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## MAXIMALLY STRETCHED LAMINATIONS ON GEOMETRICALLY FINITE HYPERBOLIC MANIFOLDS

Citations: | 12 - 7 self |

### Citations

1795 | Differential Geometry, Lie Groups and Symmetric - HELGASON - 1978 |

775 | A Haefliger, Metric spaces of non-positive curvature, manuscript of a book - Bridson |

251 | Manifolds of nonpositive curvature - Ballmann, Gromov, et al. - 1985 |

188 | The density at infinity of a discrete group of hyperbolic motions, - Sullivan - 1979 |

162 | The limit set of a Fuchsian group, - Patterson - 1976 |

136 |
The geometry of finitely generated kleinian groups,
- Marden
- 1974
(Show Context)
Citation Context ... PO(n, 1) = Isom(Hn) is open in Hom(Γ0, G) (see [B2, Prop. 4.1] for instance). The set of geometrically finite representations is open in the set of representations Γ0 → G of fixed cusp type if n ≤ 3 =-=[Ma]-=-, or if all cusps have rank ≥ n − 2 [B2, Prop. 1.8], but not in general for n ≥ 4 [B2, § 5]. In Sections 6.1.1 and 6.2 of the paper, where we examine the continuity properties of the function (j, ρ) 7... |

133 | Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. - Sullivan - 1984 |

118 |
The geometry and topology of 3-manifolds, Princeton lecture notes
- Thurston
- 1979
(Show Context)
Citation Context ... note that for a compact manifoldN , the so-called Ehresmann–Thurston principle asserts that the set of holonomies of all (not necessarily complete) (G,X)-structures on N is open in Hom(π1(N),G) (see =-=[T1]-=-). For n = 2, Klingler [Kl] proved that all (G,X)-structures on N are complete, which implies Corollary 7.8. For n > 2, it is not known whether all (G,X)-structures on N are complete; it has been conj... |

108 | The classification of Kleinian surface groups, II: The ending lamination conjecture.
- Brock, Canary, et al.
- 2012
(Show Context)
Citation Context ...y infinite representations j in dimension n = 3. Does Theorem 1.8 hold for finitely generated Γ0 but geometrically infinite j? To prove this in dimension 3, using the Ending Lamination Classification =-=[BCM]-=-, one avenue would be to extend Theorem 1.3 in a way that somehow allows the stretch locus E(j, ρ) to be an ending lamination. One would also need to prove a good quantitative rigidity statement for i... |

106 | Minimal stretch maps between hyperbolic surfaces. arXiv:9801039,
- Thurston
- 1986
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Citation Context ...e of the stretch locus E(j, ρ). In Section 9.1 we explain how, in the case that n = 2 and that j and ρ are both injective and discrete with finite covolume, Theorem 1.3 follows from Thurston’s theory =-=[T2]-=- of the asymmetric metric on Teichmüller space. More precise results in the case C(j, ρ) = 1 are given (for arbitrary n) in Section 5, leading to a reasonable understanding of the stretch locus when C... |

92 | On discontinuous groups in higher-dimensional symmetric spaces, Contributions to functional theory, - Selberg - 1960 |

88 |
Geometrical finiteness for hyperbolic groups
- Bowditch
- 1993
(Show Context)
Citation Context ...ry 1.12); in particular, C ′(j, ρ) < 1 implies C(j, ρ) < 1. In Theorem 1.8, the fact that if C(j, ρ) < 1 then (j, ρ) is left admissible easily follows from the general properness criterion of Benoist =-=[B1]-=- and Kobayashi [Ko3] (see Section 7.3). Conversely, suppose that (j, ρ) is left admissible. Then C ′(j, ρ) ≤ 1 (because (j, ρ) cannot be simultaneously left and MAXIMALLY STRETCHED LAMINATIONS 7 right... |

