Hyperbolic groups have flat-rank at most 1

TL;DR

This paper extends the concept of hyperbolic groups to the topological setting and proves that such groups have a flat-rank of at most 1, impacting the understanding of their structure.

Contribution

It generalizes hyperbolic groups to topological groups and establishes an upper bound on their flat-rank, linking to known constructions by Paulin and Haglund.

Findings

Hyperbolic groups in the topological context have flat-rank ≤ 1

The result applies to groups constructed by Paulin and Haglund

Provides a new invariant bound for topological hyperbolic groups

Abstract

The flat-rank of a totally disconnected, locally compact group G is an integer, which is an invariant of G as a topological group. We generalize the concept of hyperbolic groups to the topological context and show that a totally disconnected, locally compact, hyperbolic group has flat-rank at most 1. It follows that the simple totally disconnected locally compact groups constructed by Paulin and Haglund have flat-rank at most 1.