Amenable category and complexity

Topological volumes of fibrations: a note on open covers

“…We recall the definition of the Lusternik-Schnirelmann category (of spaces and maps) and the generalization to fundamental group constraints [12,15,18,20,22,29].…”

Amenable covers of right‐angled Artin groups

“…Another invariant of a similar spirit is Farber's topological complexity 𝖳𝖢 which is motivated by the motion planning problem in robotics [12]. In [4,Question 8.1] it is asked for which topological spaces 𝑋 the following inequality holds:…”

“…Examples of spaces and groups satisfying this inequality can be found in [4,Section 8], and no counter-example seems to be known at the time of writing. We show that all right-angled Artin groups are positive examples.…”

“…Our proofs rely on combining upper and lower bounds (Lemma 2.2) with existing results on generalised LS-category, classifying spaces for families of subgroups, and homology growth from [4,21,23,28,30,33].…”

“…The amenable category is difficult to compute in general, the usual strategy being to exhibit an explicit open cover by amenable subsets and to prove its minimality using (co)homological obstructions. The precise value of $$\operatorname{cat}_\mathcal {AME}(G)$$ is known, for example, for the following classes of groups: $$\operatorname{cat}_\mathcal {AME}(G)=0$$ if and only if $$G$$ is amenable; $$\operatorname{cat}_\mathcal {AME}(G)=1$$ if and only if $$G$$ is a non‐amenable fundamental group of a graph of amenable groups [4, Corollary 5.4]; $$\operatorname{cat}_\mathcal {AME}(G)=\operatorname{cd}(G)$$ if $$G$$ is torsion‐free non‐elementary hyperbolic [27] [4, Example 7.8]. …”