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Preprint Number 1066
1066. O. Kharlampovich, A. Miasnikov
First-order theory of group algebras
Submission date: 11 July 2016.
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). We will show that similar results hold for group algebras of limit groups.
Mathematics Subject Classification: 16B70, 03C60, 20E05
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