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## Geometry and dynamics in Gromov hyperbolic metric spaces I -- with an emphasis . . . (2014)

Citations: | 2 - 2 self |

### Citations

1795 |
Differential Geometry, Lie Groups and Symmetric
- HELGASON
- 1978
(Show Context)
Citation Context ...C 1 4 [ Q(x+ y)−Q(x− y) +∑ℓ=i,j,k (− ℓQ(x+ yℓ) + ℓQ(x− yℓ))] F = Q guarantees that (2.3.2) BQ(Tx, Ty) = BQ(x,y) ∀x,y ∈ H. Comparing with (2.2.2) shows that [T ] is an isometry. 22In the notation of =-=[92]-=-, the spaces H p R , H p C , H p Q , and H 2 O are written as SO(p, 1)/ SO(p), SU(p, 1)/ SU(p), Sp(p, 1)/ Sp(p), and (f4(−20) , so(9)), respectively. GEOMETRY AND DYNAMICS IN HYPERBOLIC METRIC SPACES ... |

775 |
A Haefliger, Metric spaces of non-positive curvature, manuscript of a book
- Bridson
(Show Context)
Citation Context ...; second, we feel it is better methodologically to present the entire theory from scratch, in order to provide a basic reference for the theory, since no such reference exists currently (the closest, =-=[37]-=-, has a fairly different emphasis). To make things easier for the reader interested in nontrivially new results, we provide a summary below. In particular, the four most significant results of this pa... |

587 | Real Analysis - Modern Techniques and Their Applications - Folland - 1999 |

392 | Classical Descriptive set theory, Graduate texts in mathematics - Kechris - 1995 |

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

237 | The Octonions.
- Baez
- 2002
(Show Context)
Citation Context ...ing the statements slightly). We leave it to an algebraist to verify this. For the reader interested in learning more about the Cayley hyperbolic plane, see [128, pp.136-139], [150], or [6]; see also =-=[12]-=- for an excellent introduction to octonions in general. Fix F ∈ {R,C,Q} and an index set J , and let us construct a ROSSONCT of type F in dimension #(J). We remark that usually we will let J = N = {1,... |

229 | Lectures on Analysis on Metric Spaces, Universitext, - Heinonen - 2001 |

172 | La propriete (T) de Kazhdan pour les groupes localement compacts,” - Harpe, P, et al. - 1989 |

166 | la Harpe, Topics in geometric group theory - de - 2000 |

137 | Shrinkwrapping and the taming of hyperbolic 3-manifolds.
- Calegari, Gabai
- 2006
(Show Context)
Citation Context ..., Maskit, Jorgensen, Sullivan, Thurston, etc.) to study three-dimensional Kleinian groups has no generalization to higher dimensions. Moreover, the recent resolution of the Ahlfors measure conjecture =-=[2, 41]-=- has more to do with threedimensional topology than with analysis and dynamics. Indeed, the conjecture remains open in higher dimensions [104, p. 526, last paragraph]. Throughout the twentieth century... |

105 | Tameness of hyperbolic 3-manifolds
- Agol
- 2004
(Show Context)
Citation Context ...state the first major theorem of this paper, which generalizes all the aforementioned results: Theorem 1.2.1. Let G ≤ Isom(X) be a nonelementary group. Suppose either that (1) G is strongly discrete, =-=(2)-=- X is a CAT(-1) space and G is moderately discrete, (3) X is a ROSSONCT and G is weakly discrete, or that (4) X is a ROSSONCT and G acts irreducibly (cf. Subsection 7.6) and is COT-parametrically disc... |

103 |
Complex hyperbolic geometry. Oxford Mathematical Monographs.
- Goldman
- 1999
(Show Context)
Citation Context ...er important models of hyperbolic geometry. Note that the Poincaré ball model, which many of the figures of later sections are drawn in, is not discussed here. References for this subsection include =-=[43, 76]-=-. 2.5.1. The (Klein) ball model. Let B = BJF = {x ∈ H := HJF : ‖x‖ < 1}, and let bordB denote the closure of B relative to H. Observation 2.5.1. The map eB,H : bordB → bordH defined by the equation eB... |

101 | Valette: Kazhdan’s property (T - Bekka, Harpe, et al. - 2008 |

99 |
The degree of polynomial growth of finitely generated nilpotent groups
- Bass
- 1972
(Show Context)
Citation Context ...counting function of Γ interpreted as acting on the metric space (Γ, dΓ) (cf. Remark 8.1.3). The following analogue of (11.2.1) was proven by H. Bass and independently by Y. Guivarch: Theorem 11.2.1 (=-=[14, 85]-=-). Let Γ be a finitely generated nilpotent group with lower central series (Γi) ∞ 1 and nilpotency class k, and let αΓ = k∑ i=1 i rank(Γi/Γi+1). Let dΓ be a Cayley metric on Γ. Then for all R ≥ 1, (11... |

