Amenability, Reiter's condition and Liouville property

TL;DR

This paper proves the equivalence of the Liouville property and Reiter's condition for semigroupoids, confirming Kaimanovich's conjecture for measured and locally compact groupoids, and extends these results to semigroups and their actions.

Contribution

It establishes the equivalence of amenability, Reiter's condition, and the Liouville property for a broad class of algebraic structures, including groupoids and semigroups.

Findings

Liouville property and Reiter's condition are equivalent for semigroupoids

Confirmed Kaimanovich's conjecture for measured and locally compact groupoids

Extended results to semigroups and their actions

Abstract

We show that the Liouville property and Reiter's condition are equivalent for semigroupoids. This result applies to semigroups as well as semigroup actions. In the special case of measured groupoids and locally compact groupoids, our result proves Kaimanovich's conjecture of the equivalence of amenability and the Liouville property.