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208
On the Deformation Theory of Representations of Fundamental Groups of Compact Hyperbolic 3Manifolds
, 1996
"... . We construct compact hyperbolic 3manifolds M 1 ; M 2 and an irreducible representation ae 1 : ß 1 (M 1 ) ! SO(3) so that the singularity of the representation variety of ß 1 (M 1 ) into SO(3) at ae 1 is not quadratic. We prove that for any semisimple Lie group G the singularity of the representat ..."
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Cited by 165 (8 self)
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. We construct compact hyperbolic 3manifolds M 1 ; M 2 and an irreducible representation ae 1 : ß 1 (M 1 ) ! SO(3) so that the singularity of the representation variety of ß 1 (M 1 ) into SO(3) at ae 1 is not quadratic. We prove that for any semisimple Lie group G the singularity of the representation variety of ß 1 (M 2 ) into G at the trivial representation is not quadratic. 1 Contents 1 Introduction 2 2 Varieties with nonquadratic singularities 3 3 Computation of H 2 (\Gamma; g) 4 4 The Massey triple product 7 5 Nonquadratic singularities for representations of subgroups of finite index 11 6 Singularities near the trivial representation 13 7 Construction of lattices 13 8 Construction of linkages 17 9 Deformations of mechanical linkages 17 10 Representation varieties with nonquadratic singularities 20 11 Deformation theory near the identity representation 22 The first author was partially supported by NSF grant DMS9306140, the second author by NSF grant DMS9205154. 1 19...
Complex earthquakes and Teichmüller theory
 J. AMER. MATH. SOC
, 1999
"... It is known that any two points in Teichmuller space are joined by an earthquake path. In this paper we show any earthquake path R ! T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmuller space, where R ae D ae C . These complex earthquakes relate WeilPeter ..."
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Cited by 71 (5 self)
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It is known that any two points in Teichmuller space are joined by an earthquake path. In this paper we show any earthquake path R ! T (S) extends to a proper holomorphic mapping of a simplyconnected domain D into Teichmuller space, where R ae D ae C . These complex earthquakes relate WeilPetersson geometry, projective structures, pleated surfaces and quasifuchsian groups. Using complex earthquakes, we prove grafting is a homeomorphism for all 1dimensional Teichmuller spaces, and we construct bending coordinates on Bers slices and their generalizations. In the appendix we use projective surfaces to show the closure of quasifuchsian space is not a topological manifold.
The monodromy groups of Schwarzian equations on closed Riemann surfaces
 ANN. OF MATH
, 2000
"... Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either ..."
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Cited by 54 (1 self)
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Let θ: π1(R) → PSL(2, C) be a homomorphism of the fundamental group of an oriented, closed surface R of genus exceeding one. We will establish the following theorem. Theorem. Necessary and sufficient for θ to be the monodromy representation associated with a complex projective stucture on R, either unbranched or with a single branch point of order 2, is that θ(π1(R)) be nonelementary. A branch point is required if and only if the representation θ does not lift to
Algebraic limits of Kleinian groups which rearrange the pages of a book
 Zbl 0874.57012 MR 1411128
, 1996
"... Dedicated to Bernard Maskit on the occasion of his sixtieth birthday ..."
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Cited by 49 (14 self)
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Dedicated to Bernard Maskit on the occasion of his sixtieth birthday
The Selberg zeta function for convex cocompact Schottky groups
 Comm. Math. Phys
"... Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the o ..."
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Cited by 45 (8 self)
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Abstract. We give a new upper bound on the Selberg zeta function for a convex cocompact Schottky group acting on H n+1: in strips parallel to the imaginary axis the zeta function is bounded by exp(Cs  δ) where δ is the dimension of the limit set of the group. This bound is more precise than the optimal global bound exp(Cs  n+1), and it gives new bounds on the number of resonances (scattering poles) of Γ\H n+1. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. We also study this problem numerically and provide evidence that the bound may be optimal. Our motivation comes from molecular dynamics and we consider Γ\H n+1 as the simplest model of quantum chaotic scattering. The proof of this result is based on the application of holomorphic L 2techniques to the study of the determinants of the Ruelle transfer operators and on the quasiselfsimilarity of limit sets. 1.
Cores of hyperbolic 3manifolds and limits of Kleinian groups II
 Amer. J. Math
, 1996
"... this paper, as well as the referee for useful comments. 2 Preliminaries ..."
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Cited by 38 (15 self)
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this paper, as well as the referee for useful comments. 2 Preliminaries
Free Kleinian groups and volumes of hyperbolic 3manifolds
 J. Differential Geom
, 1996
"... The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1. ..."
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Cited by 36 (24 self)
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The central result of this paper, Theorem 6.1, gives a constraint that must be satisfied by the generators of any free, topologically tame Kleinian group without parabolic elements. The following result is case (a) of Theorem 6.1.
Dimension Of The Limit Set And The Density Of Resonances For Convex CoCompact Hyperbolic Surfaces.
 Invent. Math
"... this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous ..."
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Cited by 35 (9 self)
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this paper is to show how the methods of Sjostrand for proving the geometric bounds for the density of resonances [28] apply to the case of convex cocompact hyperbolic surfaces. We prove that the exponent in the Weyl estimate for the number of resonances in subconic neighbourhoods of the continuous spectrum is related to the dimension of the limit set of the corresponding Kleinian group. Figure 1. Tesselation by the Schottky group generated by inversions in three symmetrically placed circles each cutting the unit circle in an 110