Higher rank confining subsets and hyperbolic actions of solvable groups

TL;DR

This paper extends the machinery of confining subsets to analyze hyperbolic actions of solvable groups with higher rank abelianizations, providing new classifications and reproofs of known invariants.

Contribution

It introduces an extended framework linking hyperbolic actions and confining subsets for higher rank solvable groups, broadening previous methods.

Findings

Complete description of hyperbolic actions for generalized solvable Baumslag-Solitar groups

Reproof of Bieri-Neumann-Strebel invariants for these groups

Extension of confining subset machinery to higher rank abelianizations

Abstract

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.