On soficity for certain fundamental groups of graphs of groups

TL;DR

This paper establishes the soficity of a class of fundamental groups of graphs of groups, including doubles of sofic groups over separable subgroups, under a new technical condition called sigma-co-sofic.

Contribution

It introduces the sigma-co-sofic condition and proves soficity for these groups, extending known results to more general constructions like doubles over separable subgroups.

Findings

Proves soficity for a broad class of groups built from sofic vertex groups.

Introduces the sigma-co-sofic condition as a key hypothesis.

Includes results on doubles of sofic groups over arbitrary separable subgroups.

Abstract

In this note we study a family of graphs of groups over arbitrary base graphs where all vertex groups are isomorphic to a fixed countable sofic group $G$, and all edge groups $H<G$ are such that the embeddings of $H$ into $G$ are identical everywhere. We prove soficity for this family of groups under a flexible technical hypothesis for $H$ called $\sigma$-co-sofic. This proves soficity for group doubles $*_H G$, where $H<G$ is an arbitrary separable subgroup and $G$ is countable and sofic. This includes arbitrary finite index group doubles of sofic groups among various other examples.