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

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.