Actions of small cancellation groups on hyperbolic spaces
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Actions of small cancellation groups on hyperbolic spaces Carolyn R. Abbott1
· David Hume2
Received: 25 February 2019 / Accepted: 11 August 2020 © Springer Nature B.V. 2020
Abstract We generalize Gruber–Sisto’s construction of the coned-off graph of a small cancellation group to build a partially ordered set T C of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber–Sisto coned-off graph. In almost all cases T C is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions [G X ] [G Y ] in this poset, there is an embeddeding ι : P(ω) → T C such that ι(∅) = [G X ] and ι(N) = [G Y ]. We use this poset to prove that there are uncountably many quasi-isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both. Keywords Hyperbolic spaces · Acylindrical actions · Small cancellation groups · Largest actions Mathematics Subject Classification 20F65 · 20F05 · 20F67
1 Introduction The study of acylindrical actions on hyperbolic spaces is a powerful tool for understanding algebraic properties of groups that admit aspects of non-positive curvature. The class of groups that admit such actions on non-elementary hyperbolic spaces, called acylindrically hyperbolic groups, is incredibly rich, including non-elementary hyperbolic and relatively hyperbolic groups, non-exceptional mapping class groups, Out(Fn ) for n ≥ 2, and nondirectly decomposable, non-virtually cyclic right-angled Artin and Coxeter groups, among many others. Moreover, the consequences of being acylindrically hyperbolic are far-reaching. Such groups are SQ-universal, have non-abelian free normal subgroups, a maximal finite normal subgroup, infinite dimensional second bounded cohomology, and a well-developed small cancellation theory [7,12,14,15].
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Carolyn R. Abbott [email protected] David Hume [email protected]
1
Department of Mathematics, Columbia University, New York, NY 10027, USA
2
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK
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Geometriae Dedicata
A single acylindrically hyperbolic group will admit many different acylindrical actions on different hyperbolic spaces, and it is natural to ask how these actions relate to each other. This kind of question was made precise in [4], where the authors and Osin define a partial order on the set of actions of a group on a metric space as follows: G X G Y if given any points x ∈ X , y ∈ Y , the map (G.y, dY ) → (G.x, d X ) given by g.y → g.x is coarsely Lipschitz1 . The largest action of a group in this partial ordering is always the action on its Cayley graph and the smallest action is the action on a point. Under this partial ordering, the set of (equivalence classes of) cobounded actions of a given group G on metric spaces has a natural poset structure; we call this poset Acb (G) [1]. More
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