97 |
Visibility manifolds,
- Eberlein, O’Neill
- 1973
(Show Context)
Citation Context ... Isom(X) is either elliptic, parabolic, or loxodromic. Classification of groups has appeared in the literature in various contexts, from Eberlein and O’Neill’s results regarding visiblility manifolds =-=[66]-=-, through Gromov’s remarks about groups acting on strictly convex spaces [81, §3.5] and word-hyperbolic groups [83, §3.1], to the more general results of Hamann [86, Theorem 2.7], Osin [135, §3], and ... |

93 | Mesures de Patterson–Sullivan sur le bord d’un espace hyperbolique au sens de Gromov. - - Coornaert - 1993 |

91 |
The combinatorial structure of cocompact discrete hyperbolic groups.
- Cannon
- 1984
(Show Context)
Citation Context ... example, Dehn proved that the word problem is solvable for finitely generated Fuchsian groups [60], and this was generalized by Cannon to groups acting cocompactly on manifolds of negative curvature =-=[42]-=-. Gromov attempted to give a geometric characterization of these groups in terms of their Cayley graphs; he tried many definitions (cf. [81, §6.4], [82, §4]) before converging to what is now known as ... |

88 |
Geometrical finiteness for hyperbolic groups
- Bowditch
- 1993
(Show Context)
Citation Context ...skit’s condition that the limit set is the union of the radial limit set Λr with the set Λbp of bounded parabolic points [19], but the situation in higher dimensions was somewhat murky until Bowditch =-=[32]-=- wrote a paper which described which equivalences remain true in higher dimensions, and which do not. The condition of a finite-sided convex fundamental domain is no longer equivalent to any other con... |

86 | Hausdorff dimension and Kleinian groups - Bishop, Jones - 1997 |

85 |
On boundaries of Teichmuller spaces and on Kleinian groups:
- Bers
- 1970
(Show Context)
Citation Context ...nown for a long time that every finitely generated Fuchsian group has a finite-sided convex fundamental domain (e.g. [106, Theorem 4.6.1]). This result does not generalize beyond two dimensions (e.g. =-=[23, 100]-=-), but subgroups of Isom(H3) with finite-sided fundamental domains came to be known as geometrically finite groups. Several equivalent definitions of geometrical finiteness in the three-dimensional se... |

82 | Fuchsian groups. Chicago Lectures in Mathematics. - Katok - 1992 |

75 | On homogeneous manifolds of negative curvature - Heintze - 1974 |

72 | Metric Diophantine approximation on manifolds - Bernik, Dodson - 1999 |

64 | CAT(-1)-spaces, divergence groups and their commensurators - Burger, Mozes - 1996 |

63 | Structure conforme au bord et flot geodesique d’un CAT(−1)-espace’, - Bourdon - 1995 |

63 |
Groups with the Haagerup property. Gromov’s a-T-menability
- Cherix, Cowling, et al.
- 2001
(Show Context)
Citation Context ...iterature is the following: For which abstract groups Γ can Γ be embedded as a strongly discrete subgroup of Isom(B)? Such a group is said to have the Haagerup property.39 For a detailed account, see =-=[48]-=-. Remark 11.1.2. The following groups have the Haagerup property: • [58, pp.73-74] Groups which admit a cocompact action on a proper R-tree. In particular this includes Fn(Z) for every n. • [99] Amena... |

58 |
Über unendliche diskontinuierliche Gruppen
- Dehn
(Show Context)
Citation Context ... geometric group theory, in particular the study of groups acting on manifolds of negative curvature. For example, Dehn proved that the word problem is solvable for finitely generated Fuchsian groups =-=[60]-=-, and this was generalized by Cannon to groups acting cocompactly on manifolds of negative curvature [42]. Gromov attempted to give a geometric characterization of these groups in terms of their Cayle... |

58 | Hyperbolic manifolds, groups and actions. In Riemann surfaces and related topics - Gromov - 1978 |

52 | Approximate subgroups of linear groups,
- Breuillard, Green, et al.
- 2011
(Show Context)
Citation Context ...(x) is virtually nilpotent. It was proven recently by E. Breuillard, B. Green, and T. C. Tao [36, Corollary 1.15] that Margulis’s lemma holds on all metric spaces with bounded packing in the sense of =-=[36]-=-. This result includes Proposition 11.1.3 as a special case. By contrast, in infinite dimensions we have the following: Observation 11.1.4. Margulis’s lemma does not hold on the space X = E = E∞. Proo... |

