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Diffeomorphism groups of tame Cantor sets and Thompson-like groups

Part of: Smooth dynamical systems: general theory Low-dimensional topology Connections with other structures, applications Special aspects of infinite or finite groups

Published online by Cambridge University Press:  03 April 2018

Louis Funar  and
Yurii Neretin
Louis Funar
Affiliation:
Institute Fourier, UMR 5582, Department of Mathematics, University Grenoble Alpes, CS40700, 38058 Grenoble, CEDEX 9, France email [email protected]
Yurii Neretin
Affiliation:
Mathematics Department, University of Vienna, Nordbergstrasse 15, Vienna, Austria Institute for Theoretical and Experimental Physics, Mech. Math. Department, Moscow State University, Kharkevich Institute for Information Transmission Problems, Moscow, Russia email [email protected]
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Abstract

The group of ${\mathcal{C}}^{1}$ -diffeomorphisms of any sparse Cantor subset of a manifold is countable and discrete (possibly trivial). Thompson’s groups come out of this construction when we consider central ternary Cantor subsets of an interval. Brin’s higher-dimensional generalizations $nV$ of Thompson’s group $V$ arise when we consider products of central ternary Cantor sets. We derive that the ${\mathcal{C}}^{2}$ -smooth mapping class group of a sparse Cantor sphere pair is a discrete countable group and produce this way versions of the braided Thompson groups.

Keywords

mapping class groupinfinite type surfacesbraided Thompson groupdiffeomorphism groupCantor setself-similar setsiterated function systems

MSC classification

Primary: 20F36: Braid groups; Artin groups 37C85: Dynamics of group actions other than ${bf Z}$ and ${bf R}$, and foliations 57S05: Topological properties of groups of homeomorphisms or diffeomorphisms 57M50: Geometric structures on low-dimensional manifolds 54H15: Transformation groups and semigroups
Type
Research Article
Information
Compositio Mathematica , Volume 154 , Issue 5 , May 2018 , pp. 1066 - 1110
Copyright
© The Authors 2018 

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