On homological properties of the Schlichting completion

TL;DR

This paper investigates how finiteness properties transfer from groups and subgroups to their Schlichting completions and provides criteria for cohomology isomorphisms, with applications to the Neretin group.

Contribution

It introduces new criteria for the transfer of finiteness properties and cohomology isomorphisms in Schlichting completions, with specific applications to the Neretin group.

Findings

Finiteness properties transfer from groups to their Schlichting completions.

Criteria established for when dense embeddings induce cohomology isomorphisms.

Continuous cohomology of the Neretin group vanishes in all positive degrees.

Abstract

We show how finiteness properties of a group and a subgroup transfer to finiteness properties of the Schlichting completion relative to this subgroup. Further, we provide a criterion when the dense embedding of a discrete group into the Schlichting completion relative to one of its subgroups induces an isomorphism in (continuous) cohomology. As an application, we show that the continuous cohomology of the Neretin group vanishes in all positive degrees.