boundaries. In this paper we investigate when the converse holds. We prove that for cocompact CAT(0) spaces, a homeomorphism of Morse boundaries is induced by a quasi-isometry if and only if the homeomorphism is quasi-mobius and 2-stable.
We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.
We show that if X is a piecewise Euclidean 2-complex with a cocompact isometry group, then every 2-quasiflat in X is at finite Hausdorff distance from a subset which is locally flat outside a compact set, and asymptotically conical.
Let X be a proper hyperbolic geodesic metric space and let G be a closed subgroup of the isometry group Iso(X) of X. We show that if G is not amenable then its second continuous bounded cohomology group with coefficients the regular representation does not vanish. This yields some structure results for such groups.
Given a locally compact Polish space X, a necessary and sufficient condition for a group G of homeomorphisms of X to be the full isometry group of (X,d) for some proper metric d on X is given. It is shown that every locally compact Polish group G acts freely on GxY as the full isometry group of GxY with respect to a certain proper metric on GxY, where Y is an arbitrary locally compact Polish space with (card(G),card(Y)) different from (1,2). Locally compact Polish groups which act effectively and almost transitively on complete metric spaces as full isometry groups are characterized. Locally c…
We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further developments on non-positively curved lattices are exposed in a companion paper: "Isometry groups of non-positively curved spaces: discrete subgroups".
Let X be a locally compact geodesically complete CAT(0) space and G be a discrete group acting properly and cocompactly on X. We show that G contains an element acting as a hyperbolic isometry on each indecomposable de Rham factor of X. It follows that if X is a product of d factors, then G contains free abelian group of rank d.
We say that a group has property $R_{\infty}$ if any group automorphism has an infinite number of twisted conjugacy classes. Fel'shtyn and Goncalves prove that the solvable Baumslag-Solitar groups BS(1,m) have property $R_{\infty}$. We define a solvable generalization $Γ(S)$ of these groups which we show to have property $R_{\infty}$. We then show that property $R_{\infty}$ is geometric for these groups, that is, any group quasi-isometric to $Γ(S)$ has property $R_{\infty}$ as well.
Let $A_Γ$ be a right-angled Artin group. Charney, Vogtmann and the author constructed an outer space for $\text{Out}(A_Γ)$ generalizing both $CV_n$ for $\text{Out}(F_n)$ and the symmetric space $\text{SL}_n(\mathbb{R})/\text{SO}_n(\mathbb{R})$ for $\text{GL}_n(\mathbb{Z})$. Points in this space are equivalence classes of pairs $(X,ρ)$ where $ρ\colon X\rightarrow \mathbb{S}_Γ$ is a homotopy equivalence from $X$ to the Salvetti complex $\mathbb{S}_Γ$ and $X$ is a locally CAT(0) space called a skewed $Γ$-complex. In this note we show that any isometry of a skewed $Γ$-complex which is homotopic to…
We announce results on the structure of CAT(0) groups, CAT(0) lattices and of the underlying spaces. Our statements rely notably on a general study of the full isometry groups of proper CAT(0) spaces. Classical statements about Hadamard manifolds are established for singular spaces; new arithmeticity and rigidity statements are obtained.
This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $Γ<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different characterizations of the subclass of Anosov subgroups, which generalize convex-cocompact subgroups of rank 1 Lie groups, in terms of various equivalent dynamical and geometric properties (such as asymptotically embedded, RCA, Morse, URU). 3. Discuss the topological dynamics of discrete su…
Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtually abelian). We are led to consider a set $\bdfine X$ which is a refinement of the visual boundary $\bd X$. For each $x \in \bdfine X$, the stabilizer $G_x$ is amenable.
Previously one of the authors constructed uncountable families of groups of type $FP$ and of $n$-dimensional Poincaré duality groups for each $n\geq 4$. We strengthen these results by showing that these groups comprise uncountably many quasi-isometry classes. We deduce that for each $n\geq 4$ there are uncountably many quasi-isometry classes of acyclic $n$-manifolds admitting free cocompact properly discontinuous discrete group actions.
