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163
Commutator Bounds For Eigenvalues, With Applications To Spectral Geometry
"... We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace Beltrami operators on manifolds. We significantly improve earlier bounds of Yang and Yau, Li, and Harrell. 1 Introduction In a recent paper [6 ..."
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We prove a purely algebraic version of an eigenvalue inequality of Hile and Protter, and derive corollaries bounding differences of eigenvalues of Laplace Beltrami operators on manifolds. We significantly improve earlier bounds of Yang and Yau, Li, and Harrell. 1 Introduction In a recent paper [6], one of us produced an algebraic version of a wellknown bound on differences between eigenvalues due to Payne, P'olya, and Weinberger [18]. They had shown that the difference of successive eigenvalues of the Dirichlet Laplacian on a domain in R is bounded by a universal constant times the sum of all the lower eigenvalues. It is easy to see, however, that if the analogous problem is considered on a manifold, then the geometry must 1 Partially supported by the US NSF through grant DMS9211624. 2 Part of this research was conducted during the USSweden Workshop on Spectral Methods sponsored by the NSF under grant INT9217529. 3 Work related to Doctoral Thesis, Georgia Institute of ...
Wave 0trace and length spectrum on convex cocompact hyperbolic manifolds
 Comm. Anal. Geom
"... Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length s ..."
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Abstract. For convex cocompact hyperbolic quotients Γ\H n+1 we obtain a formula relating the 0trace of the wave operator with the resonances and some conformal invariants of the boundary, generalizing a formula of Guillopé and Zworski in dimension 2. Then, by writing this 0trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of Γ is greater than n 2. 1.
WORD HYPERBOLIC EXTENSIONS OF SURFACE GROUPS
, 2005
"... Let S be a closed surface of genus g ≥ 2. A finitely generated group ΓS is an extension of the fundamental group π1(S) of S if π1(S) is a normal subgroup of ΓS. We show that the group ΓS is hyperbolic if and only if the orbit map for the action of the quotient group Γ = ΓS/π1(S) on the complex of c ..."
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Cited by 14 (1 self)
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Let S be a closed surface of genus g ≥ 2. A finitely generated group ΓS is an extension of the fundamental group π1(S) of S if π1(S) is a normal subgroup of ΓS. We show that the group ΓS is hyperbolic if and only if the orbit map for the action of the quotient group Γ = ΓS/π1(S) on the complex of curves is a quasiisometric embedding.
Mutually Isospectral Riemann Surfaces
, 1997
"... this paper, we address the following question: given a natural number g, how many Riemann surfaces S 1 ; : : : ; S k of genus g can there be such that S 1 ; : : : ; S k all share the same spectrum of the Laplacian? ..."
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Cited by 13 (3 self)
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this paper, we address the following question: given a natural number g, how many Riemann surfaces S 1 ; : : : ; S k of genus g can there be such that S 1 ; : : : ; S k all share the same spectrum of the Laplacian?
THE COMPLEXSYMPLECTIC GEOMETRY OF SL(2, C)CHARACTERS OVER SURFACES
"... Dedicated to M.S. Raghunathan on his sixtieth birthday Abstract. The SL(2, C)character variety X of a closed surface M enjoys a natural complexsymplectic structure invariant under the mapping class group Γ of M. Using the ergodicity of Γ on the SU(2)character variety, we deduce that every Γinvar ..."
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Cited by 12 (0 self)
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Dedicated to M.S. Raghunathan on his sixtieth birthday Abstract. The SL(2, C)character variety X of a closed surface M enjoys a natural complexsymplectic structure invariant under the mapping class group Γ of M. Using the ergodicity of Γ on the SU(2)character variety, we deduce that every Γinvariant meromorphic function on X is constant. The trace functions of closed curves on M determine regular functions which generate complex Hamiltonian flows. For simple closed curves, these complex Hamiltonian flows arise from holomorphic flows on the representation variety generalizing the FenchelNielsen twist flows on Teichmüller space and the complex quakebend flows on quasiFuchsian space. Closed curves in the complex trajectories of these flows lift to paths in the deformation space CP 1 (M) of complexprojective structures between different CP 1structures with the same holonomy (grafting). If P is a pants decomposition, then the
Selberg’s zeta function and the spectral geometry of geometrically finite hyperbolic surfaces
 Comment. Math Helv
"... Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R ..."
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Cited by 12 (1 self)
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Abstract. For hyperbolic Riemann surfaces of finite geometry, we study Selberg’s zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsionfree, discrete subgroup of SL(2, R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean [19] and Müller [22] to groups which are not necessarily cofinite. 1.
Isospectral Manifolds with Different Local Geometries
"... . We construct several new classes of isospectral manifolds with dierent local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the rst examples of isospectral four ..."
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Cited by 11 (2 self)
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. We construct several new classes of isospectral manifolds with dierent local geometries. After reviewing a theorem by Carolyn Gordon on isospectral torus bundles and presenting certain useful specialized versions (Chapter 1) we apply these tools to construct the rst examples of isospectral fourdimensional manifolds which are not locally isometric (Chapter 2). Moreover, we construct the rst examples of isospectral left invariant metrics on compact Lie groups (Chapter 3). Thereby we also obtain the rst continuous isospectral families of globally homogeneous manifolds and the rst examples of isospectral manifolds which are simply connected and irreducible. Finally, we construct the rst pairs of isospectral manifolds which are conformally equivalent and not locally isometric (Chapter 4). Contents Introduction 1. Constructions of isospectral, locally nonisometric manifolds 1.1 Isospectral torus bundles with totally geodesic bers 1.2 A rst specialization and its app...
Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
Geodesic length functions and Teichmüller spaces, Electronic Research Announcement
 31 Magnus, W.: Rings of
, 1996
"... Abstract Given a compact orientable surface with finitely many punctures Σ, let S(Σ) be the set of isotopy classes of essential unoriented simple closed curves in Σ. We determine a complete set of relations for a function from S(Σ) to R to be the geodesic length function of a hyperbolic metric with ..."
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Cited by 10 (5 self)
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Abstract Given a compact orientable surface with finitely many punctures Σ, let S(Σ) be the set of isotopy classes of essential unoriented simple closed curves in Σ. We determine a complete set of relations for a function from S(Σ) to R to be the geodesic length function of a hyperbolic metric with geodesic boundary and cusp ends on Σ. As a consequence, the Teichmüller space of hyperbolic metrics with geodesic boundary and cusp ends on Σ is reconstructed from an intrinsic (QP 1, PSL(2,Z)) structure on S(Σ). Let Σ = Σ s g,r be a compact oriented surface of genus g with r boundary components and s punctures, i.e., a surface of signature (g, r, s). The Teichmüller space of isotopy classes of hyperbolic metrics with geodesic boundary and cusp ends on Σ is denoted by T s g,r = T(Σ) and the isotopy classes of essential simple closed unoriented curves in Σ is denoted by S = S(Σ). A