48 |
The Dirichlet problem at infinity for manifolds of negative curvature.
- Anderson
- 1983
(Show Context)
Citation Context ...th (2.2.2) shows that [T ] is an isometry. 22In the notation of [92], the spaces H p R , H p C , H p Q , and H 2 O are written as SO(p, 1)/ SO(p), SU(p, 1)/ SU(p), Sp(p, 1)/ Sp(p), and (f4(−20) , so=-=(9)-=-), respectively. GEOMETRY AND DYNAMICS IN HYPERBOLIC METRIC SPACES 23 The group OF(L;Q) is quite large. In addition to containing all maps of the form T ⊕ I, where T ∈ OF(H; E) and I : F → F is the id... |

47 | Fuchsian groups and transitive horocycles - Hedlund - 1936 |

41 |
Limit points of Kleinian groups and finite sided fundamental polyhedra.
- Beardon, Maskit
- 1974
(Show Context)
Citation Context ...104, p. 526, last paragraph]. Throughout the twentieth century, there are several instances of theorems proven for three-dimensional Kleinian groups whose proofs extended easily to n dimensions (e.g. =-=[19, 128]-=-), but it seems that the theory of higher-dimensional Kleinian groups was not really considered a subject in its own right until around the 1990s. For more information on the theory of higher-dimensio... |

39 |
Propriétés statistiques des groupes de présentation finie
- Champetier
- 1995
(Show Context)
Citation Context ...of examples of hyperbolic metric spaces which are not CAT(-1) is furnished by the Cayley graphs of finitely presented groups. Indeed, we have the following: Theorem 3.3.7 ([84, p.78], [134]; see also =-=[47]-=-). Fix k ≥ 2 and an alphabet A = {a±11 , a±12 , · · · , a±1k }. Fix i ∈ N and a sequence of positive integers (n1, · · · , ni). Let N = N(k, i, n1, · · · , ni) be the number of group presentations G =... |

36 | Isometric group actions on Hilbert spaces: Growth of cocycles - Cornulier, Tessera, et al. |

33 |
Compact 3-manifolds of constant negative curvature fibering over the circle.
- Jørgensen
- 1977
(Show Context)
Citation Context ...nown for a long time that every finitely generated Fuchsian group has a finite-sided convex fundamental domain (e.g. [106, Theorem 4.6.1]). This result does not generalize beyond two dimensions (e.g. =-=[23, 100]-=-), but subgroups of Isom(H3) with finite-sided fundamental domains came to be known as geometrically finite groups. Several equivalent definitions of geometrical finiteness in the three-dimensional se... |

32 |
Linear differential equations and group theory from Riemann to Poincaré, volume 2nd ed
- Gray
- 2000
(Show Context)
Citation Context ...efore Poincaré and continued well after he had moved on to other areas, viz. that of Klein, Schottky, Schwarz, and Fricke. See [78, Chapter 3] for a brief exposition of this fascinating history, and =-=[77, 59]-=- for more in-depth presentations of the mathematics involved. We note that in finite dimensions, the theory of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-sp... |

30 | Hyperbolicity of the complex of free factors.
- Bestvina, Feighn
- 2011
(Show Context)
Citation Context ...construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex [93, 123, 94] and the free splitting and free factor complexes =-=[87, 25, 102, 94]-=-. These examples are particularly interesting as they extend the well-known dictionary [24, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging ... |

30 | Harmonic measures versus quasiconformal measures for hyperbolic groups - Blachére, Häıssinsky, et al. |

30 | Hyperbolic geometry,
- Cannon, Floyd, et al.
- 1997
(Show Context)
Citation Context ...al rank one symmetric spaces of noncompact type (ROSSONCTs). These spaces provide models of infinite-dimensional hyperbolic geometry. References for the theory of finite-dimensional ROSSONCTs include =-=[37, 43, 120]-=-; infinitedimesional symmetric spaces of noncompact type and finite rank are discussed in [64]. 2.1. The definition. Finite-dimensional ROSSONCTs come in four flavors, corresponding to the classical d... |

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

24 |
Discrete groups in space and uniformization problems, volume 40 of Mathematics and its Applications (Soviet Series
- Apanasov
- 1991
(Show Context)
Citation Context ...h equivalences remain true in higher dimensions, and which do not. The condition of a finite-sided convex fundamental domain is no longer equivalent to any other conditions in higher dimensions (e.g. =-=[10]-=-), so a higher-dimensional Kleinian group is said to be geometrically finite if it satisfies any of Bowditch’s five equivalent conditions (GF1)-(GF5). In infinite dimensions, Bowditch’s condition (GF5... |

