We study actions of higher rank lattices $\Gamma <G$ on hyperbolic spaces and we show that all such actions satisfying mild properties come from the rank-one factors of G. In particular, all non-elementary isometric actions on an unbounded hyperbolic space are of this type.

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].

We identify the Poisson boundary of the dual of the universal compact quantum group Au(F) with a measurable field of ITPFI (infinite tensor product of finite type I) factors.

We provide an alternative algebraic and geometric approach to the results of [I. Bailleul, Probab. Theory Related Fields141 (2008) 283–329] describing the asymptotic behaviour of the relativistic diffusion.

The Poisson and Martin boundaries for invariant random walks on the dual of the orthogonal quantum groups $A_{\mathrm{o}}(F)$ are identified with higher-dimensional Podleś spheres that we describe in terms of generators and relations. This provides the first such identification for random walks on non-amenable discrete quantum groups.