Poisson Boundary for Upper-Triangular Groups

We prove that finite entropy random walks on the torsion-free Baumslag group in dimension $d=2$ have non-trivial Poisson boundary. This is in contrast with the torsion case where the situation for simple random walks on Baumslag groups is the same as for the lamplighter groups of the same dimension. Our proof uses the realization of the Baumslag group as a linear group. We define and study a class of linear groups associated with multivariable polynomials which we denote $G_k(p)$. We show that the groups $G_3(p)$ have non-trivial Poisson boundary for all irreducible finite entropy measures, under a condition on the polynomial $p$ which we call the spaced polynomial property. We show that the Baumslag group has $G_3(1+x-y)$ as a subgroup, and that the polynomial $p = 1+x-y$, satisfies this property. Given any upper-triangle group of characteristic zero, we prove that one of the following must hold: 1) all finite second moment symmetric random walks on $G$ have trivial boundary 2) the group admits a block, which has a $3$ dimensional wreath product as a subgroup, and all non-degenerate random walks on $G$ have non-trivial boundary. 3) $G$ has a group $G_3(p)$ as a subgroup. We give a conjectural characterisation of all polynomials satisfying the spaced polynomial property. If this is confirmed, our result provides a characterisation of linear groups $G$ which admit a finitely supported symmetric random walk with non-trivial boundary.