Higher topological complexity of hyperbolic groups

TL;DR

The paper addresses computing higher topological complexity $TC_r(Γ)$ for groups, focusing on non-elementary torsion-free hyperbolic groups. It develops a framework using classifying spaces for families and Lueck-Weierman pushouts to translate $TC_r(Γ)$ into obstructions in Bredon cohomology, proving a key vanishing/surjectivity result that yields $TC_r(Γ)=cd(Γ^r)$. For groups of type F, it shows $cd(Γ^r)=r\,cd(Γ)$, giving the TC-generating function $f_Γ(t)=cd(Γ)\frac{(2-t)t}{(1-t)^2}$ and affirming the rationality conjecture for this class. In particular, the results confirm the Farber-Oprea question for torsion-free hyperbolic groups and extend to certain toral relatively hyperbolic groups.

Abstract

We prove for non-elementary torsion-free hyperbolic groups $Γ$ and all $r\ge 2$ that the higher topological complexity ${\sf{TC}}_r(Γ)$ is equal to $r\cdot \mathrm{cd}(Γ)$. 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.

Theorems & Definitions (11)

• Theorem A
• Corollary B
• Corollary C
• Theorem 2.1: Lück--Weiermann
• Theorem 2.2: Farber--Oprea
• Lemma 3.1
• Lemma 3.2
• proof
• proof : Proof of Theorem \ref{['thmx.main']}
• proof : Proof of Corollary \ref{['corx.growth']}