Geometric invariants of locally compact groups: the homotopical perspective

Abstract

We extend the classical theory of homotopical $\Sigma$-sets ${\Sigma }^n$ developed by Bieri, Neumann, Renz and Strebel for abstract groups, to $\Sigma$-sets ${\Sigma }_{\mathrm{top}}^n$ for locally compact Hausdorff groups. Given such a group $G$, our ${\Sigma }_{\mathrm{top}}^n(G)$ are sets of continuous homomorphisms $G\to \mathbb{R}$ ("characters"). They match the classical $\Sigma$-sets ${\Sigma }^n(G)$ if $G$ is discrete, and refine the homotopical compactness properties ${\mathrm{C}}_n$ of Abels and Tiemeyer. Moreover, our theory recovers the definition of ${\Sigma }_{\mathrm{top}}^1$ and ${\Sigma }_{\mathrm{top}}^2$ proposed by Kochloukova. Besides presenting various characterizations of ${\Sigma }_{\mathrm{top}}^n$ (particularly for $n\in \{1,2\}$), we show that characters in ${\Sigma }_{\mathrm{top}}^n(G)$ are also in ${\Sigma }_{\mathrm{top}}^n(H)$ if $H\le G$ is a closed cocompact subgroup, and we generalize several classical results. Namely, we prove that the set of nonzero elements of ${\Sigma }_{\mathrm{top}}^n(G)$ is open, we prove that characters in a group of type ${\mathrm{C}}_n$ that do not vanish on the center always lie in ${\Sigma }_{\mathrm{top}}^n(G)$, and we relate the $\Sigma$-sets of a group with those of its quotients by closed subgroups of type ${\mathrm{C}}_n$. Lastly, we describe how ${\Sigma }_{\mathrm{top}}^n(G)$ governs whether a closed normal subgroup with abelian quotient is of type ${\mathrm{C}}_n$, generalizing one of the highlights of the classical theory.

62 pages, 5 figures; version accepted for publication