23 | Introduction to Λ-trees, World Scientific - Chiswell - 2001 |

21 | Limit sets of discrete groups of isometries of exotic hyperbolic spaces,”Trans - Corlette, Iozzi |

21 | On hyperbolicity of free splitting and free factor complexes
- Kapovich, Rafi
(Show Context)
Citation Context ...construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex [93, 123, 94] and the free splitting and free factor complexes =-=[87, 25, 102, 94]-=-. These examples are particularly interesting as they extend the well-known dictionary [24, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging ... |

20 | On non-expansive mappings of Banach spaces - Edelstein - 1964 |

20 | Ergodic decomposition of quasi-invariant probability measures.
- Greschonig, Schmidt
- 2000
(Show Context)
Citation Context ... R satisfy [80, (1.1)-(1.3)]. Then by [80, Theorem 1.4], there is a measure µ̂ satisfying (15.3.1) supported on the set of ergodic probability measures which are “̺-admissible” (in the terminology of =-=[80]-=-). But by [80, (1.1)], we have b̺(g,ξ) ≍× g′(ξ)s for µ-a.e. ξ ∈ ∂X , say for all ξ ∈ ∂X \ S, where µ(S) = 0. Then every ̺-admissible measure ν satisfying ν(S) = 0 is s-quasiconformal. But by (15.3.1),... |

20 | Non-expanding maps and Busemann functions. Ergodic Theory Dynam - Karlsson |

19 |
Groupes de Lie à croissance polynomiale
- Guivarc’h
- 1970
(Show Context)
Citation Context ...counting function of Γ interpreted as acting on the metric space (Γ, dΓ) (cf. Remark 8.1.3). The following analogue of (11.2.1) was proven by H. Bass and independently by Y. Guivarch: Theorem 11.2.1 (=-=[14, 85]-=-). Let Γ be a finitely generated nilpotent group with lower central series (Γi) ∞ 1 and nilpotency class k, and let αΓ = k∑ i=1 i rank(Γi/Γi+1). Let dΓ be a Cayley metric on Γ. Then for all R ≥ 1, (11... |

19 | Slim unicorns and uniform hyperbolicity for arc graphs and curve graphs.
- Hensel, Przytycki, et al.
- 2013
(Show Context)
Citation Context ... we could simultaneously answer our own questions about H∞ and construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex =-=[93, 123, 94]-=- and the free splitting and free factor complexes [87, 25, 102, 94]. These examples are particularly interesting as they extend the well-known dictionary [24, p.375] between mapping class groups and t... |

18 |
The Hopf-Rinow theorem is false in infinite dimensions
- Atkin
- 1975
(Show Context)
Citation Context ...t of the theory of infinite-dimensional Riemannian manifolds would be too much of a digression, let us make the following points: • An infinite-dimensional analogue of the Hopf–Rinow theorem is false =-=[11]-=-, i.e. there exists an infinite-dimensional Riemannian manifold such that some two points on that manifold cannot be connected by a geodesic. However, if an infinite-dimensional Riemannian manifold X ... |

17 | Reflection groups on the octave hyperbolic plane.
- Allcock
- 1999
(Show Context)
Citation Context ...y after modifying the statements slightly). We leave it to an algebraist to verify this. For the reader interested in learning more about the Cayley hyperbolic plane, see [128, pp.136-139], [150], or =-=[6]-=-; see also [12] for an excellent introduction to octonions in general. Fix F ∈ {R,C,Q} and an index set J , and let us construct a ROSSONCT of type F in dimension #(J). We remark that usually we will ... |

15 |
Borel cocycles, approximation properties and relative property T. Ergodic Theory Dynam
- Jolissaint
(Show Context)
Citation Context ..., see [48]. Remark 11.1.2. The following groups have the Haagerup property: • [58, pp.73-74] Groups which admit a cocompact action on a proper R-tree. In particular this includes Fn(Z) for every n. • =-=[99]-=- Amenable groups. This includes solvable and nilpotent groups. A class of examples of groups without the Haagerup property is the class of infinite groups with Kazdan’s property (T). For example, if n... |

14 | A hyperbolic Out(Fn - Bestvina, Feighn |

14 | Discrete subgroups of the Lorentz group - Greenberg - 1962 |

12 | Melián : Bounded geodesics of Riemann surfaces and hyperbolic manifolds
- Fernández, V
- 1995
(Show Context)
Citation Context ...paces; see Definitions 7.1.2, 7.2.1, and 8.1.1. The radial limit set was introduced by Hedlund in 1936 in his analysis of transitivity of horocycles [88, Theorem 2.4]. After some intermediate results =-=[69, 152]-=-, Bishop and Jones [26, Theorem 1] generalized Patterson and Sullivan by proving that if G is a nonelementary Kleinian group, then dimH(Λr) = dimH(Λur) = δ. 9 Further generalization was made by Paulin... |

