We study the growth of the rank of subgroups of finite index in residually finite groups, by relating it to the notion of cost. As a by-product, we show that the 'Rank vs. Heegaard genus' conjecture… (More)

We explore the structure of the p-adic automorphism group Γ of the infinite rooted regular tree. We determine the asymptotic order of a typical element, answering an old question of Turán. We… (More)

We prove a new uniform bound for subgroup growth of a Chevalley group G over the local ringF[[t]] and also over local pro-p rings of higher Krull dimension. This is applied to the determination of… (More)

We give an explicit bound on the spectral radius in terms of the densities of short cycles in finite d-regular graphs. It follows that the a finite d-regular Ramanujan graph G contains a negligible… (More)

In this note we discuss Benjamini-Schramm (or local) convergence for sequences of finite graphs of bounded degree. We prove a continuity result on the expected spectral measure on sofic random rooted… (More)

This paper investigates the asymptotic behaviour of the minimal number of generators of finite index subgroups in residually finite groups. We analyze three natural classes of groups: amenable… (More)

We define thematching 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… (More)

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… (More)