Maximal hyperbolic towers and weight in the theory of free groups

TL;DR

This paper demonstrates that maximal hyperbolic tower structures over free groups are not unique and can vary significantly, impacting the understanding of types' weight in the theory of free groups.

Contribution

It constructs examples of groups with non-canonical hyperbolic tower structures over free subgroups, revealing variability in their ranks and analyzing implications for type weight.

Findings

Existence of groups with arbitrarily large rank ratios in hyperbolic towers

Groups share the same first order theory as free groups

Hyperbolic tower structures are non-canonical

Abstract

We show that in general for a given group the structure of a maximal hyperbolic tower over a free group is not canonical: We construct examples of groups having hyperbolic tower structures over free subgroups which have arbitrarily large ratios between their ranks. These groups have the same first order theory as non-abelian free groups and we use them to study the weight of types in this theory.