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Rational discrete first degree cohomology for totally disconnected locally compact groups



It is well known that the existence of more than two ends in the sense of J.R. Stallings for a finitely generated discrete group G can be detected on the cohomology group H1(G,R[G]), where R is either a finite field, the ring of integers or the field of rational numbers. It will be shown (cf. Theorem A*) that for a compactly generated totally disconnected locally compact group G the same information about the number of ends of G in the sense of H. Abels can be provided by dH1(G, Bi(G)), where Bi(G) is the rational discrete standard bimodule of G, and dH(G, _) denotes rational discrete cohomology as introduced in [6].

As a consequence one has that the class of fundamental groups of a finite graph of profinite groups coincides with the class of compactly presented totally disconnected locally compact groups of rational discrete cohomological dimension at most 1 (cf. Theorem B).



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† This work was supported by GNSAGA-INdAM, by Programma SIR 2014 - MIUR (Project GADYGR) Number RBSI14V2LI cup G22I15000160008 and by EPSRC Grant N007328/1 Soluble Groups and Cohomology.



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Rational discrete first degree cohomology for totally disconnected locally compact groups



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