Largest hyperbolic actions and quasi-parabolic actions in groups

TL;DR

This paper investigates the existence and structure of largest hyperbolic actions in groups, showing they are rare outside hyperbolic groups and providing detailed characterizations for specific classes like Anosov mapping tori.

Contribution

It introduces the concept of largest hyperbolic actions, proves their rarity in non-hyperbolic groups, and characterizes the poset of hyperbolic actions for certain geometric groups.

Findings

Hyperbolic groups admit largest hyperbolic actions.

Many 3-manifold and mapping class groups do not have largest hyperbolic actions.

Mapping class groups of closed surfaces have only trivial comparable hyperbolic actions.

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.