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

TL;DR

This paper explores invariants like the number of ends and rational discrete cohomological dimension in totally disconnected locally compact groups, linking them to cohomology, group actions on buildings, and Coxeter group combinatorics.

Contribution

It establishes new relationships between group invariants, cohomology, and combinatorics, and provides explicit constructions for groups with specific cohomological properties.

Findings

Number of ends expressed via rational discrete cohomology.

Relations between invariants and Weyl groups in building actions.

Construction of cocompact actions on trees for certain cohomological dimensions.

Abstract

The paper investigates two invariants for totally disconnected locally compact groups: the number of ends and the rational discrete cohomological dimension. For such a compactly generated group $G$ it is shown that its number of ends can be expressed in terms of the rational discrete cohomology of $G$. If $G$ is suitably acting on a building the number of ends and the rational cohomological dimension of $G$ are related to those of the Weyl group associated to the building. In special cases, we are also able to compare the rational discrete cohomological dimension of $G$ to the flat-rank of $G$. Moreover, examples of groups for which these two invariants coincide are given. Our approach leverages the combinatorics of Coxeter groups, yielding new results of independent interest in Coxeter theory. Finally, in the class of totally disconnected locally compact groups acting properly and cocompactly on locally finite buildings, an accessibility result is proved: we explicitly construct a cocompact proper action on a tree if the rational discrete cohomological dimension is one.