Poisson boundary of group extensions

We prove a criterion for the Liouville property of random walks, which allows us to compare this property for different group extensions, and to check whether a given element in a group extension acts non-trivially on the Poisson boundary. We reduce the question about triviality of the boundary in nilpotent-by-Abelian groups to the same question on associated metabelian groups. In the particular case of a finitely generated amenable linear group we reduce to a finite collection of $2\times 2$ matrix metabelian groups, which we call basic metabelian blocks. We introduce a "cautiousness" criterion for the Liouville property. Applying this criterion to metabelian group and combining it with our reduction theorem we deduce a sufficient condition for boundary triviality for linear groups. We conjecture that our condition is also necessary. We prove this conjecture for linear groups over characteristic $p$ fields, by developing a $\Delta$-restriction entropy criterion for non-triviality of the boundary.