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