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

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 $\mathcal{C}\mathcal{O}$-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--Poincaré characteristic (cf. Thm. H).

Final revised version. To appear in Mathematische Annalen