Results 1  10
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208
Three dimensional manifolds, Kleinian groups and hyperbolic geometry
 Bull. Amer. Math. Soc
, 1982
"... late 19th century, in which Poincaré had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater tha ..."
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Cited by 311 (3 self)
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late 19th century, in which Poincaré had a large role, was the uniformization theory for Riemann surfaces: that every conformai structure on a closed oriented surface is represented by a Riemannian metric of constant curvature. For the typical case of negative Euler characteristic (genus greater than 1) such a metric gives a hyperbolic structure: any small neighborhood in the surface is isometric to a neighborhood in the hyperbolic plane, and the surface itself is the quotient of the hyperbolic plane by a discrete group of motions. The exceptional cases, the sphere and the torus, have spherical and Euclidean structures. Threemanifolds are greatly more complicated than surfaces, and I think it is fair to say that until recently there was little reason to expect any analogous theory for manifolds of dimension 3 (or more)—except perhaps for the fact that so many 3manifolds are beautiful. The situation has changed, so that I feel fairly confident in proposing the 1.1. CONJECTURE. The interior of every compact ^manifold has a canonical
Rigidity of quasiisometries for symmetric spaces and Euclidean buildings
 Inst. Hautes Études Sci. Publ. Math
, 1997
"... 1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X ..."
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Cited by 136 (26 self)
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1.1 Background and statement of results An (L, C) quasiisometry is a map Φ: X − → X ′ between metric spaces such that for all x1, x2 ∈ X
Hyperbolic structures on 3manifolds, I: Deformation of acylindrical manifolds
 Annals of Math
, 1986
"... Abstract. This is an eprint approximation to [Thu86], which is the definitive form of this paper. This eprint is provided for convenience only; the theorem numbering of this version is different, and not all corrections are present, so any reference or quotation should refer to the published form. P ..."
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Cited by 75 (1 self)
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Abstract. This is an eprint approximation to [Thu86], which is the definitive form of this paper. This eprint is provided for convenience only; the theorem numbering of this version is different, and not all corrections are present, so any reference or quotation should refer to the published form. Parts II and III ( [Thua] and [Thub]) of this series, although accepted for publication, for many years have only existed in preprint form; they will also be made available as eprints. This is the first in a series of papers dealing with the conjecture that all compact 3manifolds admit canonical decompositions into geometric pieces. This conjecture will be discussed in detail in part IV. Here is an easily stated special case, in which no decomposition is necessary: Conjecture 0.1 (Indecomposable Implies Geometric). Let M 3 be a closed, prime, atoroidal 3manifold.
Hyperbolic geometry: the first 150 years
 Bull. Amer. Math. Soc., New Ser
, 1982
"... This will be a description of a few highlights in the early history of noneuclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3space. The mathematical literature on noneuclidean geometry begins in 1829 with ..."
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Cited by 56 (0 self)
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This will be a description of a few highlights in the early history of noneuclidean geometry, and a few miscellaneous recent developments. An Appendix describes some explicit formulas concerning volume in hyperbolic 3space. The mathematical literature on noneuclidean geometry begins in 1829 with publications by N. Lobachevsky in an obscure Russian journal. The infant subject grew very rapidly. Lobachevsky was a fanatically hard worker, who progressed quickly from student to professor to rector at his university of Kazan, on the Volga. Already in 1829, Lobachevsky showed that there is a natural unit of distance in noneuclidean geometry, which can be characterized as follows. In the right triangle of Figure 1 with fixed edge a, as the opposite vertex A moves infinitely far away, the angle 9 will increase to a limit 90 which is assumed to be strictly less than 7r/2. He showed that a =log tan(0o/2) if the unit of distance is suitably chosen. In particular, a « (TT/2) 0O if a is very small. (In the interpretation introduced by Beltrami forty years later, this unit of distance is chosen so that curvature =1.) FIGURE 1. A right triangle in hyperbolic space By early 1830, Lobachevsky was testing his "imaginary geometry " as a possible model for the real world. If the universe is noneuclidean in Lobachevsky's sense, then he showed that our solar system must be extremely small, in terms of this natural unit of distance. More precisely, taking the vertex A in Figure 1 to be the star Sirius and taking the edge a to be a suitably chosen radius of the Earth's orbit, he used the (unfortunately incorrect) estimate 7T — 20 ss 1.24 seconds of arc s 6 x 10~6 radians
A rigidity theorem for the solvable BaumslagSolitar groups
, 1996
"... this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory. ..."
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Cited by 55 (10 self)
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this paper we take the first steps towards applying some of these ideas to proving rigidity results for groups that arise most naturally not in geometry but in combinatorial group theory.
A differential geometric approach to the geometric mean of symmetric positivedefinite matrices
 SIAM J. Matrix Anal. Appl
"... Abstract. In this paper we introduce metricbased means for the space of positivedefinite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some inva ..."
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Cited by 48 (2 self)
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Abstract. In this paper we introduce metricbased means for the space of positivedefinite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization of this mean. Key words. Geometric mean, Positivedefinite symmetric matrices, Riemannian distance, Geodesics. AMS subject classifications. 47A64, 26E60, 15A48, 15A57
Fibred surfaces, varieties isogenous to a product and related moduli spaces
 Amer. J. Math
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Quasiflats and rigidity in higher rank symmetric spaces
 J. Amer. Math. Soc
, 1997
"... In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, Z),n ≥ 3: Theorem 8.1 is a major step towards th ..."
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Cited by 43 (7 self)
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In this paper we use elementary geometrical and topological methods to study some questions about the coarse geometry of symmetric spaces. Our results are powerful enough to apply to noncocompact lattices in higher rank symmetric spaces, such as SL(n, Z),n ≥ 3: Theorem 8.1 is a major step towards the proof of quasiisometric
Generic properties of Whitehead’s algorithm and isomorphism rigidity of random onerelator groups
 Pacific J. Math
"... Abstract. We show that the “hard ” part of Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly lineartime genericcase co ..."
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Cited by 40 (19 self)
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Abstract. We show that the “hard ” part of Whitehead’s algorithm for solving the automorphism problem in a fixed free group Fk terminates in linear time (in terms of the length of an input) on an exponentially generic set of input pairs and thus the algorithm has strongly lineartime genericcase complexity. We also prove that the stabilizers of generic elements of Fk in Aut(Fk) are cyclic groups generated by inner automorphisms. We apply these results to onerelator groups and show that onerelator groups are generically complete groups, that is, they have trivial center and trivial outer automorphism group. We prove that the number In of isomorphism types of kgenerator onerelator groups with defining relators of length n satisfies c1 n (2k − 1)n ≤ In ≤ c2 n (2k − 1)n, where c1 = c1(k)> 0, c2 = c2(k)> 0 are some constants independent of n. Thus In grows in essentially the same manner as the number of cyclic words of length n.