86 | Hausdorff dimension and Kleinian groups
- Bishop, Jones
- 1997
(Show Context)
Citation Context ...[Pat, S1, Ro1]. The Poincaré series ∑ γ∈Γ0 e−s d(p,j(γ)·p) converges for s > δ(j) and diverges for s ≤ δ(j). Equivalently, δ(j) is the Hausdorff dimension of the limit set of j(Γ0) [S1, S2] (see also =-=[BJ]-=-). The following remark implies that dTh is nonnegative on the level sets δ−1(r) ⊂ T (M) of the critical exponent function δ. Remark 8.2. For any j1, j2 ∈ T (M), δ(j1) δ(j2) ≤ C(j1, j2) = edTh(j1,j2).... |

82 |
Sur la sphère vide, Izvestia Akademii Nauk SSSR,
- Delaunay
- 1934
(Show Context)
Citation Context ...o M := j(Γ0)\Hn will automatically contain the convex core). The idea is to use a classical construction, the hyperbolic Delaunay decomposition (an analogue of the Euclidean Delaunay decomposition of =-=[D]-=-), and make sure that it is finite modulo j(Γ0). Let N ⊂ Hn be the preimage of the convex core of M = j(Γ0)\Hn and let N be the uniform 1-neighborhood of N . For R ≥ 0, we call R-hyperball of Hn any c... |

60 | Extending Lipschitz functions via random metric partitions.
- Lee, Naor
- 2005
(Show Context)
Citation Context ...such as Theorem 5.1 should be compared to a number of recent results in the theory of extension of Lipschitz maps: see Lang–Schröder [LS], Lang–Pavlović–Schröder [LPS], Buyalo–Schröder [BS], Lee–Naor =-=[LN]-=-, etc. We also point to [DGK] for an infinitesimal version. In fact, we can allow K to be the empty set in Theorem 1.6, in which case we define C0 to be the supremum of ratios λ(ρ(γ))/λ(j(γ)) for γ ∈ ... |

52 | Ergodicité et équidistribution en courbure négative - Roblin - 2003 |

47 | Renewal theorems in symbolic dynamics, with applications to geodesic flows, non-Euclidean tessellations and their fractal limits - Lalley - 1989 |

43 |
Proprietes asymptotiques des groups lineaires (II), Analysis on homogeneous spaces and representation theory of Lie groups,
- Benoist
- 1997
(Show Context)
Citation Context ....2.1] there are a finite subset F of Γ0 and a constant D ≥ 0 with the following property: for any γ ∈ Γ0 there is an element f ∈ F such that |µ(ρ(γf))− λ(ρ(γf))| ≤ D (the element γf is proximal — see =-=[B2]-=-). Then (7.2) and (7.3) imply µ(ρ(γ)) ≤ µ(ρ(γf)) + µ(ρ(f)) ≤ λ(ρ(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)λ(j(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)µ(j(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)µ(j(γ)) + c, where we set c := D +max f∈... |

41 |
Actions propres sur les espaces homogènes réductifs
- Benoist
- 1996
(Show Context)
Citation Context ...ry 1.12); in particular, C ′(j, ρ) < 1 implies C(j, ρ) < 1. In Theorem 1.8, the fact that if C(j, ρ) < 1 then (j, ρ) is left admissible easily follows from the general properness criterion of Benoist =-=[B1]-=- and Kobayashi [Ko3] (see Section 7.3). Conversely, suppose that (j, ρ) is left admissible. Then C ′(j, ρ) ≤ 1 (because (j, ρ) cannot be simultaneously left and MAXIMALLY STRETCHED LAMINATIONS 7 right... |

40 | Cusps are dense.’
- McMullen
- 1991
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Citation Context ...typically the fiber group of a hyperbolic surface bundle over the circle) may have small deformations j′(Γ0 that are not even discrete (e.g. small perturbations of a nearby cusp group in the sense of =-=[Mc]-=-). MAXIMALLY STRETCHED LAMINATIONS 73 7.8. Interpretation of Theorem 1.9 in terms of (G,X)-structures. We can translate Theorem 1.9 in terms of geometric structures, in the sense of Ehresmann and Thur... |

