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

Olga Kharlampovich, Alexei Myasnikov

Research output: Contribution to journalArticlepeer-review

6 Scopus citations

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.

Original languageEnglish
Pages (from-to)523-547
Number of pages25
JournalAnnals of Pure and Applied Logic
Volume169
Issue number6
DOIs
StatePublished - Jun 2018

Keywords

  • First-order theory
  • Group algebra
  • Limit group

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