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Diffeomorphism groups of tame Cantor sets and Thompson-like groups

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

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 louis.funar@univ-grenoble-alpes.fr
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 neretin@mccme.ru
Corresponding
E-mail address:
louis.funar@univ-grenoble-alpes.fr
neretin@mccme.ru
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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 group infinite type surfaces braided Thompson group diffeomorphism group Cantor set self-similar sets iterated 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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