Submission date: 10 July 2017

Abstract:

Let F be a finitely generated non-abelian free group and Q a finite quotient. Denote by L_Q the language obtained by adding unary predicates P_q, q in Q to the language of groups. Using a slight generalization of some of the techniques involved in Zlil Sela's solution to Tarski's problem on the elementary theory of non-abelian free groups, we provide a few basic results on the validity of first order entences in the L_Q-expansion of F in which every P_q is interpreted as the preimage of q in F. In particular we prove an analogous result to Sela's generalization of Merzlyakov's theorem on ∀∃-sentences and show that the positive theory depends only on Q and neither on the rank of F nor the particular quotient map.

Mathematics Subject Classification: 0E05, 20F65, 03Cxx

Keywords and phrases:

Full text arXiv 1707.03066: pdf, ps.