Some invariants of totally disconnected locally compact groups: cohomology and combinatorics

Abstract

The paper investigates two invariants for totally disconnected locally compact (t.d.l.c.) groups: the number of ends and the rational discrete cohomological dimension. Firstly, for a compactly generated t.d.l.c. group $G$, it is shown that its number of ends can be expressed in terms of the low-dimensional rational discrete cohomology of $G$, generalising a notable theorem by E. Specker to the t.d.l.c. setting. Secondly, we focus on a t.d.l.c. group $G$ acting under suitable conditions on a building $\Delta$, showing that both the number of ends and the rational cohomological dimension of $G$ are related to those of the Weyl group associated to $\Delta$. In special cases, we compare the rational discrete cohomological dimension of $G$ to its flat-rank, and in the appendix we provide examples for which these two invariants coincide. Finally, for t.d.l.c. groups with rational discrete cohomological dimension one that act properly and chamber-transitively on locally finite buildings, we explicitly construct a proper cocompact action on a tree out of the action on the building. All the results of independent interest in Coxeter theory have been included in the appendix.

arXiv comment40 pages, 1 figure. Changed auxiliary results and their proofs in the new Sect. 3.3 to fix a mistake in the old Example 2.2; the main result of the section (Thm 3.3) is unchanged. We integrated the referees' suggestions. Notably, we expanded the preliminaries, moved some results to an appendix, and adapted the abstract and the introduction accordingly. Version accepted by Rev. Math. Iberoam