squares of characters and finite groups abstract let g be a finite group and be an irreducible complex character we study the character 2 in the case that 1 is a prime power

growth of weil petersson volumes and random hyperbolic surfaces of large genus in this paper we investigate the geometric properties of random hyperbolic surfaces of large genus we describe the relationship between the behavior of lengths of simple closed geodesics on a hyperbolic surface and properties of the moduli space of such surfaces first we study the asymptotic behavior of weil petersson volume vg n of the moduli spaces of hyperbolic surfaces of genus g with n punctures as g then we discuss basic geometric properties of a random hyperbolic surface of genus g with respect to the weil petersson measure as g

the weil petersson hessian of length on teichmuller space we present a brief but nearly self contained proof of a formula for the weil petersson hessian of the geodesic length of a closed curve either simple or not simple on a hyperbolic surface the formula is the sum of the integrals of two naturally defined positive functions over the geodesic proving convexity of this functional over teichmuller space due to wolpert 1987 we then estimate this hessian from below in terms of local quantities and distance along the geodesic the formula extends to proper arcs on punctured hyperbolic surfaces and the estimate to laminations wolpert s result that the thurston metric is a multiple of the weil petersson metric directly follows on taking a limit of the formula over an appropriate sequence of curves we give further applications to upper bounds of the hessian especially near pinching loci recover through a geometric argument wolpert s result on the convexity of length to the half power and give a lower bound for growth of length in terms of twist

on the period matrix of a riemann surface of large genus with an appendix by j h conway and n j a sloane riemann showed that a period matrix of a compact riemann surface of genusg 1 satisfies certain relations we give a further simple combinatorial property related to the length of the shortest non zero lattice vector satisfied by such a period matrix see 1 13 in particular it is shown that for large genus the entire locus of jacobians lies in a very small neighborhood of the boundary of the space of principally polarized abelian varieties

uniform bounds for weil petersson curvatures we find bounds for weil petersson holomorphic sectional curvature and the weil petersson curvature operator in several regimes that do not depend on the topology of the underlying surface among other results we show that the minimal weil petersson holomorphic sectional curvature of a sufficiently thick hyperbolic surface is comparable to 1 independently of the genus this provides a counterexample to some suggestions that the weil petersson metric becomes asymptotically flat as the genus g goes to infinity in some loci in teichmuller space adopting a different perspective on curvature we also show that the minimal most negative eigenvalue of the curvature operator at any point in the teichmuller space teich sg of a closed surface sg of genus g is uniformly bounded away from zero restricting to a thick part of teich sg we show that the minimal eigenvalue is uniformly bounded below by an explicit constant which does not depend on the topology of the surface but only on the given bound on injectivity radius

the weil petersson metric and volumes of 3 dimensional hyperbolic convex cores we present a coarse interpretation of the weil petersson distance dwp x y between two finite area hyperbolic riemann surfaces x and y using a graph of pants decompositions introduced by hatcher and thurston the combinatorics of the pants graph reveal a connection between riemann surfaces and hyperbolic 3 manifolds conjectured by thurston the volume of the convex core of the quasi fuchsian manifold q x y with x and y in its conformal boundary is comparable to the weil petersson distance dwp x y in applications we relate the weil petersson distance to the hausdorff dimen sion of the limit set and the lowest eigenvalue of the laplacian for q x y and give a new finiteness criterion for geometric limits mathematics department university of chicago 5734 s university ave chicago il 60637 e mail address brock math uchicago edu

weil petersson volumes and intersection theory on the moduli space of curves in this paper we establish a relationship between the weil petersson volume vgin b of the moduli space mg n b of hyperbolic riemann surfaces with geodesic boundary components of lengths b bn and the intersection numbers of tauto logical classes on the moduli space mg n of stable curves as a result by using the recursive formula for vg n b obtained in 22 we derive a new proof of the virasoro constraints for a point this result is equivalent to the witten kontsevich formula 14

periodic solutions of homogeneous equations abstract stability conditions for periodic spiral solutions of homogeneous systems in the nonautonomous and the autonomous case are derived as well as a concept of lyapunov functions and a bendixson criterion

behavior of geodesic length functions on teichmuller space let t be the teichmuller space of marked genus g n punctured riemann surfaces with its bordification t the augmented teichmuller space of marked riemann surfaces with nodes abi77 ber74 pro vided with the wp metric t is a complete cat 0 metric space dw03 wol03 yam04 an invariant of a marked hyperbolic struc ture is the lengthof the geodesic in a free homotopy class a basic feature of teichmuller theory is the interplay of two dimensional hyperbolic geometry weil petersson wp geometry and the behavior of geodesic length functions our goal is to develop the understanding of the intrinsic local wp geometry through a study of the gradient and hessian of geodesic length functions considerations include ex pansions for the wp pairing of gradients expansions for the hessian and covariant derivative comparability models for the wp metric as well as the behavior of wp geodesics including a description of the alexandrov tangent cone at the augmentation approximations and applications for geodesics close to the augmentation are developed an application for fixed points of group actions is described bound ing configurations and functions on the hyperbolic plane is basic to our approach considerations include analyzing the orbit of a dis crete group of isometries and bounding sums of the inverse square exponential distance

canonical metrics on the moduli space of riemann surfaces i in this paper we continue our study of the canonical metrics on the moduli space of curves we first prove the bounded geometry of the ricci and perturbed ricci metrics by carefully choosing the pertubation constant and by studying the asymptotics we show that the ricci and holomorphic sectional curvatures of the perturbed ricci metric are bounded from above and below by negative constants based on our understanding of the kahler einstein metric we show that the logarithmic cotangent bundle over the deligne mumford moduli space is stable with respect to the canonical polarization finally in the last section we prove the strongly bounded geometry of the kahler einstein metric by using the kahler ricci flow and a priori estimates of the complex monge ampere equation

the weil petersson geodesic flow is ergodic we prove that the geodesic flow for the weil petersson metric on the moduli space of riemann surfaces is ergodic and in fact bernoulli and has finite positive metric entropy

a type of moduli for saddle connections of planar diffeomorphisms on considere un diffeomorphisme f sur une variete a 2 dimensions m ayant une paire de points fixes hyperboliques p et q de type selle de sorte que l une des composantes connexes de w u p p est une composante connexe de w s q q q on etudie les conjugaisons hors de la connexion selle on introduit la notion de conjugaison pincee

the riemannian sectional curvature operator of the weil petersson metric and its application fix a number g gt 1 let s be a close surface of genus g and let mathrm teich s be the teichmuller space of s endowed with the weil petersson metric in this paper we show that the riemannian sectional curvature operator of mathrm teich s is non positive definite as an application we show that any twist harmonic map from rank one hyperbolic spaces h q m sp m 1 sp m cdot sp 1 or h o 2 f 4 20 so 9 into mathrm teich s is a constant

moduli of stability for germs of homogeneous vectorfields on r3 abstract in this paper we will show that there are an uncountable number of homogeneous quadratic vectorfields on r 3 so that no two of these vectorfields are topologically equivalent

transition functions and moduli of stability for 3 dimensional homogeneous vector fields with a hyperbolic blowing up abstract in the first part of the paper we introduce the normal transition function for saddle connections of planar diffeomorphisms it is a positive multiple of the usual transition function but in its definition we do not need c 1 linearizing coordinates among other nice properties it is found to be analytic when the diffeomorphism is the second part of the paper deals with the existence of a modulus of stability for germs on r 3 of homogeneous vector fields with a hyperbolic blowing up we show that inside a specific class of examples the modulus occurs for a sufficiently high degree