37 | Nonstandard Lorentz space forms.
- Goldman
- 1985
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Citation Context ...plete if the developing map is a covering; this is equivalent to a notion of geodesic completeness for the natural pseudo-Riemannian structure induced by the Killing form of the Lie algebra of G (see =-=[Go]-=-). For n > 2, the fundamental group of G0 = PO(n, 1)0 is finite, hence completeness is equivalent to the fact that the (G,X)-structure identifies N with the quotient of X by some discrete subgroup Γ o... |

37 |
Lorentz space forms and Seifert fiber spaces.
- Kulkarni, Raymond
(Show Context)
Citation Context ...e form (1.3) Γj,ρ0 = {(j(γ), ρ(γ)) | γ ∈ Γ0} where Γ0 is a discrete group and j, ρ ∈ Hom(Γ0, G) are representations with j injective and discrete (up to switching the two factors): this was proved in =-=[KR]-=- for n = 2, and in [Ka2] (strengthening partial results of [Ko2]) for general rank-one groups G. The group Γ is thus isomorphic to the fundamental group of the hyperbolic n-manifold M := j(Γ0)\Hn, and... |

34 | Semigroups containing proximal linear maps - Abels, Margulis, et al. - 1995 |

33 |
Completude des varietes lorentziennes a courbure constante.
- Klingler
- 1996
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Citation Context ...of [Z]) where Γ is contained in G×{1}; for n = 3, this was initially proved by Ghys [Gh]. The general case for n = 2 follows from the completeness of compact anti-de Sitter manifolds, due to Klingler =-=[Kl]-=-, and from the Ehresmann–Thurston principle on the deformation of holonomies of (G,X)-structures on compact manifolds. An interpretation of Theorem 1.9 in terms of (G,X)-structures will be given in Se... |

32 |
Kirszbraun’s theorem and metric spaces of bounded curvature.
- Lang, Schroeder
- 1997
(Show Context)
Citation Context ...–Ascoli theorem (see Section 5.4). Theorem 1.6 and its refinements such as Theorem 5.1 should be compared to a number of recent results in the theory of extension of Lipschitz maps: see Lang–Schröder =-=[LS]-=-, Lang–Pavlović–Schröder [LPS], Buyalo–Schröder [BS], Lee–Naor [LN], etc. We also point to [DGK] for an infinitesimal version. In fact, we can allow K to be the empty set in Theorem 1.6, in which case... |

31 |
Uber die zusammenziehende und Lipschitzsche Transformationen,”
- Kirszbraun
- 1934
(Show Context)
Citation Context ... Lipschitz maps in Hn. In order to prove Theorem 1.3, following the approach of [Ka1], we develop the extension theory of Lipschitz maps in Hn and, more precisely, refine an old theorem of Kirszbraun =-=[Kir]-=- and Valentine [V], which states that any Lipschitz map from a compact subset of Hn to Hn with Lipschitz constant ≥ 1 can be extended to a map from Hn to itself with the same Lipschitz constant. We pr... |

31 | action on a homogeneous space of reductive type - Kobayashi, Proper - 1989 |

25 | Séries de Poincaré des groupes géométriquement finis - Dal’bo, Otal, et al. |

24 |
Extensions of Lipschitz maps into Hadamard spaces
- Lang, Pavlović, et al.
(Show Context)
Citation Context ...5.4). Theorem 1.6 and its refinements such as Theorem 5.1 should be compared to a number of recent results in the theory of extension of Lipschitz maps: see Lang–Schröder [LS], Lang–Pavlović–Schröder =-=[LPS]-=-, Buyalo–Schröder [BS], Lee–Naor [LN], etc. We also point to [DGK] for an infinitesimal version. In fact, we can allow K to be the empty set in Theorem 1.6, in which case we define C0 to be the suprem... |