11 | Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature - Ancona - 1994 |

10 | Cantat: Dynamical Degrees of Birational transformations of projective surfaces
- Blanc, Serge
(Show Context)
Citation Context ... extend the well-known dictionary [24, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging between mapping class groups and Cremona groups, see =-=[28, 62]-=-. We speculate that developing the Patterson–Sullivan theory in the three areas would be fruitful and may lead to new connections between these areas. There is a longer story of which this paper is on... |

10 | Equivariant embeddings of trees into hyperbolic spaces
- Burger, Iozzi, et al.
- 2005
(Show Context)
Citation Context ...o deaf ears, and the geometry and representation theory of infinitedimensional hyperbolic space H∞ and its isometry group have been studied in the last decade by a handful of mathematicians, see e.g. =-=[38, 61, 127]-=-. However, infinite-dimensional hyperbolic geometry has come into prominence most spectacularly through the recent resolution of a long-standing conjecture in algebraic geometry due to Enriques from t... |

10 | The hyperbolicity of the sphere complex via surgery paths.
- Hilion, Horbez
- 2012
(Show Context)
Citation Context ... we could simultaneously answer our own questions about H∞ and construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex =-=[93, 123, 94]-=- and the free splitting and free factor complexes [87, 25, 102, 94]. These examples are particularly interesting as they extend the well-known dictionary [24, p.375] between mapping class groups and t... |

10 |
Fuchsian groups and ergodic theory
- Hopf
- 1936
(Show Context)
Citation Context ...rticular . . . ” but it follows easily from the equivalence of (A) and (C). Remark 1.4.3. Theorem 1.4.2 has a long history. The equivalence (B)⇔ (C) was first proven by E. Hopf in the case δ = d− 114 =-=[97, 98]-=- (1936, 1939). The equivalence (A) ⇔ (B) was proven by Z. Yûjôbô in the case δ = d−1 = 1 [170] (1949), following an incorrect proof by M. Tsuji [162] (1944).15 Sullivan proved (A) ⇔ (C) in the case... |

8 |
A theorem on triangles in a metric space and some of its applications
- Aleksandrov
- 1951
(Show Context)
Citation Context ...urved Riemannian manifolds. The idea was to describe the most important consequences of negative curvature in terms of the metric structure of the manifold. This approach was pioneered by Aleksandrov =-=[5]-=-, who discovered for each κ ∈ R an inequality regarding triangles in a metric space with the property that a Riemannian manifold satisfies this inequality if and only if its sectional curvature is bou... |

8 |
The Hausdorff dimension of singular sets of properly discontinuous groups
- Beardon
- 1966
(Show Context)
Citation Context ...e “thickness” of the limit set of a Fuchsian group: in 1941 Myrberg [130] showed that the limit set Λ of a nonelementary Fuchsian group has positive logarithmic capacity; this was improved by Beardon =-=[15]-=- who showed that Λ has positive Hausdorff dimension, thus deducing Myrberg’s result as a corollary (since positive Hausdorff dimension implies positive logarithmic capacity for compact subsets of R2 [... |

8 | On the growth of quotients of kleinian groups, Ergodic Theory and Dynamical Systems 31 - Dal’bo, Peigné, et al. |

8 | Diophantine approximation and the geometry of limit sets in Gromov hyperbolic metric spaces
- Fishman, Simmons, et al.
(Show Context)
Citation Context ...existence of a δ-quasiconformal measure for groups of divergence type, even if the space they are acting on is not proper. We note that weaker versions of Theorem 1.2.1 already appeared in the papers =-=[55, 70]-=-, each of which has a two-author intersection with the present paper. Remark 1.0.1. In [70] we also included as standing assumptions that G was strongly discrete and of general type (see Definitions 5... |

7 | The nonsolvability of the Dirichlet problem on negatively curved manifolds. Differential Geometry and its Applications 8: 217 - Borbély - 1998 |

7 |
finiteness with variable negative curvature
- Geometrical
- 1995
(Show Context)
Citation Context ...(GF5). In infinite dimensions, Bowditch’s condition (GF5) does not make sense, as it relies on the notion of volume. (GF3) seems unlikely to yield a good definition in any general context; indeed, in =-=[34]-=-, Bowditch showed that it does not even generalize to the setting of finite-dimensional CAT(-1) manifolds. It is easy to show that (GF1) implies (GF4) in all CAT(-1) spaces, but the reverse direction ... |

