Finitary approximations of groups and their applications

TL;DR

This paper surveys recent advances in finitary approximation properties of infinite groups, exploring restrictions and applications to group theory and related fields.

Contribution

It provides a comprehensive overview of recent results on finitary approximations of groups and their implications for understanding group equations.

Findings

Groups can be approximated by finite solvable groups and unitary groups.

Approximation properties have applications in the equational theory of groups.

The survey highlights recent progress and open problems in the area.

Abstract

In these notes we will survey recent results on various finitary approximation properties of infinite groups. We will discuss various restrictions on groups that are approximated for example by finite solvable groups or finite-dimensional unitary groups with the Frobenius metric. Towards the end, we also briefly discuss various applications of those approximation properties to the understanding of the equational theory of a group.