Volume vs. Complexity of Hyperbolic Groups

TL;DR

This paper establishes a lower bound on the size of quotient spaces of hyperbolic graphs based on their simplices count, and applies this to show certain hyperbolic groups cannot have isomorphic finite-index subgroups of different indices.

Contribution

It introduces a new lower bound relating quotient sizes to simplices in hyperbolic groups and demonstrates a rigidity property for hyperbolic cubulated groups.

Findings

Lower bound on quotient size in hyperbolic graphs

Hyperbolic cubulated groups cannot have isomorphic finite-index subgroups of different indices

Application of the bound to groups with stable cylinders

Abstract

We prove that for a one-ended hyperbolic graph $X$, the size of the quotient $X/G$ by a group $G$ acting freely and cocompactly bounds from below the number of simplices in an Eilenberg-MacLane space for $G$. We apply this theorem to show that one-ended hyperbolic cubulated groups (or more generally, one-ended hyperbolic groups with globally stable cylinders \`a la Rips-Sela) cannot contain isomorphic finite-index subgroups of different indices.