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

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