Characterization of sofic groups and equations over groups

TL;DR

This paper characterizes sofic and weakly sofic groups by their ability to solve systems of equations that are solvable in finite or alternating groups, providing a new perspective on their algebraic properties.

Contribution

It offers a novel characterization of sofic and weakly sofic groups through the solvability of equations over these groups, linking group properties to solutions of algebraic systems.

Findings

Sofic groups are characterized by solvability of equations solvable in alternating groups.

Weakly sofic groups are characterized by solvability of equations solvable in finite groups.

Provides an algebraic criterion for identifying sofic and weakly sofic groups.

Abstract

We give the following characterization of sofic (weakly sofic) groups: a group $G$ is sofic (weakly sofic) if and only if any system of equations solvable in any alternating group (any finite group) is solvable over $G$.