This paper is concerned with higher topological complexity of certain hyperbolic groups.
Let \(r \geq 2\) be an integer. Consider the path-fibration \(p : X^{[0,1]} \rightarrow X^{r}\) that maps a path \(\omega: [0, 1] \rightarrow X\) to the tuple \((\omega(0), \omega\frac{1}{r-1}, \dots ,\omega \frac{r-2}{r-1}, \omega(1))\). Then \(TC_{r} (X)\) is defined as the minimal integer \(n\) for which \(X^{r}\) can be covered by \(n + 1\) many open subsets \(U_{0}, \dots ,U_{n}\) such that \(p\) admits a local section over each \(U_{i}\). If no such \(n\) exists, one sets \(TC_{r}(X) :=\infty\). Note that \(TC_{2}(X)\) recovers the usual topological complexity.
Let \(\Gamma\) be a non-elementary torsion-free hyperbolic group. Since the higher topological complexities are homotopy invariants, one obtains interesting invariants of groups \(\Gamma\) by setting \(TC_{r} (\Gamma) := TC_{r}(K(\Gamma, 1))\), where \(K(\Gamma, 1)\) is an Eilenberg-MacLane space. In a celebrated result by A. Dranishnikov [Proc. Am. Math. Soc. 148, No. 10, 4547–4556 (2020; Zbl 1447.55002)] (see also [M. Farber and S. Mescher, J. Topol. Anal. 12, No. 2, 293–319 (2020; Zbl 1455.55003)]), the topological complexity \(TC_{2}(\Gamma)\) of groups with cyclic centralisers, such as hyperbolic groups, is equal \(cd(\Gamma \times \Gamma)\). Here \(cd\) denotes the cohomological dimension. The authors generalise this result to all higher topological complexities \(TC_{r} (\Gamma)\) for \(r \geq 2\), as well as to a larger class of groups containing certain toral relatively hyperbolic groups.
Theorem 1.1. Let \(r\geq 2\) and let \(\Gamma\) be a torsion-free group with \(cd(\Gamma) \geq 2\). Suppose that \(\Gamma\) admits a malnormal collection of abelian subgroups \(\lbrace P_i \mid i \in I \rbrace\) satisfying \(cd(P^{r}_{ i }) < cd(\Gamma^{r})\) such that the centraliser \(C_{\Gamma}(g)\) is cyclic for every \(g \in \Gamma\) that is not conjugate into any of the \(P_{i}\). Then \(TC_{r} (\Gamma) = cd(\Gamma^{r})\).
For a space \(X\), the \(TC\)-generating function \(f_{X} (t)\) is defined as the formal power series \[ f_{X} (t) := \sum^{\infty}_{r=1}TC_{r+1}(X) \cdot t^{r} . \] The \(TC\)-generating function of a group \(\Gamma\) is set to be \(f_{\Gamma}(t) := f_{K(\Gamma,1)}(t)\). Recall that a group \(\Gamma\) is said to be of type \(F\) (or geometrically finite) if it admits a finite model for \(K(\Gamma, 1)\).
Following [M. Farber and J. Oprea, Topology Appl. 258, 142–160 (2019; Zbl 1412.55003)], we say that a finite \(CW\)-complex \(X\) (resp. a group \(\Gamma\) of type \(F\)) satisfies the rationality conjecture if the \(TC\)-generating function \(f_{X} (t)\) (resp. \(f_{\Gamma}(t))\) is a rational function of the form \(\frac{P(t)}{(1-t)^{2}}\), where \(P(t)\) is an integer polynomial with \(P(1) = cat(X)\) (resp. \(P(1) = cd(\Gamma))\). Here cat denotes the Lusternik-Schnirelmann category. While a counter-example to the rationality conjecture for finite \(CW\)-complexes was found in [M. Farber et al., Topology Appl. 278, Article ID 107235, 4 p. (2020; Zbl 1440.55001)], the rationality conjecture for groups of type \(F\) remains open. The result in the present paper extends the class of groups for which the rationality conjecture holds as follows:
Corollary 1.2. Let \(\Gamma\) be a group as in Theorem 1.1. If \(\Gamma\) is of type \(F\), then \[ f_{\Gamma}(t) = cd(\Gamma) \frac{(2-t) t}{(1-t)^{2}}. \] In particular, the rationality conjecture holds for \(\Gamma\).
Corollary 1.3 The rationality conjecture holds for torsion-free hyperbolic groups.