Hyperbolic towers and independent generic sets in the theory of free groups

TL;DR

This paper explores the model theory of free groups using hyperbolic towers, revealing properties of generic types, independence, and homogeneity in finitely generated models.

Contribution

It introduces hyperbolic towers as a tool to analyze the generic type and independence in free groups, showing new phenomena in model-theoretic properties.

Findings

All finitely generated models realize the generic type p_0.

Existence of a finitely generated model omitting p_0^{(2)}.

Different sizes of independent sets of realizations of p_0 in the same model.

Abstract

We use hyperbolic towers to answer some model theoretic questions around the generic type in the theory of free groups. We show that all the finitely generated models of this theory realize the generic type $p_0$, but that there is a finitely generated model which omits $p_0^{(2)}$. We exhibit a finitely generated model in which there are two maximal independent sets of realizations of the generic type which have different cardinalities. We also show that a free product of homogeneous groups is not necessarily homogeneous.