Expansion of coset graphs of PSL_2(F_p)

TL;DR

This paper studies bipartite coset graphs of PSL_2(F_p), demonstrating their potential for large girth and spectral gap, which are useful in geometric group theory constructions.

Contribution

It establishes the existence of bipartite coset graphs with large girth and spectral gap for PSL_2(F_p), advancing understanding of their combinatorial properties.

Findings

Graphs with large girth and spectral gap exist for PSL_2(F_p)

These graphs are bipartite and constructed from cosets of subgroups

Applications in constructing infinite groups in geometric group theory

Abstract

Let $G$ be a finite group and let $H_1,H_2<G$ be two subgroups. In this paper, we are concerned with the bipartite graph whose vertices are $G/H_1\cup G/H_2$ and a coset $g_1H_1$ is connected with another coset $g_2H_2$ if and only if $g_1H_1\cap g_2 H_2\neq\varnothing$. The main result of the paper establishes the existence of such graphs with large girth and large spectral gap. Lubotzky, Manning and Wilton use such graphs to construct certain infinite groups of interest in geometric group theory.