21 | Criterion of proper action on homogeneous space of reductive type,
- Kobayashi
- 1996
(Show Context)
Citation Context ...ular, C ′(j, ρ) < 1 implies C(j, ρ) < 1. In Theorem 1.8, the fact that if C(j, ρ) < 1 then (j, ρ) is left admissible easily follows from the general properness criterion of Benoist [B1] and Kobayashi =-=[Ko3]-=- (see Section 7.3). Conversely, suppose that (j, ρ) is left admissible. Then C ′(j, ρ) ≤ 1 (because (j, ρ) cannot be simultaneously left and MAXIMALLY STRETCHED LAMINATIONS 7 right admissible, as ment... |

20 | Discontinuous groups acting on homogeneous spaces of reductive type,
- Kobayashi
- 1992
(Show Context)
Citation Context ...rete group and j, ρ ∈ Hom(Γ0, G) are representations with j injective and discrete (up to switching the two factors): this was proved in [KR] for n = 2, and in [Ka2] (strengthening partial results of =-=[Ko2]-=-) for general rank-one groups G. The group Γ is thus isomorphic to the fundamental group of the hyperbolic n-manifold M := j(Γ0)\Hn, and the quotient of G by Γ = Γj,ρ0 is compact if and only if M is c... |

19 |
Deformations des structures complexes sur les espaces homogenes de SL(2,
- Ghys
- 1995
(Show Context)
Citation Context ...icular case of Theorem 1.9 was proved by Kobayashi [Ko4], namely the so-called “special standard” case (terminology of [Z]) where Γ is contained in G×{1}; for n = 3, this was initially proved by Ghys =-=[Gh]-=-. The general case for n = 2 follows from the completeness of compact anti-de Sitter manifolds, due to Klingler [Kl], and from the Ehresmann–Thurston principle on the deformation of holonomies of (G,X... |

17 |
Varietes anti-de Sitter de dimension 3 exotiques. Annales de l’Institut Fourier,
- Salein
- 2000
(Show Context)
Citation Context ...ed on G, namely quotients of G by discrete subgroups of G × G acting properly discontinuously and freely on G by left and right multiplication: (g1, g2) · g = g1gg−12 . This link was first noticed in =-=[Sa]-=-, then developed in [Ka1]. For n = 2, the manifolds locally modeled on PO(2, 1)0 ∼= PSL2(R) are the anti-de Sitter 3-manifolds, or Lorentzian 3-manifolds of constant negative curvature, which are Lore... |

16 |
Deformation of compact Clifford-Klein forms of indefinite Riemannian homogeneous manifolds,
- Kobayashi
- 1998
(Show Context)
Citation Context ...tural inclusion such that for all ϕ ∈ U , the group ϕ(Γ) is discrete in G × G and acts properly discontinuously, freely, and cocompactly on G. A particular case of Theorem 1.9 was proved by Kobayashi =-=[Ko4]-=-, namely the so-called “special standard” case (terminology of [Z]) where Γ is contained in G×{1}; for n = 3, this was initially proved by Ghys [Gh]. The general case for n = 2 follows from the comple... |

15 | Sur la fonction orbitale des groupes discrets en courbure negative, - Roblin - 2002 |

13 | Dégénérescences de sous-groupes discrets de groupes de Lie semisimples et actions de groupes sur des immeubles affines - Parreau |

12 | Shortening all the simple closed geodesics on surfaces with boundary
- Papadopoulos, Théret
(Show Context)
Citation Context ...LAMINATIONS 99 which can be bounded away from 0 (again by compactness: αi exits the convex core and G must not). This implies C ′s(j, ρ) < 1. Therefore E(j, ρ) = ∅. A similar argument can be found in =-=[PT]-=-. (This is an example where C ′s(j, ρ) < 1 = C(j, ρ) = C ′(j, ρ), the last equality coming from Lemma 7.4.) 10.9. A nonreductive, non-cusp-deteriorating ρ with C ′(j, ρ) < 1 = C(j, ρ) (and E(j, ρ) = ∅... |

