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Title: Largest hyperbolic actions and quasi-parabolic actions in groups
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 the 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 the 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.  more » « less
Award ID(s):
2106906
PAR ID:
10334996
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Journal of Topology and Analysis
ISSN:
1793-5253
Page Range / eLocation ID:
1 to 45
Format(s):
Medium: X
Sponsoring Org:
National Science Foundation
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