Sofic actions on graphs

TL;DR

This paper develops a comprehensive theory of sofic actions on graphs, establishing new stability properties, characterizations, and examples, significantly advancing the understanding of sofic groups and their actions.

Contribution

It introduces a unified framework for sofic actions on graphs, proves preservation under various operations, and provides new examples, including graph wreath products, enriching the class of known sofic groups.

Findings

Soficity is preserved by graph join operations.

Group actions on Cayley graphs are sofic iff the group is sofic.

Actions of amenable and free groups on graphs are sofic.

Abstract

We develop a theory of soficity for actions on graphs and obtain new applications to the study of sofic groups. We establish various examples, stability and permanence properties of sofic actions on graphs, in particular soficity is preserved by taking several natural graph join operations. We prove that an action of a group on its Cayley graph is sofic if and only if the group is sofic. We show that arbitrary actions of amenable groups on graphs are sofic. Using a graph theoretic result of E. Hrushovski, we also show that arbitrary actions of free groups on graphs are sofic. Notably we show that arbitrary actions of sofic groups on graphs, with amenable stabilizers, are sofic, settling completely an open problem from \cite{gao2024soficity}. We also show that soficity is preserved by taking limits under a natural Gromov-Hausdorff topology, generalizing prior work of the first author \cite{gao2024actionslerfgroupssets}. Our work sheds light on a family of groups called graph wreath products, simultaneously generalizing graph products and generalized wreath products. Extending various prior results in this direction including soficity of generalized wreath products \cite{gao2024soficity}, B. Hayes and A. Sale \cite{HayesSale}, and soficity of graph products \cite{CHR, charlesworth2021matrix}, we show that graph wreath products are sofic if the action and acting groups are sofic. These results provide several new examples of sofic groups in a systematic manner.