What does a group algebra of a free group know about the group?

TL;DR

This paper investigates how much algebraic and geometric information about free groups can be recovered from their group algebras over infinite fields, showing definability of key properties and classification results.

Contribution

It demonstrates that group algebras of free groups encode significant geometric and algebraic information, enabling elementary classification and definability of bases and geodesics.

Findings

Group algebras of free groups over infinite fields are elementarily equivalent iff the groups are isomorphic and fields are elementarily equivalent.

The set of all free bases of a free group is 0-definable in its group algebra.

Many geometric properties of free groups are definable within their group algebras.

Abstract

We describe solutions to the problem of elementary classification in the class of group algebras of free groups. We will show that unlike free groups, two group algebras of free groups over infinite fields are elementarily equivalent if and only if the groups are isomorphic and the fields are equivalent in the weak second order logic. We will show that the set of all free bases of a free group $F$ is 0-definable in the group algebra $K(F)$ when $K$ is an infinite field, the set of geodesics is definable, and many geometric properties of $F$ are definable in $K(F)$. Therefore $K(F)$ knows some very important information about $F$. We will show that similar results hold for group algebras of limit groups.