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

Rich groups, weak second order logic, and applications

“…Theorem 18. [22] Let F = (X) be a non-abelian f.g. free group and K an infinite field. Then the group algebra K(F ) is bi-interpretable with S(K, N), with parameters X, P , where P is a non-invertible polynomial with at least three monomials, uniformly in K, X, P .…”

“…Interestingly, as we mentioned above free and torsion-free hyperbolic groups are not rich. Actually they are very far from being rich, however, their group algebras over infinite fields are rich [22]. This shows how much more expressive is the first order ring language of a group algebra of a free group compared with the first order language of the group.…”

Diophantine problems in solvable groups

“…This ring constitutes an important feature of f , and in some sense it provides an "approximation" to interpreting (in (A, B; f )) multiplication of constant elements from N and M by integer variables. It has been used successfully to study different first order theoretic aspects of different types of structures, including rings whose additive group is finitely generated [32], free algebras [22,23,24], and nilpotent groups [33,34]. For us the most important property of R(f ) is that it is e-interpretable in (A, B; f ): Theorem 3.1 (Theorem 3.5 of [16]).…”

Diophantine problems in rings and algebras: undecidability and reductions to rings of algebraic integers

“…From bilinear maps to commutative algebra Next, we briefly explain the approach taken in order to pass from possibly non-associative, non-commutative, and nonunitary algebras to algebras of scalars (and similarly for rings). The ideas we present here were introduced by the second author in [26], and they have been used successfully to study different first order theoretic aspects of different types of structures, including rings whose additive group is finitely generated [27], free algebras [16,18,19], and nilpotent groups [28,29].…”

“…In Corollary 1.8 we obtain further results of this type. Many results regarding the first order theory of free algebras can be found in [16,18,19].…”