Subshifts with slow complexity and simple groups with the Liouville property

Nicolás Matte Bon

Published 2014-02-10, updated 2014-05-23Version 3

We study random walk on topological full groups of subshifts, and show the existence of infinite, finitely generated, simple groups with the Liouville property. Results by Matui and Juschenko-Monod have shown that the derived subgroups of topological full groups of minimal subshifts provide the first examples of finitely generated, simple amenable groups. We show that if the (not necessarily minimal) subshift has a complexity function that grows slowly enough (e.g. linearly), then every symmetric and finitely supported probability measure on the topological full group has trivial Poisson-Furstenberg boundary. We also get explicit upper bounds for the growth of F{\o}lner sets.