Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups

Groff, Bradley

doi:10.1142/s1793525313500192

Cited by 16 publications

(7 citation statements)

References 40 publications

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“…As mentioned above, Bowditch showed that for a one‐ended hyperbolic group

G

which is not cocompact Fuchsian the existence of a local cut point is equivalent to a splitting of

G

over a two‐ended subgroup. Generalizations of Bowditch's results have been studied by Papasoglu–Swenson , and Groff for

CAT (0)

and relatively hyperbolic groups, respectively.…”

Section: Introductionmentioning

confidence: 89%

“…Using cut pairs instead of local cut points Groff obtains a partial extension of Bowditch's JSJ tree construction for relatively hyperbolic groups, and Guralnik observed that in the special case that the relative boundary

\partial (Γ, P)

has no global cut points, then many of Bowditch's results about the valence of local cut points in the boundary of a hyperbolic group translate directly to the relatively hyperbolic setting. Their results were subsequently used by Groves and Manning to show that if

\partial (Γ, P)

has no global cut points and all the peripheral subgroups are one‐ended, then the existence of a local cut point in

\partial (Γ, P)

is equivalent to the existence of a splitting of

normalΓ

relative to

double-struckP

over a non‐parabolic 2‐ended subgroup.…”

Section: Introductionmentioning

confidence: 99%

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Boundary classification and two-ended splittings of groups with isolated flats

Haulmark

2018

Journal of Topology

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In this paper we provide a classification theorem for one-dimensional boundaries of groups with isolated flats. Given a group Γ acting geometrically on a CAT(0) space X with isolated flats and one-dimensional boundary, we show that if Γ does not split over a virtually cyclic subgroup, then ∂X is homeomorphic to a circle, a Sierpinski carpet, or a Menger curve. This theorem generalizes a theorem of Kapovich-Kleiner, and resolves a question due to Kim Ruane.We also study the relationship between local cut points in ∂X and splittings of Γ over two-ended subgroups. In particular, we generalize a theorem of Bowditch by showing that the existence of a local point in ∂X implies that Γ splits over a two-ended subgroup.

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“…As mentioned above, Bowditch showed that for a one‐ended hyperbolic group

G

which is not cocompact Fuchsian the existence of a local cut point is equivalent to a splitting of

G

over a two‐ended subgroup. Generalizations of Bowditch's results have been studied by Papasoglu–Swenson , and Groff for

CAT (0)

and relatively hyperbolic groups, respectively.…”

Section: Introductionmentioning

confidence: 89%

\partial (Γ, P)

\partial (Γ, P)

has no global cut points and all the peripheral subgroups are one‐ended, then the existence of a local cut point in

\partial (Γ, P)

is equivalent to the existence of a splitting of

normalΓ

relative to

double-struckP

over a non‐parabolic 2‐ended subgroup.…”

Section: Introductionmentioning

confidence: 99%

Boundary classification and two-ended splittings of groups with isolated flats

Haulmark

2018

Journal of Topology

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“…(See [6] for a complete proof.) A recent paper of Groff [15] extends this theorem to show that the Bowditch boundary of a relatively hyperbolic group is also well-defined up to quasi-isometry.…”

Section: Introductionmentioning

confidence: 90%

Contracting boundaries of CAT(0) spaces

Charney

Sultan

2014

Journal of Topology

107

201

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As demonstrated by Croke and Kleiner, the visual boundary of a CAT(0) group is not well‐defined since quasi‐isometric CAT(0) spaces can have non‐homeomorphic boundaries. We introduce a new type of boundary for a CAT(0) space, called the contracting boundary, made up of rays satisfying one of five hyperbolic‐like properties. We prove that these properties are all equivalent and that the contracting boundary is a quasi‐isometry invariant. We use this invariant to distinguish the quasi‐isometry classes of certain right‐angled Coxeter groups.

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“…Moreover the maximal parabolic groups are precisely the peripheral subgroups; by Theorem 2.9 and since conical limit points cannot be parabolic these are all bounded, and in particular the stabiliser of each bounded parabolic point is finitely-generated (Proposition 6. 13) We summarize all of this in a statement that will be used again in the next section: Proposition 6.14. -Given a representation ρ : Γ → GL(d, R) which is 1-dominated relative to P, ρ(Γ) acts on Λ rel as a geometrically-finite convergence group, with P Γ as the set of maximal parabolic subgroups.…”

Section: Summary Of Argumentmentioning

confidence: 99%

Relatively dominated representations

Zhu¹

2022

Annales De l'Institut Fourier

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Anosov representations give a higher-rank analogue of convex cocompactness in a rank-one Lie group which shares many of its good geometric and dynamical properties; geometric finiteness in rank one may be seen as a controlled weakening of convex cocompactness to allow for isolated failures of hyperbolicity. We introduce relatively dominated representations as a relativization of Anosov representations, or in other words a higher-rank analogue of geometric finiteness. We prove that groups admitting relatively dominated representations must be relatively hyperbolic, that these representations induce limit maps with good properties, provide examples, and draw connections to work of Kapovich-Leeb which also introduces higher-rank analogues of geometric finiteness.Résumé. -Les représentations Anosov fournissent une classe de sous-groupes discrets des groupes de Lie qui généralisent les sous-groupes convexes-cocompacts d'un groupe de Lie de rang un. En rang un, la classe des sous-groupes géométriquement finis est une généralisation de la classe des sous-groupes convexes-cocompacts, qui autorise des défauts isolés d'hyperbolicité. Nous introduisons les représentations relativement dominées comme une relativisation des représentations Anosov, autrement dit un analogue de la finitude géométrique en rang supérieur. Nous montrons qu'un groupe qui admet une représentation relativement dominée est nécessairement relativement hyperbolique et que ces représentations induisent des applications de bord satisfaisant des bonnes propriétés. Nous donnons des exemples et faisons des connexions avec le travail de Kapovich-Leeb sur d' autres analogues de la finitude géometrique en rang supérieur. ANNALES DE L'INSTITUT FOURIERRELATIVELY DOMINATED REPRESENTATIONS 5 Relatively hyperbolic groupsRelative hyperbolicity is a group-theoretic notion-originally suggested by Gromov in [14], and further developed by Bowditch [6], Farb [11], Yaman [25], Groves-Manning [15], and others-of non-positive curvature inspired by the geometry of cusped hyperbolic manifolds and free products.The geometry of a relatively hyperbolic group is akin to the geometry of a cusped hyperbolic manifold in that it is negatively-curved outside of certain regions, which, like the cusps in a cusped hyperbolic manifold, can be more or less separated from each other.There are various ways to make this intuition precise, resulting in various equivalent characterizations of relatively hyperbolic groups. We will use a definition of Bowditch, in the tradition of Gromov:Consider a finite-volume cusped hyperbolic manifold with an open neighborhood of each cusp removed: call the resulting truncated manifold M . The universal cover M of such a M is hyperbolic space with a countable set of horoballs removed. The universal cover M is not Gromov-hyperbolic; distances along horospheres that bound removed horoballs are distorted. If we glue the removed horoballs back in to the universal cover, however, the resulting space will again be hyperbolic space.We can do a similar thing from a gr...

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Quasi-Isometries, Boundaries and JSJ-Decompositions of Relatively Hyperbolic Groups

Cited by 16 publications

References 40 publications

Boundary classification and two-ended splittings of groups with isolated flats

Boundary classification and two-ended splittings of groups with isolated flats

Contracting boundaries of CAT(0) spaces

Relatively dominated representations

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