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

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.