Geometric invariants of locally compact groups: the homotopical perspective

TL;DR

This paper extends classical homotopical invariants of groups to locally compact groups using topological characters, providing new characterizations and relations with group properties, thus broadening the scope of geometric group theory.

Contribution

It introduces a topological version of Sigma-sets for locally compact groups, connecting them with classical invariants and establishing new properties and relations.

Findings

Characters in Sigma_top^n(G) are stable under passage to cocompact subgroups.

The set of nonzero characters in Sigma_top^n(G) is open.

Characters in groups of type C_n that do not vanish on the center are in Sigma_top^n(G).

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.