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

Subshifts with slow complexity and simple groups with the Liouville property