  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.
