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Topological Full Groups of Étale Groupoids

  • Conference paper
Operator Algebras and Applications

Part of the book series: Abel Symposia ((ABEL,volume 12))

Abstract

This is a survey of the recent development of the study of topological full groups of étale groupoids on the Cantor set. Étale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal \(\mathbb{Z}\)-actions, minimal \(\mathbb{Z}^{N}\)-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so on. Homology groups of étale groupoids, groupoid C ∗-algebras and their K-groups are also investigated.

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Author information

Authors and Affiliations

  1. Graduate School of Science, Chiba University, Inage-ku, Chiba, 263-8522, Japan

    Hiroki Matui

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  1. Hiroki Matui
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Correspondence to Hiroki Matui .

Editor information

Editors and Affiliations

  1. Department of Science and Technology, University of the Faroe Islands, Tórshavn, Faroe Islands

    Toke M. Carlsen

  2. Department of Mathematics, University of Oslo, Oslo, Norway

    Nadia S. Larsen

  3. Department of Mathematics, University of Oslo, Oslo, Norway

    Sergey Neshveyev

  4. Department of Mathematical Sciences, Norwegian University of Science and Technology, Trondheim, Norway

    Christian Skau

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Matui, H. (2016). Topological Full Groups of Étale Groupoids. In: Carlsen, T.M., Larsen, N.S., Neshveyev, S., Skau, C. (eds) Operator Algebras and Applications. Abel Symposia, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-319-39286-8_10

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