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On a problem raised by Alperin and Bass. I: Group actions on groupoids. (English) Zbl 0741.20017
R. Alperin and H. Bass [in Combinatorial Group Theory and Topology, Alta 1984, Ann. Math. Stud. 111, 265-378 (1987; Zbl 0647.20016)] regard as a fundamental problem “to find the group-theoretic information carried by a \(\Lambda\)-tree action, analogous to that presented in J.-P. Serre’s book Trees (1980; Zbl 0548.20018) for the case \(\Lambda =\mathbb{Z}\)”. The author [in Proc. Workshop Arboreal Group Theory, Berkeley 1988, 35-68 (1991)] provides a possible answer if \(\Lambda\) is a totally-ordered abelian group. This paper deals with the group-theoretic aspects concerning group actions on groupoids.
Section 2 deals with non-abelian cohomology \(H^ 0\), \(H^ 1\), \(H^ 2\) of a group \(G\) with coefficients in a groupoid \(X\) on which \(G\) acts. {For \(H^ 0\), \(H^ 1\), there is some overlap with the reviewer’s paper [Proc. Lond. Math. Soc., III. Ser. 25, 413-426 (1972; Zbl 0245.20045)].} Section 3 introduces the notion of a cover morphism \((G,{\mathbf X})\to (G^ 1,{\mathbf X}^ 1)\) of group actions. The main result is that if \(G\) acts on a connected groupoid \(\mathbf X\), then there is a universal cover \((\widehat G,\widehat{\mathbf X})\), unique up to isomorphism, and for which \(\widehat{\mathbf X}\) is an indiscrete groupoid. Section 4 gives a combinatorial description of the group \(\widehat G\) using a notion of a ‘graph of groups’.
Reviewer: R.Brown (Bangor)

MSC:
20E08 Groups acting on trees
20L05 Groupoids (i.e. small categories in which all morphisms are isomorphisms)
20J05 Homological methods in group theory
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[1] Alperin, R.; Bass, H., Length functions of group actions on λ-trees, (), 265-377, Annals of Mathematics Studies · Zbl 0978.20500
[2] Basarab, Ş., On a problem raised by Alperin and bass, (), also in: Proceedings of the Workshop on Arboreal Group Theory, Berkeley 1988, to appear.
[3] Serre, J.-P., Trees, (1980), Springer New York
[4] Springer, T.A., Nonabelian H2 in Galois cohomology, (), 164-182
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