Free groupoids, trees, and free groups
Let \({\mathcal F}: E\rightrightarrows V\) be a directed graph (E the set of edges and V the set of vertices). Adding formal inverses to the elements of E and degenerate loops we get the graph \({\mathcal F}(E^{-1})\). The free groupoid \({\mathbb{F}}({\mathcal F})\) on the graph \({\mathcal F}\) is described as the groupoid whose objects are the vertices of \({\mathcal F}\) and whose arrows are the equivalence classes of paths of length \(\geq 0\) of the graph \({\mathcal F}(E^{-1})\). It is proved that if F: \({\mathbb{G}}\to {\mathbb{F}}\) is an essential equivalence of groupoids in which \({\mathbb{F}}\) is free on a graph \({\mathcal F}\), then F admits a quasi-inverse G such that \({\mathbb{G}}\) is free on the directed graph \({\mathcal G}\rightrightarrows {\mathcal O}b({\mathbb{G}})\) which consists of the non-identity images under G of the generators of \({\mathbb{F}}\) together with the natural isomorphisms \(\beta_ X: GF(X)\overset \sim \rightarrow X\), \(X\in {\mathcal O}b({\mathbb{G}})\) of the equivalence. It follows that if F: \({\mathbb{G}}\to {\mathbb{F}}({\mathcal F})\) is an essential equivalence of a group \({\mathbb{G}}\) with a free groupoid \({\mathbb{F}}({\mathcal F})\), then \({\mathbb{G}}\) is a free group freely generated by the set of all paths of the form \(g={\mathcal T}_ Y^{-1}f{\mathcal T}_ X\neq id(F(e))\), where \(f\in {\mathcal F}\), e is any (fixed) element and \({\mathcal T}_ X\) and \({\mathcal T}_ Y\) are the unique reduced paths from F(e) to X and Y whose generators lie in a fixed maximal tree \({\mathcal T}\subseteq {\mathcal F}\). Using this result, the author proves 1) the Nielsen-Schreier theorem: any subgroup \({\mathbb{G}}\) of a free group \({\mathbb{F}}({\mathcal F})\) is free, with free generators the set of elements \(t^{-1}gs\neq e\), where s,t\(\in S\), a Schreier transversal, \(s\in s{\mathbb{G}}\), \(g\in {\mathcal F}\), \(t\in gs{\mathbb{G}}\), and 2) the Serre theorem: a group \({\mathbb{G}}\) is free if and only if \({\mathbb{G}}\) can be made to act freely on a tree.
scientific article; zbMATH DE number 4198290
scientific article