this paper we use the argument of [3] to estimate the above number M . Improving these arguments we show that M can be taken to be 21g. The same argument can be, word by word, applied also to Riemann surfaces without symmetries to obtain the same value for M . That is an essential improvement for the estimate derived in [3]. The set of 3g \Gamma 3 disjoint closed geodesic curves on the Riemann surface of genus

We construct a geometric model for the mapping class group M(S) of a non-exceptional oriented surface S of genus g with m punctures and use it to show that the action of M(S) on the compact Hausdorff space of complete geodesic laminations for S is topologically amenable. As a consequence, the Novikov higher signature conjecture holds for every subgroup of M(S).

Abstract. The subject of this paper is the relationship among the marked length spectrum, the length spectrum, the Laplace spectrum on functions, and the Laplace spectrum on forms on Riemannian nilmanifolds. In particular, we show that for a large class of three-step nilmanifolds, if a pair of nilmanifolds in this class has the same marked length spectrum, they necessarily share the same Laplace spectrum on functions. In contrast, we present the first example of a pair of isospectral Riemannian manifolds with the same marked length spectrum but not the same spectrum on one-forms. Outside of the standard spheres vs. the Zoll spheres, which are not even isospectral, this is the only example of a pair of Riemannian manifolds with the same marked length spectrum, but not the same spectrum on forms. This partially extends and partially contrasts the work of Eberlein, who showed that on two-step nilmanifolds, the same marked length spectrum implies the same Laplace spectrum both on functions and on forms. Section 1: Introduction. The spectrum of a closed Riemannian manifold (M, g), denoted spec(M, g),

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

For a cocompact group of SL2(R) we fix a non-zero harmonic 1-form a. We normalize and order the values of the Poincaré pairing hg; ai according to the length of the corresponding closed geodesic lðgÞ. We prove that these normalized values have a Gaussian distribution.

Abstract. We prove that the modular symbols appropriately normalized and ordered have an asymptotical normal distribution for all cocompact subgroups of SL2(R). We introduce hyperbolic Eisenstein series in order to calculate the moments of the modular symbols. 1.

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Abstract. In this article we consider natural counting problems for closed geodesics on negatively curved surfaces. We present asymptotic estimates for pairs of closed geodesics, the differences of whose lengths lie in a prescribed family of shrinking intervals. Related pair correlation problems have been studied in both Quantum Chaos and number theory. One of the most striking properties of negatively curved surfaces is the regularity of the distribution of the lengths of their closed geodesics. This is shown by the wellknown prime geodesic theorem. More precisely, let V denote a compact surface with a C ∞ Riemannian metric of strictly negative curvature. Given any closed geodesic

Let I ⊂ R denote a compact interval symmetric about 0 and let c · I denote the dilation of I by a factor of c. Let M and N, be compact oriented smooth manifolds of dimensions d and d + 1 respectively. We suppose that we have an embedded copy of 3 · I × M inside of N. (See Figure 1). Let N 0 denote the complement of

Using train tracks on a non-exceptional oriented surface S of finite type in a systematic way we give a proof that the complex of curves C(S) of S is a hyperbolic geodesic metric space. We also discuss the relation between the geometry of the complex of curves and the geometry of Teichmüller space.