In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.
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…
It is shown that every accessible group which is integrable orbit equivalent to a free group is virtually free. Moreover, we also show that any integrable orbit-equivalence between finitely generated groups extends to their end compactifications.
We prove that for any two continuous minimal (topologically free) actions of the infinite dihedral group on an infinite compact Hausdorff space, they are continuously orbit equivalent only if they are conjugate. We also show the above fails if we replace the infinite dihedral group with certain other virtually cyclic groups, e.g. the direct product of the integer group with any non-abelian finite simple group.
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…
Let $G$ be a group. A function $l:G\rightarrow \lbrack 0,\infty )$ is called a length function if (1) $l(g^{n})=|n|l(g)$ for any $g\in G$ and $n\in \mathbb{Z};$ (2) $l(hgh^{-1})=l(g)$ for any $h,g\in G;$ and (3) $l(ab)\leq l(a)+l(b)$ for commuting elements $a,b.$ Such length functions exist in many branches of mathematics, mainly as stable word lengths, stable norms, smooth measure-theoretic entropy, translation lengths on $\mathrm{CAT}(0)$ spaces and Gromov $δ$% -hyperbolic spaces, stable norms of quasi-cocycles, rotation numbers of circle homeomorphisms, dynamical degrees of birational maps …
A result due to M. Gromov states that any two finitely generated groups Γ and Λ are quasi-isometric if and only if they admit a topological coupling, i.e., a commuting pair of proper continuous cocompact actions $Γ\curvearrowright X\curvearrowleft Λ$ on a locally compact Hausdorff space. This result is extended here to all (compactly generated) locally compact second countable groups.
We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $Γ_1,\dots,Γ_n\in {\mathscr{M}}$ and $Γ_1\times\dots\timesΓ_n$ is measure equivalent to a product $Λ_1\times\dots\timesΛ_m$ of infinite icc groups, then $n \ge m$, and if $n = m$ then, after permutation of the indices, $Γ_i$ is measure equivalent to $Λ_i$, for all $1\leq i\leq n$. This provides an analogue of Monod and Shalom's theorem \cite{MS02} for groups that belong to ${\mathscr{M}}$. Class ${\mathscr{M}}$ is constructed using groups whose von Neumann algebras admit an…
We study equivalence relations $\mathcal R(Γ\curvearrowright G)$ that arise from left translation actions of countable groups on their profinite completions. Under the assumption that the action $Γ\curvearrowright G$ is free and has spectral gap, we describe precisely when $\mathcal R(Γ\curvearrowright G)$ is orbit equivalent or Borel reducible to another such equivalence relation $\mathcal R(Λ\curvearrowright H)$. As a consequence, we provide explicit uncountable families of free ergodic probability measure preserving (p.m.p.) profinite actions of $SL_2(\mathbb Z)$ and its non-amenable subgro…
In [BBM21], Belk, Bleak and Matucci proved that hyperbolic groups can be seen as subgroups of the rational group. In order to do so, they associated a tree of atoms to each hyperbolic group. Not so many connections between this tree and the literature on hyperbolic groups were known. In this paper, we prove an atom-version of the fellow traveler property and exponential divergence, together with other similar results. These leads to several consequences: a bound from above of the topological dimension of the Gromov boundary, the definition of an augmented tree which is quasi-isometric to the C…
We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups, or admits infinitely many subgroups of finite index which are pairwise non elementarily equivalent.
Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…
