Finite index rigidity of hyperbolic groups

TL;DR

This paper establishes a linear relationship between the topological complexity of finite index subgroups of hyperbolic groups and their index, revealing a form of rigidity in their structure.

Contribution

It proves finite index rigidity for hyperbolic groups, showing isomorphic finite index subgroups must have the same index, and relates quotient size to minimal cell counts in classifying spaces.

Findings

Topological complexity is linear in subgroup index.

Isomorphic finite index subgroups have equal indices.

Size of quotient relates to minimal cell counts.

Abstract

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.