Minor-excluded graphs and soficity

TL;DR

This paper proves that certain infinite graphs, specifically one-ended unimodular graphs excluding a fixed minor, are sofic, meaning they can be approximated by finite graphs, with results extending to quasi-transitive graphs.

Contribution

It establishes the soficity of one-ended unimodular graphs excluding a fixed minor, generalizing previous results and removing the end count restriction under quasi-transitivity.

Findings

One-ended unimodular graphs excluding a fixed minor are sofic.

The end count restriction can be removed for quasi-transitive graphs.

The results extend the class of graphs known to be sofic.

Abstract

A random rooted graph is said to be sofic if it is the Benjamini-Schramm limit of a sequence of finite graphs. Given any finite graph $H$, we prove that every one-ended, unimodular random rooted graph that does not have H as a minor must be sofic. The hypothesis regarding the number of ends can be dropped under the additional assumption that the graph is quasi-transitive.