Unimodular totally disconnected locally compact groups of rational discrete cohomological dimension one

TL;DR

This paper extends the Stallings--Swan theorem to totally disconnected locally compact groups, showing that certain groups of cohomological dimension one are fundamental groups of finite graphs of profinite groups.

Contribution

It generalizes Dunwoody's rational Stallings--Swan theorem to the setting of t.d.l.c. groups, providing a structural classification based on cohomological dimension.

Findings

Groups of cohomological dimension ≤1 are fundamental groups of finite graphs of profinite groups.

Unimodular t.d.l.c. groups with dimension 1 have non-positive Euler–Poincaré characteristic.

The theorem broadens the understanding of the structure of t.d.l.c. groups with low cohomological dimension.

Abstract

It is shown that a Stallings--Swan theorem holds in a totally disconnected locally compact (= t.d.l.c.) context (cf. Thm. B). More precisely, a compactly generated $\mathcal{CO}$-bounded t.d.l.c. group $G$ of rational discrete cohomological dimension less than or equal to $1$ must be isomorphic to the fundamental group of a finite graph of profinite groups. This result generalises Dunwoody's rational version of the classical Stallings--Swan theorem to t.d.l.c. groups. The proof of Theorem B is based on the fact that a compactly generated unimodular t.d.l.c. group with rational discrete cohomological dimension $1$ has necessarily non-positive Euler--Poincar\'e characteristic (cf. Thm. H).