Elementary equivalence vs commensurability for hyperbolic groups

TL;DR

This paper investigates the logical equivalence between hyperbolic groups and their finite index subgroups, revealing structural conditions and the existence of infinitely many non-elementarily equivalent subgroups.

Contribution

It characterizes when hyperbolic limit groups are elementarily equivalent to their finite index subgroups and identifies conditions leading to non-equivalence.

Findings

Hyperbolic limit groups are either free products of cyclic and surface groups or have infinitely many non-elementarily equivalent finite index subgroups.

The paper establishes criteria for elementary equivalence in hyperbolic groups.

It demonstrates the diversity of subgroup structures in hyperbolic groups regarding elementary properties.

Abstract

We study to what extent torsion-free (Gromov)-hyperbolic groups are elementarily equivalent to their finite index subgroups. In particular, we prove that a hyperbolic limit group either is a free product of cyclic groups and surface groups, or admits infinitely many subgroups of finite index which are pairwise non elementarily equivalent.