Relative cubulations and groups with a 2-sphere boundary

Einstein, Eduard; Groves, Daniel

doi:10.1112/s0010437x20007095

Cited by 10 publications

(27 citation statements)

References 24 publications

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“…We use the special case where elements of P are abelian to prove the Relative Cannon Conjecture for groups which are cubulated (Corollary 6.2) and for those which admit a weakly relatively geometric action on a CAT(0) cube complex (Theorem 6.1). This second result strengthens one of Einstein-Groves [15]. See Definition 1.5 for the definition of a weakly relatively geometric action.…”

Section: Introductionsupporting

confidence: 70%

“…The following result is a generalization of [15,Theorem 1.1]. Given the work we have already done, the proof is very similar.…”

Section: Application To the Relative Cannon Conjecturementioning

confidence: 61%

“…The notion of a relatively geometric action was defined in [15]; it can be obtained from the above by strengthening (3) to the requirement that every cell stabilizer is either finite or full parabolic.…”

Section: Introductionmentioning

confidence: 99%

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Specializing cubulated relatively hyperbolic groups

Groves¹,

Manning²

2020

Preprint

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In [1], Agol proved the Virtual Haken and Virtual Fibering Conjectures by confirming a conjecture of Wise: Every cubulated hyperbolic group is virtually special. We extend this result to cocompactly cubulated relatively hyperbolic groups with minimal assumptions on the parabolic subgroups. Our proof proceeds by first recubulating to obtain an improper action with controlled stabilizers (a weakly relatively geometric action), and then Dehn filling to obtain many cubulated hyperbolic quotients. We apply our results to prove the Relative Cannon Conjecture for certain cubulated or partially cubulated relatively hyperbolic groups.One of our main results (Theorem A) recovers via different methods a theorem of Oregón-Reyes [31].

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Section: Introductionsupporting

confidence: 70%

“…The following result is a generalization of [15,Theorem 1.1]. Given the work we have already done, the proof is very similar.…”

Section: Application To the Relative Cannon Conjecturementioning

confidence: 61%

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Specializing cubulated relatively hyperbolic groups

Groves¹,

Manning²

2020

Preprint

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“…Unfortunately, fineness and other important finiteness properties of the action of pG, Pq on the coned-off Cayley graph are not quasi-isometry invariants. In [EG20a], the first and second author introduced the notion of a relatively hyperbolic pair pG, Pq acting relatively geometrically on a CAT(0) cube complex r X. In this situation, a result of Charney and Crisp, [CC07, Theorem 5.1], implies that r X and its 1-skeleton r X p1q is quasi-isometric to the coned-off Cayley graph of pG, Pq.…”

Section: Introductionmentioning

confidence: 99%

Separation and Relative Quasi-convexity Criteria for Relatively Geometric Actions

Einstein¹,

Groves²,

Ng³

2021

Preprint

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Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasiconvexity as subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.

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“…On the other hand, work of Agol shows that hyperbolic groups acting properly and cocompactly on CAT(0) cube complexes are residually finite [Ago13, Theorem 1.1]. This article is concerned with the more general context of certain improper actions by relatively hyperbolic groups on CAT(0) cube complexes called relatively geometric actions developed by Einstein and Groves [EG20a]. The additional cubical geometry makes studying residual properties of relatively cubulated groups more tractable than for arbitrary relatively hyperbolic groups [EG20b,GM20].…”

Section: Introductionmentioning

confidence: 99%

Relative Cubulation of Small Cancellation Free Products

Einstein¹,

Ng²

2021

Preprint

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We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under C ′ ( 1 6 )-small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on Gromov hyperbolic CAT(0) cube complexes.Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion of Einstein and Groves and proving that wall stabilizers form a sufficiently rich family of full relatively quasi-convex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. For example, we prove that C ′ ( 1 6 )-small cancellation free products of residually finite groups are residually finite.

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Relative cubulations and groups with a 2-sphere boundary

Cited by 10 publications

References 24 publications

Specializing cubulated relatively hyperbolic groups

Specializing cubulated relatively hyperbolic groups

Separation and Relative Quasi-convexity Criteria for Relatively Geometric Actions

Relative Cubulation of Small Cancellation Free Products

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