(partly joint work with D.Osin, A.Rasmussen and C.Abbott) Recent papers of Balasubramanya and Abbott-Rasmussen have classified the hyperbolic actions of several families of classically studied solvable groups. I shall first talk about the known results in this context.
A key tool for these investigations is the machinery of confining subsets. This machinery applies to solvable groups with virtually cyclic abelianizations. I will then talk about recent work that extends this machinery to solvable groups with higher rank abelianizations.