Expanders have a spanning Lipschitz subgraph with large girth

TL;DR

This paper proves that regular graphs with good local expansion contain spanning Lipschitz subgraphs with large girth and degree, connecting expansion properties with girth and addressing conjectures in graph theory.

Contribution

It introduces a new proof that strengthens the Gaboriau-Lyons result, providing finite analogues and answering questions about geometric random subgroups and girth in graphs.

Findings

Existence of spanning Lipschitz subgraphs with large girth in regular graphs with good local expansion

Strengthening of the Gaboriau-Lyons theorem and finite analogues of dynamical results

Application to Thomassen's conjecture on large girth and average degree in finite graphs

Abstract

We show that every regular graph with good local expansion has a spanning Lipschitz subgraph with large girth and minimum degree. In particular, this gives a finite analogue of the dynamical solution to the von Neumann problem by Gaboriau and Lyons. We give a new proof and strengthen the Gaboriau-Lyons result, that allows us to answer two questions of Monod about geometric random subgroups. Our finite theorems are kind of converse to the theorem of Bourgain and Gamburd showing that large girth implies expansion for Cayley graphs of SL_2(F_p). We apply these to the regular case of Thomassen's conjecture stating that every finite graph with large average degree has a subgraph with large girth and average degree. Our main tool is an infinite version of the Lovasz Local Lemma developed in this paper.