7 |
Dynamical systems and categories
- Dimitrov, Haiden, et al.
(Show Context)
Citation Context ... extend the well-known dictionary [24, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging between mapping class groups and Cremona groups, see =-=[28, 62]-=-. We speculate that developing the Patterson–Sullivan theory in the three areas would be fruitful and may lead to new connections between these areas. There is a longer story of which this paper is on... |

7 | groups as geometric objects - Infinite - 1983 |

7 | Bowen-Margulis and Patterson measures on negatively curved compact manifolds, Dynamical Systems and Related Topics
- Kaimanovich
- 1990
(Show Context)
Citation Context ...ausdorff dimension, the Patterson–Sullivan measure also relates to the spectral theory of the Laplacian (e.g. [137, Theorem 3.1], [155, Proposition 28]) and the geodesic flow on the quotient manifold =-=[101]-=-. An important property of Patterson–Sullivan measures is conformality. Given s > 0, a measure µ on ∂Bd is said to be s-conformal with respect to a discrete group G ≤ Isom(Bd) if (1.4.1) µ(g(A)) = ∫ A... |

6 | Geodesics in Hadamard spaces. Algebra i Analiz - Buyalo - 1998 |

6 |
de Saint-Gervais, Uniformisation des surfaces de Riemann. Retour sur un théorème centenaire. ENS Éditions
- P
- 2010
(Show Context)
Citation Context ...efore Poincaré and continued well after he had moved on to other areas, viz. that of Klein, Schottky, Schwarz, and Fricke. See [78, Chapter 3] for a brief exposition of this fascinating history, and =-=[77, 59]-=- for more in-depth presentations of the mathematics involved. We note that in finite dimensions, the theory of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-sp... |

6 |
The free splitting complex of a free group
- Handel, Mosher
(Show Context)
Citation Context ...construct a theoretical framework for those who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex [93, 123, 94] and the free splitting and free factor complexes =-=[87, 25, 102, 94]-=-. These examples are particularly interesting as they extend the well-known dictionary [24, p.375] between mapping class groups and the groups Out(FN ). In another direction, a dictionary is emerging ... |

5 | Conformal measures associated to ends of hyperbolic n-manifolds
- Anderson, Falk, et al.
(Show Context)
Citation Context ...mensions. Although the Patterson–Sullivan theorem guarantees the existence of a δconformal measure, it does not guarantee its uniqueness. Indeed, the δ-conformal measure is often not unique; see e.g. =-=[8]-=-. However, it turns out that the hypothesis of divergence type is enough to guarantee uniqueness. In fact, the condition of divergence type turns out to be quite important in the theory of conformal m... |

5 | Embeddings of Gromov hyperbolic spaces’, Geom. Funct - Bonk, Schramm |

5 |
On the Lipschitz equivalence of Cantor sets, Mathematika 39
- Falconer, Marsh
- 1992
(Show Context)
Citation Context ...mal iterated function systems [125] (or in fact even graph directed Markov systems [126]) satisfying the strong separation condition (also known as the disconnected open set condition [144]; see also =-=[68]-=-, where the limit sets of iterated function systems satisfying the strong separation condition are called dust-like). Indeed, the notion of a partition structure was intended primarily to generalize t... |

5 | Algebras of differentiable functions on Riemannian manifolds - Garrido, Jaramillo, et al. |

5 |
On the absence of Sullivan’s cusp finiteness theorem in higher dimensions
- Kapovich
- 1995
(Show Context)
Citation Context ...ss Theorem [156], which applies to all finitely generated subgroups of Isom(H3) (not just the geometrically finite ones). However, the Cusp Finiteness Theorem does not generalize to higher dimensions =-=[103]-=-. Proof. Let H be the collection of horoballs defined in the proof of (B3) ⇒ (A), i.e. H = {g(Hp) : p ∈ P} for some finite set P . We claim that Λbp = G(P ). Indeed, fix ξ ∈ Λbp. By the proof of (A) ⇒... |

4 |
Normal subgroups in the Cremona group. With an appendix by Yves de
- Cantat, Lamy
(Show Context)
Citation Context ...eometry has come into prominence most spectacularly through the recent resolution of a long-standing conjecture in algebraic geometry due to Enriques from the late nineteenth century. Cantat and Lamy =-=[45]-=- proved that the Cremona group (i.e. the group of birational transformations of the complex projective plane) has uncountably many non-isomorphic normal subgroups, thus disproving Enriques’ conjecture... |

