Kesten's theorem for Invariant Random Subgroups

By Miklós Abért, Yair Glasner, B'alint Vir'ag. 2012

186 citations recorded by the source corpus.

An invariant random subgroup of the countable group {\Gamma} is a random subgroup of {\Gamma} whose distribution is invariant under conjugation by all elements of {\Gamma}. We prove that for a nonamenable invariant random subgroup H, the spectral radius of every finitely supported random walk on {\Gamma} is strictly less than the spectral radius of the corresponding random walk on {\Gamma}/H. This generalizes a result of Kesten who proved this for normal subgroups. As a byproduct, we show that for a Cayley graph G of a linear group with no amenable normal subgroups, any sequence of finite quotients of G that spectrally approximates G converges to G in Benjamini-Schramm convergence. In particular, this implies that infinite sequences of finite d-regular Ramanujan Schreier graphs have essentially large girth.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/750a30bcbb9ce01f7c9433f65623073dbab2bc72

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In Discrete Mathematics and Combinatorics

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Citations in the corpus

Finite Factor Representations of Higman-Thompson groups (2012, result): “Combining this result with [1], Proposition 14, we obtain the following.”
The measurable Kesten theorem (2011, background): “As the readership of this result is expected to be different from that of the current paper (and the current paper is already long), we decided to publish it in a separate article; see [1].”
C*-simplicity and the unique trace property for discrete groups (2014, background): “The notion of an invariant random subgroup was introduced in [1], and the problem was raised whether every amenable invariant random subgroup is concentrated on the amenable radical.”
Character rigidity for special linear groups (2013, background): “We thank the unknown referee for careful proof-reading and a many helpful comments.”
Approximate lattices (2016, background): “In the context of nonabelian groups, the study of invariant random subgroups and uniformly recurrent subgroups [1, 2, 25] has led to an intensive study of invariant measures on conjugation hulls, i.”
On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, result): “We refer the reader to [6], [29], [112], [61], [62] for other recent works that make use of this notion.”
An intrinsic characterization of C*-simplicity (2015, methodology): “This notion was introduced by Abért, Glasner and Virág [1].”
Stabilizers of Ergodic Actions of Lattices and Commensurators (2013, background): “s and Stuck [AS93] Section 4). Bergeron and Gaboriau [BG04] observed that invariant random subgroups behave similarly to normal subgroups and this topic has attracted much recent attention: [ABB+11], [AGV14], [Bow12], [GS12], [Gri11], [Ver11], [Ver12]. Our work, like that of Stuck and Zimmer, rules ”
Invariant random subgroups of the free group (2012, background): “Kg−1,S) is isomorphic to Sch(K,S) even though they are typically not root-isomorphic. Precisely, Sch(gKg−1,S) is root-isomorphic to Sch(K,S) after moving the root in Sch(K,S) from Kto Kg−1. The paper [AGV12] further develops this correspondence between Sub(Fr) and SC(Fr,S). 2.2 Measured equivalence ”
Charmenability of arithmetic groups of product type (2020, background): “An IRS of is a -invariant probability measure on Sub( ) (see [2]).”