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158
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 126 (29 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 62 (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.
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 51 (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 43 (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
"... This article is dedicated to the memory of Fernando Serrano Abstract. A fibration of an algebraic surface S over a curve B, with fibres of genus at least 2, has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group G. A variety isogenous to a (hi ..."
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Cited by 41 (18 self)
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This article is dedicated to the memory of Fernando Serrano Abstract. A fibration of an algebraic surface S over a curve B, with fibres of genus at least 2, has constant moduli iff it is birational to the quotient of a product of curves by the action of a finite group G. A variety isogenous to a (higher) product is the quotient of a product of curves of genus at least 2 by the free action of a finite group. Theorem B gives a characterization of surfaces isogenous to a higher product in terms of the fundamental group and of the Euler number. Theorem C classifies the groups thus occurring and shows that, after fixing the group and the Euler number, one obtains an irreducible moduli space. The result of Theorem B is extended to higher dimension in Theorem G, thus generalizing (cf. also Theorem H) results of JostYau and Mok concerning varieties whose universal cover is a polydisk. Theorem A shows that fibrations where the fibre genus and the genus of the base B are at least 2 are invariants of the oriented differentiable structure. The main Theorems D and E characterize surfaces carrying constant moduli fibrations as surfaces having a Zariski open set satisfying certain topological conditions (e.g., having the right Euler number, the right fundamental group and the right fundamental group at infinity). 0. Introduction. The study of fibrations of a smooth algebraic surface S
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 38 (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.
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 36 (6 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
The classification of puncturedtorus groups
 ANNALS OF MATH
, 1999
"... Thurston’s ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for puncturedtorus grou ..."
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Cited by 28 (3 self)
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Thurston’s ending lamination conjecture proposes that a finitely generated Kleinian group is uniquely determined (up to isometry) by the topology of its quotient and a list of invariants that describe the asymptotic geometry of its ends. We present a proof of this conjecture for puncturedtorus groups. These are free twogenerator Kleinian groups with parabolic commutator, which should be thought of as representations of the fundamental group of a punctured torus. As a consequence we verify the conjectural topological description of the deformation space of puncturedtorus groups (including Bers ’ conjecture that the quasiFuchsian groups are dense in this space) and prove a rigidity theorem: two puncturedtorus groups are quasiconformally conjugate if and only if they are topologically conjugate.
The Poisson Boundary Of The Mapping Class Group
, 1995
"... . A theory of random walks on the mapping class group and its nonelementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Pois ..."
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Cited by 27 (2 self)
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. A theory of random walks on the mapping class group and its nonelementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmuller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmuller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no nonelementary subgroup of the mapping class group can be a lattice in a higher rank semisimple Lie group. Contents 0. Introduction 1. Asymptotic properties of Teichmuller space 1.1. The space of projective measured foliations 1.2. The mapping class group 1.3. Teichmu...