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It is shown that a Stallings-Swan theorem holds in a totally disconnectedlocally compact (= t.d.l.c.) context. More precisely, a compactly generatedCO-bounded t.d.l.c. group of rational discrete cohomological dimension less orequal to $1$ must be isomorphic to the fundamental group of a finite graph ofprofinite groups. This result generalizes Dunwoody's rational version of theclassical Stallings-Swan theorem to t.d.l.c. groups. The proof is based on thefact that a compactly generated unimodular t.d.l.c. group with rationaldiscrete cohomological dimension $1$ has necessarily non-positiveEuler-Poincar\'e characteristic.

A Stallings-Swan theorem for unimodular totally disconnected locally compact groups