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Diffeomorphism groups of tame Cantor sets and Thompson-like groups

Published online by Cambridge University Press:  03 April 2018

Louis Funar
Institute Fourier, UMR 5582, Department of Mathematics, University Grenoble Alpes, CS40700, 38058 Grenoble, CEDEX 9, France email
Yurii Neretin
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


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.

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