Partial actions of groups on cell complexes
Given a 2-complex \(L\), let \(I(L)\) denote the inverse monoid of partial automorphisms of \(L\), that is isomorphisms between subcomplexes of \(L\), with the usual composition of relations. A partial action of a group \(G\) on \(L\) is a dual prehomomorphism \(f:G \to I(L)\), that means a map satisfying the properties: \(f(1) = \text{id}_L\); \(f(g^{-1}) = f(g)^{-1}\); \(f(g_1) f(g_2) \leq f(g_1 g_2)\) with respect to the restriction order. An ordinary group action \(F:G \to \text{Aut}(K)\) on a complex \(K\) is called a globalization of \(f\) if \(L\) is a subcomplex of \(K\) (up to isomorphism) such that \(F(G)(L) = K\) and \(F(g) \cap L^2 = f(g)\) as a relation. By \textit{J. Kellendonk} and \textit{M. V. Lawson} [Partial actions of groups, to appear in Int. J. Algebra Comput.] any partial action has a (universal) globalization. In the paper under review, the author proposes partial actions and their globalizations as a natural way to approach several classical problems concerning group actions on complexes. As a main tool in this perspective, he proves the following theorem: a partial action \(f:G \to I(L)\) has a connected globalization iff it is true, that \(\{g \in G |f(g) \neq \emptyset\}\) generates \(G\); in this case any globalization of \(f\) is connected. Using this result, the author gives a combinatorial proof of the well-known fact that a group acting without fixed points on a connected 2-complex with finite quotient is finitely generated. In fact, he proves this statement in the more general setting of true partial actions. On the other hand, in contrast with the Bass-Serre theory of groups acting on graphs [\textit{J. P. Serre}, Trees, Springer (1980; Zbl 0548.20018)], finitely generated groups are characterized as those groups having a true partial action without fixed points on a finite tree. Finally, the Bass-Serre theory is used for determining, in terms of fundamental groups of graphs of groups, when a partial action on a graph admits a tree globalization. For example, an amalgamated free product or HNN-extension has such an action on a segment. The paper ends with some open problems and suggestions for further developments. In particular, the interesting idea of approximating ordinary actions on infinite complexes with partial actions induced on finite subcomplexes for computational purposes is presented.
scientific article; zbMATH DE number 1903793