On expansions of non-abelian free groups by cosets of a finite index subgroup

TL;DR

This paper investigates the logical properties of free groups expanded by cosets of finite index subgroups, showing that certain first-order theories depend only on the finite quotient and not on the free group's rank.

Contribution

It generalizes techniques from Sela's solution to Tarski's problem to analyze the first-order theory of free groups expanded by cosets, establishing independence from the group's rank.

Findings

First-order sentences' validity depends only on the finite quotient Q.

Proves an analogue of Sela's generalization of Merzlyakov's theorem for these expansions.

Shows the positive theory is determined solely by Q, not by the free group's rank or quotient map.

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 $\forall\exists$-sentences and show that the positive theory depends only on $Q$ and neither on the rank of $F$ nor the particular quotient map.