On a problem raised by Alperin and Bass. I: Group actions on groupoids.

*(English)*Zbl 0741.20017R. 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’.

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 |

##### Keywords:

\(\Lambda\)-tree action; group actions on groupoids; non-abelian cohomology; cover morphism; universal cover; graph of groups
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\textit{Ş. A. Basarab}, J. Pure Appl. Algebra 73, No. 1, 1--12 (1991; Zbl 0741.20017)

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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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