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 $$\Sigma _\textrm{top}^n$$ for locally compact Hausdorff groups. Given such a group G , our $$\Sigma _\textrm{top}^n(G)$$ are sets of continuous homomorphisms $$G \rightarrow \mathbb {R}$$ (“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 $$\Sigma _\textrm{top}^1$$ and $$\Sigma _\textrm{top}^2$$ proposed by Kochloukova. Besides presenting various characterizations of $$\Sigma _\textrm{top}^n$$ (particularly for $$n\in \{1,2\}$$ ), we show that characters in $$\Sigma _\textrm{top}^n(G)$$ are also in $$\Sigma _\textrm{top}^n(H)$$ 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 $$\Sigma _\textrm{top}^n(G)$$ is open, we prove that characters in a group of type $$\mathrm C_n$$ that do not vanish on the center always lie in $$\Sigma _\textrm{top}^n(G)$$ , 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 $$\Sigma _\textrm{top}^n(G)$$ governs whether a closed normal subgroup with abelian quotient is of type $$\mathrm C_n$$ , generalizing one of the highlights of the classical theory.