Approximation of groups, characterizations of sofic groups, and equations over groups

TL;DR

This paper introduces new characterizations of sofic groups, linking their properties to subgroups of quotients of direct products of symmetric groups and solvability of equations over groups.

Contribution

It provides novel characterizations of sofic groups, including their relation to equations solvable in alternating groups and logical descriptions via $orallorall$-sentences.

Findings

Sofic groups are subgroups of quotients of direct products of symmetric groups.

Equations solvable in all alternating groups are solvable over sofic groups.

Soficity can be expressed using $orallorall$-sentences in logic.

Abstract

We give new characterizations of sofic groups: -- A group $G$ is sofic if and only if it is a subgroup of a quotient of a direct product of alternating or symmetric groups. -- A group $G$ is sofic if and only if any system of equations solvable in all alternating groups is solvable over $G$. The last characterization allows to express soficity of an existentially closed group by $\forall\exists$-sentences. Keywords: sofic groups, approximations, equations over groups.