Non-Liouville groups with return probability exponent at most 1/2

abstract of the talk of Michał Kotowski

(joint work with Bálint Virág)

We construct a finitely generated group without the Liouville property such that the return probability of a random walk satisfies . Recent results suggest that is indeed the smallest possible return probability exponent for non-Liouville groups. Our construction is based on permutational wreath products over tree-like Schreier graphs and the analysis of large deviations of inverted orbits on such graphs. Joint work with Balint Virag.