Foundations of Hyperbolic Manifolds

Ratcliffe, John G.

doi:10.1007/978-1-4757-4013-4

Cited by 851 publications

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“…In this section we set out some basic facts about Möbius maps and models of H m , in order to establish notation. We will use standard properties of hyperbolic geometry without comment, and for more detail the reader may wish to consult [3] or [12].…”

Section: Möbius Maps and Hyperbolic Spacementioning

confidence: 99%

Conical limit sets and continued fractions

Crane¹,

Short

2007

Conform. Geom. Dyn.

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Abstract. Inspired by questions of convergence in continued fraction theory, Erdős, Piranian and Thron studied the possible sets of divergence for arbitrary sequences of Möbius maps acting on the Riemann sphere, S 2 . By identifying S 2 with the boundary of three-dimensional hyperbolic space, H 3 , we show that these sets of divergence are precisely the sets that arise as conical limit sets of subsets of H 3 . Using hyperbolic geometry, we give simple geometric proofs of the theorems of Erdős, Piranian and Thron that generalise to arbitrary dimensions. New results are also obtained about the class of conical limit sets, for example, that it is closed under locally quasisymmetric homeomorphisms. Applications are given to continued fractions.

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Section: Möbius Maps and Hyperbolic Spacementioning

confidence: 99%

Conical limit sets and continued fractions

Crane¹,

Short

2007

Conform. Geom. Dyn.

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“…3 Moving around a tiling of H n Let G be a discrete subgroup of Isom H n that is generated by side-pairings of its finite-sided fundamental polyhedron P (see [7] or [4]). If S is a side of P , let s be the side-pairing that sends S to its pair S (thus, sS = S ).…”

Section: Hyperbolic Manifolds As Complementsmentioning

confidence: 99%

“…Then s −1 will pair S to S. For every point x in P let [x], the cycle of x, be the set of all points in P that are obtained by sending x around P via side-pairing transformations. As in [7], for every x ∈ P let ω(x) be the measure of the "spatial angle" that P subtends at x, that is ω(x) = Vol(B(x, r) ∩ P )/ Vol B(x, r), where B(x, r) is a ball around x with radius r small enough so that B(x, r) intersects only the sides on which x lies. Set ω[x] = y∈[x] ω(y).…”

Section: Hyperbolic Manifolds As Complementsmentioning

confidence: 99%

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Hyperbolic structure on a complement of tori in the 4-sphere

2004

Advg

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It is well known that many noncompact hyperbolic 3-manifolds are topologically complements of links in the 3-sphere. We extend this phenomenon to dimension 4 by exhibiting an example of a noncompact hyperbolic 4-manifold that is topologically the complement of 5 tori in the 4-sphere. We also exhibit examples of hyperbolic manifolds that are complements of 5n tori in a simply-connected 4-manifold with Euler characteristic 2n. All the examples are based on a construction of Ratcliffe and Tschantz, who produced 1171 noncompact hyperbolic manifolds with Euler characteristic 1. Our examples are finite covers of the Ratcliffe-Tschantz manifold with the biggest symmetry group.

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“…Normal forms of Euc(n) can be found in [RT11] with a slight different notion of normal form than the one presented here. Isometries of the hyperbolic space are often studied by means of the Möbius group, by considering the upper-half plane model of the hyperbolic space (see for example [Ahl85a], [Ahl85b], [Gon11], [Rat06]). This may possibly be a cause for the lack of references concerning normal forms of the Lorentz group O(1, n).…”

Section: Introductionmentioning

confidence: 99%

Classification of isometries of spaces of constant curvature and invariant subspaces

Cirici

2014

Linear Algebra and its Applications

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Abstract. We study the varieties of invariant totally geodesic submanifolds of isometries of the spherical, Euclidean and hyperbolic spaces in each finite dimension. We show that the dimensions of the connected components of these varieties determine the orbit type (or the z-class) of the isometry. For this purpose, we introduce the Segre symbol of an isometry, a discrete invariant encoding the structure of its normal form, which parametrizes z-classes. We then provide a description of the isomorphism type of the varieties of invariant subspaces in terms of the Segre symbol.

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Foundations of Hyperbolic Manifolds

Cited by 851 publications

References 0 publications

Conical limit sets and continued fractions

Conical limit sets and continued fractions

Hyperbolic structure on a complement of tori in the 4-sphere

Classification of isometries of spaces of constant curvature and invariant subspaces

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