4 |
Some Results on Infinite-Dimensional Convexity
- Fonf, Lindenstrauss
- 1998
(Show Context)
Citation Context ...ngly discrete is crucial for Proposition 12.1.5. In general, tiling Hilbert spaces turns out to be a very subtle problem and has been studied (among others) by Klee [111, 112], Fonf and Lindenstrauss =-=[72]-=- and most recently by Preiss [140]. 12.2. Cobounded and convex-cobounded groups. Before studying geometrically finite groups, we begin by considering the simpler case of cobounded and convex-cobounded... |

4 |
Group actions on metric spaces: fixed points and free subgroups
- Hamann
(Show Context)
Citation Context ...amann [86, Theorem 2.7], Osin [135, §3], and Caprace, de Cornulier, Monod, and Tessera [46, §3.A] regarding geodesic hyperbolic metric spaces.6 Many of these theorems have similar statements to ours (=-=[86]-=- and [46] seem to be the closest), but we have not kept track of this carefully, since our proof appears to be sufficiently different to warrant independent interest anyway. 5In Sections 6-10, we work... |

3 | On volumes of hyperbolic orbifolds
- Adeboye, Ilesanmi, et al.
(Show Context)
Citation Context ...n have to be shown independent of which sequences were chosen.) The naive definition (3.4.2) does not work, because the limit (3.4.2) does not necessarily exist: Example 3.4.6. Let X = {x ∈ R2 : x2 ∈ =-=[0, 1]-=-} be interpreted as a subspace of R2 with the L1 metric. Then X is a hyperbolic metric space, since it contains the cobounded hyperbolic metric space R × {0}. Its Gromov boundary consists of two point... |

3 |
Poincaré theta series and singular sets of Schottky groups
- Akaza
- 1964
(Show Context)
Citation Context ... Let G ≤ Isom(X) be a nonelementary group. Suppose either that (1) G is strongly discrete, (2) X is a CAT(-1) space and G is moderately discrete, (3) X is a ROSSONCT and G is weakly discrete, or that =-=(4)-=- X is a ROSSONCT and G acts irreducibly (cf. Subsection 7.6) and is COT-parametrically discrete. Then there exists σ > 0 such that dimH(Λr) = dimH(Λur) = dimH(Λur ∩ Λr,σ) = δ (cf. Definitions 7.1.2 an... |

3 | results on the geometry of convex hulls in manifolds of pinched negative curvature - Some - 1994 |

3 | The Cremona group in two variables - Cantat - 2013 |

3 |
Kähler groups, real hyperbolic spaces and
- Delzant, Py
(Show Context)
Citation Context ...o deaf ears, and the geometry and representation theory of infinitedimensional hyperbolic space H∞ and its isometry group have been studied in the last decade by a handful of mathematicians, see e.g. =-=[38, 61, 127]-=-. However, infinite-dimensional hyperbolic geometry has come into prominence most spectacularly through the recent resolution of a long-standing conjecture in algebraic geometry due to Enriques from t... |

3 |
Infinite-dimensional nonpositively curved symmetric spaces of finite rank
- Duchesne
(Show Context)
Citation Context ...mensional hyperbolic geometry. References for the theory of finite-dimensional ROSSONCTs include [37, 43, 120]; infinitedimesional symmetric spaces of noncompact type and finite rank are discussed in =-=[64]-=-. 2.1. The definition. Finite-dimensional ROSSONCTs come in four flavors, corresponding to the classical division algebras R, C, Q (quaternions), and O (octonions).17 The first three division algebras... |

3 | A Scientific Biography - Poincaré |

2 |
hyperbolic groups, Internat
- Relatively
(Show Context)
Citation Context ...dent of the basepoint o, but this follows from Theorems 12.4.5 and 12.4.14 below. Remark 12.4.4. Geometrical finiteness is closely related to the notion of relative hyperbolicity of a group; see e.g. =-=[35]-=-. The main differences are: 1. Relative hyperbolicity is a property of an abstract group, whereas geometrical finiteness is a property of an isometric group action (equivalently, of a subgroup of an i... |

2 | is ... a quasiconformal mapping - What |

2 |
der geodätischen Linien in Mannigfaltigkeiten negativer Krümmung
- Statistik
(Show Context)
Citation Context ...umption was demonstrated by Patterson [138], who showed that there exist Kleinian groups of the first kind (i.e. with limit set equal to ∂Hd) with arbitrarily small Poincaré exponent [138] (see also =-=[98]-=- or [151, Example 8] for an earlier example of the same phenomenon). Generalizing these theorems beyond the geometrically finite case requires the introduction of the radial and uniformly radial limit... |

