Miklós Abért

Benjamini-Schramm convergence and the distribution of chromatic roots for sparse graphs

We define the chromatic measure of a finite simple graph as the uniform distribution on its chromatic roots. We show that for a Benjamini-Schramm convergent sequence of finite graphs, the chromatic measures converge in holomorphic moments.As a corollary, for a convergent sequence of finite graphs, we prove that the normalized log of the chromatic polynomial converges to an analytic function outside a bounded disc. This generalizes a recent result of Borgs, Chayes, Kahn and Lovász, who proved convergence at large enough positive integers and answers a question of Borgs.Our methods also lead to explicit estimates on the number of proper colorings of graphs with large girth.

Congruence subgroup growth of arithmetic groups in positive characteristic

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 congruence subgroup growth of arithmetic groups over global fields of positive characteristic. In particular, we show that the subgroup growth of SLn(Fp[t]) (n ≥ 3) is of type nlogn. This was one of the main problems left open by A. Lubotzky in his article [ 5]. The essential tool for proving the results is the use of graded Lie algebras. We sharpen Lubotzky’s bounds on subgroup growth via a result on subspaces of a Chevalley Lie algebra L over a finite field F. This theorem is proved by algebraic geometry and can be modified to obtain a lower bound on the codimension of proper Lie subalgebras of L.

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 matching 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.

Symmetric presentations of Abelian groups

We characterise the abelianisation of a group that has a presentation for which the set of relations is invariant under the full symmetric group acting on the set of generators. This improves a result of Emerson.