Topological full groups of minimal subshifts with subgroups of intermediate growth

Nicolás Matte  Bon

doi:10.3934/jmd.2015.9.67

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

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Topological full groups of minimal subshifts with subgroups of intermediate growth

Nicolás Matte Bon^1,

1.
Laboratoire de Mathémathiques d’Orsay, Université Paris-Sud, F-91405 Orsay Cedex & DMA, École Normale Supérieure, 45 Rue d’Ulm, 75005, Paris

Received: July 31, 2014

Revised: December 31, 2014

Published: April 2015

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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”.

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Mathematics Subject Classification: Primary: 37B10; Secondary: 20F69, 20F65.

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