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Journal of Modern Dynamics (JMD)
 

Topological full groups of minimal subshifts with subgroups of intermediate growth

Pages: 67 - 80, Volume 9, 2015      doi:10.3934/jmd.2015.9.67

 
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Nicolás Matte Bon - Laboratoire de Mathémathiques d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex & DMA, École Normale Supérieure, 45 Rue d’Ulm, 75005, Paris, France (email)

Abstract:
This work is partially supported by the ERC starting grant GA 257110 “RaWG”. We show that every Grigorchuk group $G_\omega$ embeds in (the commutator subgroup of) the topological full group of a minimal subshift. In particular, the topological full group of a Cantor minimal system can have subgroups of intermediate growth, a question raised by Grigorchuk; moreover it can have finitely generated infinite torsion subgroups, answering a question of Cornulier. By estimating the word-complexity of this subshift, we deduce that every Grigorchuk group $G_\omega$ can be embedded in a finitely generated simple group that has trivial Poisson boundary for every simple random walk.

    This work is partially supported by the ERC starting grant GA 257110 “RaWG”.

Keywords:  Cantor minimal systems, growth of groups, periodic groups.
Mathematics Subject Classification:  Primary: 37B10; Secondary: 20F69, 20F65.

Received: August 2014;      Revised: January 2015;      Available Online: May 2015.

 References