  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.
