# Higher topological complexity of hyperbolic groups

Hughes, S
Li, K

11 October 2021

## Last Updated:

2021-11-12T09:54:19.163+00:00

## 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.

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