For a proper geodesic metric space $X$, the Morse boundary $\partial_*X$ focuses on the hyperbolic-like directions in the space $X$. It is a quasi-isometry invariant. That is, a quasi-isometry between two hyperbolic spaces induces a homeomorphism on their boundaries. In this paper, we investigate additional structures on the Morse boundary $\partial_*X$ which determine $X$ up to a quasi-isometry. We prove that, for $X$ and $Y$ proper, cocompact spaces, a homeomorphism $f$ between their Morse boundaries is induced by a quasi-isometry if and only if $f$ and $f^{-1}$ are bihölder, or quasi-symmet…
We will show that if a proper complete CAT(0) space X has a visual boundary homeomorphic to the join of two Cantor sets, and X admits a geometric group action by a group containing a subgroup isomorphic to Z^2, then its Tits boundary is the spherical join of two uncountable discrete sets. If X is geodesically complete, then X is a product, and the group has a finite index subgroup isomorphic to a lattice in the product of two isometry groups of bounded valence bushy trees.
We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the closure of f(c), instersected with the boundary of Y, is homeomorphic to Z. This characterizes all homeomorphism types of "geodesic boundary images" that arise in this manner.
Let $\mathcal{X}_S$ denote the class of spaces homeomorphic to two closed orientable surfaces of genus greater than one identified to each other along an essential simple closed curve in each surface. Let $\mathcal{C}_S$ denote the set of fundamental groups of spaces in $\mathcal{X}_S$. In this paper, we characterize the abstract commensurability classes within $\mathcal{C}_S$ in terms of the ratio of the Euler characteristic of the surfaces identified and the topological type of the curves identified. We prove that all groups in $\mathcal{C}_S$ are quasi-isometric by exhibiting a bilipschitz …
We prove that any group acting essentially without a fixed point at infinity on an irreducible finite-dimensional CAT(0) cube complex contains a rank one isometry. This implies that the Rank Rigidity Conjecture holds for CAT(0) cube complexes. We derive a number of other consequences for CAT(0) cube complexes, including a purely geometric proof of the Tits Alternative, an existence result for regular elements in (possibly non-uniform) lattices acting on cube complexes, and a characterization of products of trees in terms of bounded cohomology.
In this paper, we investigate an equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition to obtain a $G$-equivariant homeomorphism of the two boundaries $\partial X$ and $\partial Y$ as a continuous extension of the quasi-isometry $ϕ:Gx_0\to Gy_0$ defined by $ϕ(gx_0)=gy_0$, where $x_0\in X$ and $y_0\in Y$.
In this paper we extend the construction of complementary series representations to convex-cocompact isometry groups of CAT(-1) spaces with conditionally negative metrics. Our approach is purely dynamical and generalizes the constructions known for negatively curved algebraic groups as $SO(n,1)$, $SU(n,1)$ or $SL_2(\mathbb{Q}_p)$ and their lattices to new examples as non-linear groups coming from lattices of certain hyperbolic buildings.
We present a general framework for automatic continuity results for groups of isometries of metric spaces. In particular, we prove automatic continuity property for the group of isometries of the Urysohn space and the Urysohn sphere, i.e. that any homomorphism from either of these groups into a separable group is continuous. As a consequence, we get that the group of isometries of the Urysohn space has unique Polish group topology and the group of isometries of the Urysohn sphere has unique separable group topology. Moreover, as an application of our framework we obtain new proofs of the autom…
It is shown that a topological group G is topologically isomorphic to the isometry group of a (complete) metric space iff G coincides with its G-delta-closure in the Rajkov completion of G (resp. if G is Rajkov-complete). It is also shown that for every Polish (resp. compact Polish; locally compact Polish) group G there is a complete (resp. proper) metric d on X inducing the topology of X such that G is isomorphic to Iso(X,d) where X = l_2 (resp. X = Q; X = Q\{point} where Q is the Hilbert cube). It is demonstrated that there are a separable Banach space E and a nonzero vector e in E such that…
