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$H(\mathbb{Z})$ of Monod
We give sufficient conditions for the non-triviality of the Poisson boundary of random walks on 
$H(\mathbb{Z})$ and its subgroups. The group 
$H(\mathbb{Z})$ is the group of piecewise projective homeomorphisms over the integers defined by Monod [Groups of piecewise projective homeomorphisms. Proc. Natl Acad. Sci. USA110(12) (2013), 4524–4527]. For a finitely generated subgroup 
$H$ of 
$H(\mathbb{Z})$, we prove that either 
$H$ is solvable or every measure on 
$H$ with finite first moment that generates it as a semigroup has non-trivial Poisson boundary. In particular, we prove the non-triviality of the Poisson boundary of measures on Thompson’s group 
$F$ that generate it as a semigroup and have finite first moment, which answers a question by Kaimanovich [Thompson’s group 
$F$ is not Liouville. Groups, Graphs and Random Walks (London Mathematical Society Lecture Note Series). Eds. T. Ceccherini-Silberstein, M. Salvatori and E. Sava-Huss. Cambridge University Press, Cambridge, 2017, pp. 300–342, 7.A].
