Higher rank confining subsets and hyperbolic actions of solvable groups

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 a family of groups related to Baumslag-Solitar groups.

Valuations, completions, and hyperbolic actions of metabelian groups

“…We note that both [16, Proposition 4.5] and [2, Section 3.2] consider only quasi-parabolic actions, but the discussion and proofs go through verbatim for lineal actions: both require only that G ñ BX has (at least one) global fixed point. By the discussion after [16, Theorem 4.1] (see also [2,Theorem 1.1]), the confining subset Q cannot be strictly confining since G ñ X is not quasi-parabolic. Thus Q " A.…”

“…The definition of a confining subset in this paper (Definition 2.10) is only for groups with abelianizations of rank 1. Nonetheless, groups of the more general form H ¸Zn , for H an abelian group and n ě 1, were considered by the authors in [2], in which an analogous concept of confining subsets was defined and some of the results of [16] were generalized. In this section, we work in this more general setting and introduce pseudo-valuations, which are functions that are a slight weakening of valuations.…”

“…It is thus natural to attempt to solve this classification problem for groups with fewer hyperbolic actions. The authors made the first steps towards this goal in [8,3,2] by completely classifying the hyperbolic actions of certain classically studied solvable groups. See also [7] for a recent classification of hyperbolic actions of certain lattices.…”

“…See also [7] for a recent classification of hyperbolic actions of certain lattices. The present paper was born from a realization that all of the classification results of [8,3,2] fit within a common framework. By identifying this common framework and developing new tools for working with groups that fit within it, we are able to classify the hyperbolic actions of numerous metabelian, and specifically abelian-by-cyclic, groups.…”

“…The partial order is roughly given by considering one action to be smaller than another when the smaller action may be obtained by an operation of collapsing an equivariant family of subspaces in the larger action. The equivalence classes in HpGq are called hyperbolic structures on G. The results of [8,3,2] all give a complete description of the poset HpGq for the group G in question. In this paper, for any group G fitting into our common framework, we give a partial description of the poset HpGq, while for groups G satisfying certain additional properties we give a complete description of HpGq.…”