Pragmatic finiteness properties of locally compact groups

TL;DR

This paper shows that three different proposed finiteness properties for locally compact groups are equivalent, unifying the understanding of their finiteness conditions despite ongoing uncertainties about their geometric interpretations.

Contribution

It proves the equivalence of three different families of finiteness properties for locally compact groups, clarifying their relationships and justifying their unified use.

Findings

All three properties define the same notion for locally compact groups.

The results clarify the literature on finiteness properties of these groups.

The work consolidates known arguments to establish the equivalence.

Abstract

We compare finiteness properties of locally compact groups that generalize the properties of being compactly generated and of being compactly presented. Three such families of properties have been proposed: Abels--Tiemeyer's type $C_n$, coarse $(n-1)$-connectedness, and Castellano--Corob-Cook's type $F_n$. The first was defined for locally compact groups, the second can be defined for general topological groups, while the third was defined only for tdlc groups. We prove that all three families lead to the same notion for locally compact groups. This justifies working with these properties despite the fact that it is still unclear in which sense they describe finiteness properties of free classifying spaces for locally compact groups. Various parts of the arguments are well-known to various experts. By putting them together we hope to clarify the literature.