  Recent papers of the authors have completely described the hyperbolic actions
of several families of classically studied solvable groups. A key tool for
these investigations is the machinery of confining subsets of Caprace,
Cornulier, Monod, and Tessera, which applies, in particular, to solvable groups
with virtually cyclic abelianizations. In this paper, we extend this machinery
and give a correspondence between the hyperbolic actions of certain solvable
groups with higher rank abelianizations and confining subsets of these more
general groups. We then apply this extension to give a complete description of
the hyperbolic actions of generalized solvable Baumslag-Solitar groups and to
reprove a result of Sgobbi-Wong computing their Bieri-Neumann-Strebel
invariants.
