Simple groups of dynamical origin

TL;DR

This paper introduces new normal subgroups within the topological full group of an etale groupoid, demonstrating their simplicity, finite generation under expansiveness, and their relation to homology, thus linking groupoid dynamics to algebraic properties.

Contribution

It defines the subgroups S(G) and A(G) as analogs of symmetric and alternating groups, proving A(G) is simple and finitely generated in certain cases, and relates S(G)/A(G) to homology.

Findings

A(G) is simple for minimal groupoids of germs.

A(G) is finitely generated if G is expansive.

S(G)/A(G) is a quotient of H_0(G, Z/2Z).

Abstract

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 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 also show that S(G)/A(G) is a quotient of H_0(G, Z/2Z).