arXiv subjects

Damien Gaboriau

Publications and source records attributed to Damien Gaboriau.

21 records · Page 2Linked to original sources

arXiv2005

Invariant Percolation and Harmonic Dirichlet Functions

The main goal of this paper is to answer question 1.10 and settle conjecture 1.11 of Benjamini-Lyons-Schramm [BLS99] relating harmonic Dirichlet functions on a graph to those of the infinite clusters in the uniqueness phase of Bernoulli percolation. We extend the result to more general invariant percolations, including the Random-Cluster model. We prove the existence of the nonuniqueness phase for the Bernoulli percolation (and make some progress for Random-Cluster model) on unimodular transitive locally finite graphs admitting nonconstant harmonic Dirichlet functions. This is done by using the device of $\ell^2$ Betti numbers.

math.PR↗

arXiv2005

Examples of Groups that are Measure Equivalent to the Free Group

Measure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups.

math.DS↗

arXiv2005

An Uncountable Family of Non Orbit Equivalent Actions of $\Bbb F_n$

For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $α_t$ of the free group $\Bbb F_n$ on the standard probability space $(X, μ)$ such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid'' (in the sense of [Po01]), with the II$_1$ factors $L^\infty(X, μ)\rtimes_{α_t} \Bbb F_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$

math.GR↗