11 |
Quotients compacts d’espaces homogenes reels ou p-adiques.
- Kassel
- 2009
(Show Context)
Citation Context ...ontinuation of [Ka1, Chap. 5], which focused on the case n = 2 and j convex cocompact. Some of our main results, in particular Theorems 1.8 and 1.11, Corollary 1.12, and Theorem 7.1, were obtained in =-=[Ka1]-=- in this case. 1.1. Equivariant maps of Hn with minimal Lipschitz constant. Let Γ0 be a discrete group. We say that a representation j ∈ Hom(Γ0, G) of Γ0 in G = PO(n, 1) is convex cocompact (resp. geo... |

10 |
actions on corank-one reductive homogeneous spaces
- Kassel, Proper
(Show Context)
Citation Context ...j(γ), ρ(γ)) | γ ∈ Γ0} where Γ0 is a discrete group and j, ρ ∈ Hom(Γ0, G) are representations with j injective and discrete (up to switching the two factors): this was proved in [KR] for n = 2, and in =-=[Ka2]-=- (strengthening partial results of [Ko2]) for general rank-one groups G. The group Γ is thus isomorphic to the fundamental group of the hyperbolic n-manifold M := j(Γ0)\Hn, and the quotient of G by Γ ... |

9 |
On closed anti-de Sitter spacetimes
- Zeghib
- 1998
(Show Context)
Citation Context ... in G × G and acts properly discontinuously, freely, and cocompactly on G. A particular case of Theorem 1.9 was proved by Kobayashi [Ko4], namely the so-called “special standard” case (terminology of =-=[Z]-=-) where Γ is contained in G×{1}; for n = 3, this was initially proved by Ghys [Gh]. The general case for n = 2 follows from the completeness of compact anti-de Sitter manifolds, due to Klingler [Kl], ... |

8 | The Hausdorff dimension of singular sets of properly discontinuous groups - Beardon - 1966 |

8 | Deformation of proper actions on reductive homogeneous spaces - Kassel |

7 |
Ergodic theory and rigidity on the symmetric spaces of non-compact type, Ergodic Theory Dynam
- Kim
(Show Context)
Citation Context ...−1(r) ⊂ T (M) of the critical exponent function is an asymmetric metric. The point of Proposition 1.13 is that dTh(j, ρ) = logC(j, ρ) = 0 implies j = ρ for j, ρ ∈ δ−1(r). For convex cocompact M , Kim =-=[Kim]-=- proved that logC ′(j, ρ) = 0 implies j = ρ, which yields Proposition 1.13 once Corollary 1.12 is proved. Here we give a direct proof in the general geometrically finite case. In dimension n ≤ 3 the a... |

7 | Contractions in non-Euclidean spaces - Valentine - 1944 |

5 |
Extension of Lipschitz maps into 3-manifolds
- Buyalo, Schröder
(Show Context)
Citation Context ...ts refinements such as Theorem 5.1 should be compared to a number of recent results in the theory of extension of Lipschitz maps: see Lang–Schröder [LS], Lang–Pavlović–Schröder [LPS], Buyalo–Schröder =-=[BS]-=-, Lee–Naor [LN], etc. We also point to [DGK] for an infinitesimal version. In fact, we can allow K to be the empty set in Theorem 1.6, in which case we define C0 to be the supremum of ratios λ(ρ(γ))/λ... |