1 |
Ahlfors,Möbius transformations in several dimensions
- V
- 1981
(Show Context)
Citation Context ...ralizes all the aforementioned results: Theorem 1.2.1. Let G ≤ Isom(X) be a nonelementary group. Suppose either that (1) G is strongly discrete, (2) X is a CAT(-1) space and G is moderately discrete, =-=(3)-=- X is a ROSSONCT and G is weakly discrete, or that (4) X is a ROSSONCT and G acts irreducibly (cf. Subsection 7.6) and is COT-parametrically discrete. Then there exists σ > 0 such that dimH(Λr) = dimH... |

1 |
Hyberbolic orbifolds of small volume
- Belolipetsky
- 2014
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Citation Context ... sequence cobounded subgroups of Isom(Hd) as d→∞. The most promising candidate for such a direct limit has been the direct limit of a sequence of arithmetic cocompact subgroups of Isom(Hd). (See e.g. =-=[21]-=- for the definition of an arithmetic subgroup of Isom(Hd).) Nevertheless, such innocent hopes are dashed by the following result: Proposition 12.2.3. If Gd ≤ Isom(Hd) is a sequence of arithmetic subgr... |

1 |
Amenable hyperbolic groups, http://arxiv.org/abs/1202.3585v1, preprint 2012
- Caprace, Cornulier, et al.
(Show Context)
Citation Context ..., Theorem 2.7], Osin [135, §3], and Caprace, de Cornulier, Monod, and Tessera [46, §3.A] regarding geodesic hyperbolic metric spaces.6 Many of these theorems have similar statements to ours ([86] and =-=[46]-=- seem to be the closest), but we have not kept track of this carefully, since our proof appears to be sufficiently different to warrant independent interest anyway. 5In Sections 6-10, we work in the s... |

1 | Geometry of limit sets of discrete groups acting on real infinite-dimensional hyperbolic space, http://www.urbanskimath.com/wp-content/uploads/2014/07/DaStUrBishopJones_2014_07_23C.pdf, preprint 2014 - Das, Stratmann, et al. |

1 | Lectures on geometric group theory, https://www.math.ucdavis.edu/~kapovich/EPR/kapovich_drutu.pdf - Druţu, Kapovich |

1 |
On the Poincaré series of dimension -1. (über die Poincaréschen Reihen der (-1)-ten Dimension., Festschrift zum 70. Geburtstag R. Dedekinds, Verlag Vieweg und Sohn
- Fricke
- 1901
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Citation Context ... series” was later used to refer to such objects.8 The question of for which m < 2 the Poincaré series still converges was investigated by Schottky, Burnside, Fricke, and Ritter; cf. Fricke’s survey =-=[73]-=-. In what would initially appear to be an unrelated development, mathematicians began to study the “thickness” of the limit set of a Fuchsian group: in 1941 Myrberg [130] showed that the limit set Λ o... |

1 | la Harpe, Espaces métriques hyperboliques (hyperbolic metric spaces), Sur les groupes hyperboliques d’aprés Mikhael Gromov - Ghys, de - 1988 |

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Patterson-Sullivan measure and groups of divergence type
- Hong
- 1993
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Citation Context .... P. J. Nicholls finished the proof by showing (A) ⇔ (B) in full generality [133, Theorems 8.2.2 and 8.2.3]. Afterwards, S. Hong re-proved (A) ⇒ (B) in full generality twice in two independent papers =-=[95, 96]-=-, apparently unaware of any previous results. Another proof of (A) ⇒ (B) in full generality, which was conceptually similar to Thurston’s proof, was given by P. Tukia [164, Theorem 3A]. Further genera... |

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limit points and groups of divergence type
- Conical
- 1994
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Citation Context .... P. J. Nicholls finished the proof by showing (A) ⇔ (B) in full generality [133, Theorems 8.2.2 and 8.2.3]. Afterwards, S. Hong re-proved (A) ⇒ (B) in full generality twice in two independent papers =-=[95, 96]-=-, apparently unaware of any previous results. Another proof of (A) ⇒ (B) in full generality, which was conceptually similar to Thurston’s proof, was given by P. Tukia [164, Theorem 3A]. Further genera... |

1 |
groups in higher dimensions, Geometry and dynamics of groups and spaces
- Kleinian
- 2008
(Show Context)
Citation Context ...mensional Kleinian groups was not really considered a subject in its own right until around the 1990s. For more information on the theory of higher-dimensional Kleinian groups, see the survey article =-=[104]-=-, which describes the state of the art up to the last decade, emphasizing connections with homological algebra. But why stop at finite n? Dennis Sullivan, in his IHÉS Seminar on Conformal and Hyperbo... |