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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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##### References:
 [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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