Right-angled Artin groups, polyhedral products and the -generating function

Abstract

For a graph $\Gamma$ , let $K(H_{\Gamma },\,1)$ denote the Eilenberg–Mac Lane space associated with the right-angled Artin (RAA) group $H_{\Gamma }$ defined by $\Gamma$ . We use the relationship between the combinatorics of $\Gamma$ and the topological complexity of $K(H_{\Gamma },\,1)$ to explain, and generalize to the higher TC realm, Dranishnikov's observation that the topological complexity of a covering space can be larger than that of the base space. In the process, for any positive integer $n$ , we construct a graph $\mathcal {O}_n$ whose TC-generating function has polynomial numerator of degree $n$ . Additionally, motivated by the fact that $K(H_{\Gamma },\,1)$ can be realized as a polyhedral product, we study the LS category and topological complexity of more general polyhedral product spaces. In particular, we use the concept of a strong axial map in order to give an estimate, sharp in a number of cases, of the topological complexity of a polyhedral product whose factors are real projective spaces. Our estimate exhibits a mixed cat-TC phenomenon not present in the case of RAA groups.

Higher topological complexity of hyperbolic groups

“…The topological complexities TC r ( ) have been computed for several classes of groups (see e.g. Farber and Mescher 2020;Dranishnikov 2020 for r = 2, Farber andAguilar-Guzmán et al 2021;González et al 2016 for r ≥ 2, and references therein). In a celebrated result of Dranishnikov (2020) (see also Farber and Mescher 2020), the topological complexity TC 2 ( ) of groups with cyclic centralisers, such as hyperbolic groups, was shown to equal cd( × ).…”