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A Tits alternative for topological full groups

Part of: Topological dynamics Abstract harmonic analysis Smooth dynamical systems: general theory Special aspects of infinite or finite groups

Published online by Cambridge University Press:  27 August 2019

NÓRA GABRIELLA SZŐKE Open the ORCID record for NÓRA GABRIELLA SZŐKE [Opens in a new window]
NÓRA GABRIELLA SZŐKE*
Affiliation:
Institut Fourier, Université Grenoble Alpes, France email [email protected]
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Abstract

We prove a Tits alternative for topological full groups of minimal actions of finitely generated groups. On the one hand, we show that topological full groups of minimal actions of virtually cyclic groups are amenable. By doing so, we generalize the result of Juschenko and Monod for $\mathbf{Z}$-actions. On the other hand, when a finitely generated group $G$ is not virtually cyclic, then we construct a minimal free action of $G$ on a Cantor space such that the topological full group contains a non-abelian free group.

Keywords

group actionstopological dynamics

MSC classification

Primary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 43A07: Means on groups, semigroups, etc.; amenable groups
Secondary: 20F65: Geometric group theory 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 41 , Issue 2 , February 2021 , pp. 622 - 640
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Copyright
© Cambridge University Press, 2019

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