Simple groups of dynamical origin

Volodymyr Nekrashevych

Published 2015-11-25Version 1

We associate with every etale groupoid $G$ two normal subgroups $S(G)$ and $A(G)$ of the topological full group of $G$, which are analogs of the symmetric and alternating groups. We prove that if $G$ is a minimal groupoid of germs (e.g., of a group action), then $A(G)$ is simple and is contained in every non-trivial normal subgroup of the full group. We also show that if $G$ is expansive (e.g., is the groupoid of germs of an expansive action of a group), then $A(G)$ is finitely generated. We show that $S(G)/A(G)$ is a quotient of $H_0(G, Z/2Z)$.