Largest hyperbolic actions and quasi-parabolic actions in groups

Abstract

The set of equivalence classes of cobounded actions of a group on different hyperbolic metric spaces carries a natural partial order. The resulting poset thus gives rise to a notion of the "best" hyperbolic action of a group as the largest element of this poset, if such an element exists. We call such an action a largest hyperbolic action. While hyperbolic groups admit largest hyperbolic actions, we give evidence in this paper that this phenomenon is rare for non-hyperbolic groups. In particular, we prove that many families of groups of geometric origin do not have largest hyperbolic actions, including for instance many 3-manifold groups and most mapping class groups. Our proofs use the quasi-trees of metric spaces of Bestvina-Bromberg-Fujiwara, among other tools. In addition, we give a complete characterization of the poset of hyperbolic actions of Anosov mapping torus groups, and we show that mapping class groups of closed surfaces of genus at least two have hyperbolic actions which are comparable only to the trivial action.

Higher rank confining subsets and hyperbolic actions of solvable groups

“…The second author initiated this study by giving a complete description of the cobounded hyperbolic actions of the lamplighter groups (Z/nZ) Z for n ≥ 2 in [5]. The first and third authors then completely described the hyperbolic actions of Anosov mapping torus groups in [4] and the hyperbolic actions of solvable Baumslag-Solitar groups in [3].…”

“…The first two authors and Osin showed in [1] that the set H(G) of equivalence classes of cobounded hyperbolic actions of a group G admits a partial order, which roughly corresponds to collapsing equivariant families of subspaces to obtain one hyperbolic action from another (see Section 2 for the precise definition). Each of the papers [5], [3], and [4] gives a complete description of the poset H(G) for the group G in question.…”

“…The starting point for the techniques developed in [5,3,4] is a theory developed by Caprace, Cornulier, Monod, and Tessera in [7] that describes the cobounded hyperbolic actions of certain groups with rank one abelianizations, that is, groups whose abelianizations are virtually infinite cyclic. This includes groups G = H Z, with Z corresponding to a finite index subgroup of the abelianization.…”

“…However, the techniques developed in [5,3,4] do not immediately extend to solvable groups whose abelianizations have higher rank. In this paper we begin the work necessary to extend the theory of Caprace, Cornulier, Monod, and Tessera in [7] to finitely generated solvable groups in general.…”

“…Specifically, to prove Theorem 1.3 we utilize the classification of cobounded hyperbolic actions of the Baumslag-Solitar group BS (1, k) given in [3]. In light of this and the classification of cobounded hyperbolic actions of wreath products (Z/nZ) Z given in [5], a natural next step would be to apply Theorems 1.1 and 1.2 and the techniques developed in [5,2,4] to attempt to classify the hyperbolic actions of wreath products A B and extensions A B when A and B are finitely generated abelian groups. *…”