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

Higher rank confining subsets and hyperbolic actions of solvable groups