Simple groups of dynamical origin

Abstract

We associate with every étale groupoid$\mathfrak{G}$two normal subgroups$\mathsf{S}(\mathfrak{G})$and$\mathsf{A}(\mathfrak{G})$of the topological full group of$\mathfrak{G}$, which are analogs of the symmetric and alternating groups. We prove that if$\mathfrak{G}$is a minimal groupoid of germs (e.g., of a group action), then$\mathsf{A}(\mathfrak{G})$is simple and is contained in every non-trivial normal subgroup of the full group. We show that if$\mathfrak{G}$is expansive (e.g., is the groupoid of germs of an expansive action of a group), then$\mathsf{A}(\mathfrak{G})$is finitely generated. We also show that$\mathsf{S}(\mathfrak{G})/\mathsf{A}(\mathfrak{G})$is a quotient of$H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.