On a problem raised by Alperin and Bass. I: Group actions on groupoids (Q1176700)

\textit{R. Alperin} and \textit{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 \textit{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'.