5 | Geometry and topology of complete Lorentz spacetimes of constant curvature
- Danciger, Guéritaud, et al.
- 2013
(Show Context)
Citation Context ...e compared to a number of recent results in the theory of extension of Lipschitz maps: see Lang–Schröder [LS], Lang–Pavlović–Schröder [LPS], Buyalo–Schröder [BS], Lee–Naor [LN], etc. We also point to =-=[DGK]-=- for an infinitesimal version. In fact, we can allow K to be the empty set in Theorem 1.6, in which case we define C0 to be the supremum of ratios λ(ρ(γ))/λ(j(γ)) for γ ∈ Γ0 with j(γ) hyperbolic, wher... |

3 |
Spaces of geometrically finite representations
- Bowditch
- 1998
(Show Context)
Citation Context ....2.1] there are a finite subset F of Γ0 and a constant D ≥ 0 with the following property: for any γ ∈ Γ0 there is an element f ∈ F such that |µ(ρ(γf))− λ(ρ(γf))| ≤ D (the element γf is proximal — see =-=[B2]-=-). Then (7.2) and (7.3) imply µ(ρ(γ)) ≤ µ(ρ(γf)) + µ(ρ(f)) ≤ λ(ρ(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)λ(j(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)µ(j(γf)) +D + µ(ρ(f)) ≤ C ′(j, ρ)µ(j(γ)) + c, where we set c := D +max f∈... |

3 |
Groupes du ping-pong et géodésiques fermées en courbure −1, Ann. Inst. Fourier 46
- Dal’bo, Peigné
- 1996
(Show Context)
Citation Context ..., (8.3) ∫ X ψ dνγ ≤ ‖ψ‖∞ ·Nγ diam(Bp) λ(j(γ)) = ∆ Nγ λ(j(γ)) . For R > 0, let ΓR,j0 be the set of elements γ ∈ Γ0 such that j(γ) is primitive hyperbolic and λ(j(γ)) ≤ R. By [Ro2, Th. 5.1.1] (see also =-=[L, DP]-=- for special cases), (8.4) δ(j)R e−δ(j)R ∑ γ∈ΓR,j0 ∫ X ψ dνγ −→ R→+∞ ∫ X ψ dν = ε. Moreover, this convergence is still true if we replace ψ with the constant function equal to 1 on X [Ro2, Cor. 5.3], ... |

2 | Global rigidity of holomorphic Riemannian metrics on compact complex 3-manifolds
- Dumitrescu, Zeghib
(Show Context)
Citation Context ...d on PO(3, 1)0 ∼= PSL2(C) are the 3-dimensional complex holomorphic-Riemannian manifolds of constant nonzero curvature, which can be considered as complex analogues of the hyperbolic 3-manifolds (see =-=[DZ]-=- for details). For n = 2, all compact manifolds locally modeled on G are quotients of G by discrete subgroups of G × G, up to a finite covering [Kl, KR]; for n = 3, a similar property has been conject... |

2 |
Quotients compacts des groupes ultramétriques de rang un
- Kassel
(Show Context)
Citation Context ...n), over convex cocompact (resp. geometrically finite) hyperbolic manifolds, up to a finite covering (see Proposition 7.2 or [DGK, Th. 1.2]). We prove the following extension of Theorem 1.9 (see also =-=[Ka3]-=- for a p-adic analogue). Theorem 1.11. Let G = PO(n, 1) and let Γ be a discrete subgroup of G × G acting properly discontinuously on G, with a convex cocompact quotient. There is a neighborhood U ⊂ Ho... |

2 |
Discrete spectrum for non-Riemannian locally symmetric spaces
- Kassel, Kobayashi
(Show Context)
Citation Context ... to groups Γj,ρ0 with geometrically finite j and cusp-deteriorating ρ (Theorem 7.7); it is not true for n > 3. Theorem 1.8 implies that any geometrically finite quotient of G is sharp in the sense of =-=[KK]-=-; moreover, by Theorem 1.11, if the quotient is convex cocompact, then it remains sharp after any small deformation of the discrete group Γ inside G × G (see Section 7.7). This has analytic consequenc... |