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Simple groups of dynamical origin

Published online by Cambridge University Press:  17 August 2017

V. NEKRASHEVYCH
V. NEKRASHEVYCH*
Affiliation:
Texas A&M University, Mathematics, College Station, Texas 77843-3368, USA email [email protected]
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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 $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 3 , March 2019 , pp. 707 - 732
Copyright
© Cambridge University Press, 2017 

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