Boundaries of Dehn fillings

Groves, Daniel; Manning, Jason Fox; Sisto, Alessandro

doi:10.2140/gt.2019.23.2929

Cited by 19 publications

(28 citation statements)

References 46 publications

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“…We now take a sequence of longer and longer fillings of this form, obtaining a collection {G G i } of quotients so that each G i is Kleinian. As in the proof of [19,Corollary 1.4] this induces a sequence of representations ρ i : G → Isom(H 3 ), and precisely as in [19] this sequence converges to a discrete faithful representation of G into Isom(H 3 ), so G is Kleinian, as required.…”

Section: Application To the Relative Cannon Conjecturementioning

confidence: 82%

“…To any relatively hyperbolic pair (G, P) is associated its Bowditch boundary ∂(G, P) [4]. The Relative Cannon Conjecture asserts that if P is a non-empty collection of free abelian groups and ∂(G, P) is homeomorphic to a 2-sphere, then G is Kleinian (see [19,Conjecture 1.3] and the discussion in that paper). The usual Cannon Conjecture makes the same assertion when P is empty and G has no non-trivial finite normal subgroup (see [9,Conjecture 11.34] and [10,Conjecture 5.1]).…”

Section: Application To the Relative Cannon Conjecturementioning

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: Application To the Relative Cannon Conjecturementioning

confidence: 82%

Section: Application To the Relative Cannon Conjecturementioning

confidence: 99%

Specializing cubulated relatively hyperbolic groups

Groves¹,

Manning²

2020

Preprint

Self Cite

View full text Add to dashboard Cite

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“…It appears, for instance, in the solution of the virtually Haken conjecture [AGM13], the study of Farrell-Jones conjecture and isomorphism problem of relatively hyperbolic groups [ACG18,DG18], and the construction of purely pseudo-Anosov normal subgroups of mapping class groups [DGO17]. Other applications of Dehn fillings can be found, for example, in [AGM16,GMS16].…”

Section: Group Theoretic Dehn Fillingsmentioning

confidence: 99%

“…The above property was first considered by Cohen-Lyndon [CL63], hence the name "Cohen-Lyndon triple". The interested reader is referred to [EH87, GMS16,Sun18] for more results about such triples.…”

Section: Cohen-lyndon Triplesmentioning

confidence: 99%

Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems

Sun

2020

Journal of Algebra

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This is the third paper in a series of three papers studying cohomology of group theoretic Dehn fillings. In the present paper, we apply the spectral sequence constructed in the previous two papers [Sun18, Sun19] to prove several results concerning cohomological properties of Dehn fillings. As a further application, we improve known results on SQ-universality and common quotients of acylindrically hyperbolic groups by adding cohomological finiteness conditions.

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“…It is known that for any hyperbolic space X there exists a 0 > 1 such that for every a ∈ (1, a 0 ) the sequential boundary ∂X admits a visual metric with parameter a, see [GMS19] for example. If X is an R-tree then a 0 = ∞, and for every parameter a > 1 there is a canonical visual metric given by d ∂X (p, q) = a −(p,q)x 0 .…”

Section: Visual Metrics On the Boundaries Of Quasi-treesmentioning

confidence: 99%

Tree approximation in quasi-trees

Kerr¹

2020

Preprint

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In this paper we investigate the geometric properties of quasi-trees, and prove some equivalent criteria. We give a general construction of a tree that approximates the ends of a geodesic space, and use this to prove that every quasi-tree is (1, C)-quasi-isometric to a simplicial tree. As a consequence we show that Gromov's Tree Approximation Lemma for hyperbolic spaces [Gro87] can be improved in the case of quasi-trees to give a uniform approximation for any set of points, independent of cardinality. From this we show that having uniform tree approximation for finite subsets is equivalent to being able to uniformly approximate the entire space by a tree. As another consequence we note that the boundary of a quasi-tree is isometric to the boundary of its approximating tree under a certain choice of visual metric, and that this gives a natural extension of the canonical metric on the boundary of a tree.

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Boundaries of Dehn fillings

Cited by 19 publications

References 46 publications

Specializing cubulated relatively hyperbolic groups

Specializing cubulated relatively hyperbolic groups

Cohomology of group theoretic Dehn fillings I: Cohen-Lyndon type theorems

Tree approximation in quasi-trees

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