Related papers: Measure equivalence rigidity of the mapping class …

We obtain some classification result for the mapping class groups of compact orientable surfaces in terms of measure equivalence. In particular, the mapping class groups of different closed surfaces can not be measure equivalent. Moreover,…

We establish orbit equivalence rigidity for any ergodic, essentially free and measure-preserving action on a standard Borel space with a finite positive measure of the mapping class group for a compact orientable surface with higher…

Let $V$ be a connected $3$-dimensional handlebody of finite genus at least $3$. We prove that the handlebody group $\mathrm{Mod}(V)$ is superrigid for measure equivalence, i.e. every countable group which is measure equivalent to…

This article is concerned with measure equivalence and uniform measure equivalence of locally compact, second countable groups. We show that two unimodular, locally compact, second countable groups are measure equivalent if and only if they…

We undertake a comprehensive study of measure equivalence between general locally compact, second countable groups, providing operator algebraic and ergodic theoretic reformulations, and complete the classification of amenable groups within…

A discrete countable group \Gamma is said to be ME rigid if any discrete countable group that is measure equivalent to \Gamma is virtually isomorphic to \Gamma. In this paper, we construct ME rigid groups by amalgamating two groups…

We initiate a quantitative study of measure equivalence (and orbit equivalence) between finitely generated groups, which extends the classical setting of $\mathrm L^p$ measure equivalence. In this paper, our main focus will be on amenable…

We get three types of results on measure equivalence rigidity; direct product groups of Ozawa's class $\mathcal{S}$ groups, wreath product groups and amalgamated free products. We prove measure equivalence factorization results on direct…

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…

Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion is measure equivalent to $G$, with an $L^1$-integrable measure equivalence cocycle towards $G$, then $H$ is…

We establish new results and introduce new methods in the theory of measurable orbit equivalence, using bounded cohomology of group representations. Our rigidity statements hold for a wide (uncountable) class of groups arising from negative…

We single out a large class of groups ${\mathscr{M}}$ for which the following unique prime factorization result holds: if $\Gamma_1,\dots,\Gamma_n\in {\mathscr{M}}$ and $\Gamma_1\times\dots\times\Gamma_n$ is measure equivalent to a product…

By recognizing them as fundamental groups of developable complexes of groups we prove that mapping class groups of compact orientable surfaces have finite asymptotic dimension.

We study the large-scale geometry of mapping class groups of surfaces of infinite type, using the framework of Rosendal for coarse geometry of non locally compact groups. We give a complete classification of those surfaces whose mapping…

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…

Given a measure equivalence coupling between two finitely generated groups, Delabie, Koivisto, Le Ma\^itre and Tessera have found explicit upper bounds on how integrable the associated cocycles can be. We extend these results to the broader…

It is proved that the mapping class group of any closed surface with finitely many marked points is quasiisometric to a CAT(0) cube complex. We provide two distinct proofs, one tailored to mapping class groups, and one applying to a larger…

We prove that all (generalized) Higman groups on at least $5$ generators are superrigid for measure equivalence. More precisely, let $k\ge 5$, and let $H$ be a group with generators $a_1,\dots,a_k$, and Baumslag-Solitar relations given by…

We introduce the coherent algebra of a compact metric measure space by analogy with the corresponding concept for a finite graph. As an application we show that upon topologizing the collection of isomorphism classes of compact metric…

We study graph products of groups from the viewpoint of measured group theory. We first establish a full measure equivalence classification of graph products of countably infinite groups over finite simple graphs with no transvection and no…