Simple groups of dynamical origin

By Volodymyr V. Nekrashevych. Ergodic Theory and Dynamical Systems, 2015

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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 $H_{0}(\mathfrak{G},\mathbb{Z}/2\mathbb{Z})$ .

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/38ac5bf24b77a4e5ab2f48eebcf024a233439392

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Citations in the corpus

Etale groupoids arising from products of shifts of finite type (2015, background): “]] ab is deﬁned by sending the equivalence class of f to the equivalence class of τ 1τ 2...τn. Notice that it is not clear at all if this is well-deﬁned. One can ﬁnd a proof of the well-deﬁnedness in [31]. We introduce two properties of G which are related to the AH conjecture. We call τ ∈ [[G]] a t”