Higher topological complexity of hyperbolic groups

TL;DR

This paper proves that for non-elementary torsion-free hyperbolic groups, the higher topological complexity equals the number of motion steps times the cohomological dimension, confirming a conjecture for these groups.

Contribution

It establishes the exact value of higher topological complexity for a broad class of hyperbolic groups and confirms the rationality conjecture for their $ ext{TC}$-generating functions.

Findings

Higher topological complexity equals r times the cohomological dimension for hyperbolic groups.

Confirms the rationality conjecture on the $ ext{TC}$-generating function for these groups.

Extends results to certain toral relatively hyperbolic groups.

Abstract

We prove for non-elementary torsion-free hyperbolic groups $\Gamma$ and all $r\ge 2$ that the higher topological complexity ${\sf{TC}}_r(\Gamma)$ is equal to $r\cdot \mathrm{cd}(\Gamma)$. In particular, hyperbolic groups satisfy the rationality conjecture on the $\sf{TC}$-generating function, giving an affirmative answer to a question of Farber and Oprea. More generally, we consider certain toral relatively hyperbolic groups.