Random permutations, word maps and Schreier graph expansion

Abstract: Given a word \(w\) in the free group on \(r\) generators, one can obtain a word map for any finite group, \(w \colon\thinspace G^r\to G\), by substitutions. By uniformly randomly sampling \(r\) random permutations in \(S_n\) and evaluating their image under this word map, we obtain a '\(w\)-random permutation'. Recent studies of these random permutations have exposed some deep connections with various other areas of mathematics. I will discuss the current asymptotic bounds we have for the expected irreducible characters of \(w\)-random permutations, and an application towards showing that a large family of random Schreier graphs have a near-optimal spectral gap with high probability.

algebraic topologyfunctional analysisgroup theorygeometric topologyoperator algebras

Audience: researchers in the topic

Vienna Geometry and Analysis on Groups Seminar