Dynamical properties of profinite actions

We study profinite actions of residually finite groups in terms of weak containment. We show that two strongly ergodic profinite actions of a group are weakly equivalent if and only if they are isomorphic. This allows us to construct continuum many pairwise weakly inequivalent free actions of a large class of groups, including free groups and linear groups with property (T). We also prove that for chains of subgroups of finite index, Lubotzky's property ($\tau$) is inherited when taking the intersection with a fixed subgroup of finite index. That this is not true for families of subgroups in general leads to answering the question of Lubotzky and Zuk, whether for families of subgroups, property ($\tau$) is inherited to the lattice of subgroups generated by the family. On the other hand, we show that for families of normal subgroups of finite index, the above intersection property does hold. In fact, one can give explicite estimates on how the spectral gap changes when passing to the intersection. Our results also have an interesting graph theoretical consequence that does not use the language of groups. Namely, we show that an expander covering tower of finite regular graphs is either bipartite or stays bounded away from being bipartite in the normalized edge distance.Comment: Corrections made based on the referee's comment

Matching measure, Benjamini-Schramm convergence and the monomer-dimer free energy

We define the matching measure of a lattice L as the spectral measure of the tree of self-avoiding walks in L. We connect this invariant to the monomer-dimer partition function of a sequence of finite graphs converging to L. This allows us to express the monomer-dimer free energy of L in terms of the measure. Exploiting an analytic advantage of the matching measure over the Mayer series then leads to new, rigorous bounds on the monomer-dimer free energies of various Euclidean lattices. While our estimates use only the computational data given in previous papers, they improve the known bounds significantly.Comment: 18 pages, 3 figure

Bernoulli actions are weakly contained in any free action

Let Γ be a countable group and let f be a free probability measure-preserving action of Γ. We show that all Bernoulli actions of Γ are weakly contained in f. It follows that for a finitely generated group Γ, the cost is maximal on Bernoulli actions for Γ and that all free factors of i.i.d. (independent and identically distributed) actions of Γ have the same cost. We also show that if f is ergodic, but not strongly ergodic, then f is weakly equivalent to f×I, where Idenotes the trivial action of Γ on the unit interval. This leads to a relative version of the Glasner-Weiss dichotomy. © 2012 Cambridge University Press

A spectral strong approximation theorem for measure-preserving actions

Let be a finitely generated group acting by probability measure-preserving maps on the standard Borel space. We show that if is a subgroup with relative spectral radius greater than the global spectral radius of the action, then acts with finitely many ergodic components and spectral gap on. This answers a question of Shalom who proved this for normal subgroups. © Cambridge University Press, 2018

Convergence of normalized Betti numbers in nonpositive curvature

We study the convergence of volume-normalized Betti numbers in Benjamini-Schramm convergent sequences of non-positively curved manifolds with finite volume. In particular, we show that if X is an irreducible symmetric space of noncompact type, X 6 = H3, and (Mn) is any Benjamini-Schramm convergent sequence of finite volume X-manifolds, then the normalized Betti numbers bk(Mn)/vol(Mn) converge for all k. As a corollary, if X has higher rank and (Mn) is any sequence of distinct, finite volume X-manifolds, the normalized Betti numbers of Mn converge to the L2 Betti numbers of X. This extends our earlier work with Nikolov, Raimbault and Samet in [1], where we proved the same convergence result for uniformly thick sequences of compact X-manifolds. One of the novelties of the current work is that it applies to all quotients M = Γ\X where Γ is arithmetic; in particular, it applies when Γ is isotropic

Matchings in Benjamini–Schramm convergent graph sequences

We introduce the matching measure of a finite graph as the uniform distribution on the roots of the matching polynomial of the graph. We analyze the asymptotic behavior of the matching measure for graph sequences with bounded degree. A graph parameter is said to be estimable if it converges along every Benjamini– Schramm convergent sparse graph sequence. We prove that the normalized loga-rithm of the number of matchings is estimable. We also show that the analogous statement for perfect matchings already fails for d–regular bipartite graphs for any fixed d ≥ 3. The latter result relies on analyzing the probability that a randomly chosen perfect matching contains a particular edge. However, for any sequence of d–regular bipartite graphs converging to the d– regular tree, we prove that the normalized logarithm of the number of perfect matchings converges. This applies to random d–regular bipartite graphs. We show that the limit equals to the exponent in Schrijver’s lower bound on the number of perfect matchings. Our analytic approach also yields a short proof for the Nguyen–Onak (also Elek– Lippner) theorem saying that the matching ratio is estimable. In fact, we prove the slightly stronger result that the independence ratio is estimable for claw-free graphs