arXiv research

The study connects translation length to manifold structure, proving bounds and identifying finite types.

New bounds on homological eigenvalues relate to Weil-Petersson length.

Thanks to a theorem of Brock on comparison of Weil-Petersson translation distances and hyperbolic volumes of mapping tori for pseudo-Anosovs, we prove that the entropy of a surface automorphism in general has linear bounds in terms of Gromov norm of its mapping torus from below and in bounded geometry case from above. …

New results on the convexity of geodesic-length functions on Teichmüller space are presented. A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-length functions to Weil-Petersson distance is described. Applicat…

Moduli spaces of hyperbolic surfaces with geodesic boundary components of fixed lengths may be endowed with a symplectic structure via the Weil-Petersson form. We show that, as the boundary lengths are sent to infinity, the Weil-Petersson form converges to a piecewise linear form first defined by Kontsevich. The proof …

Given a closed hyperbolic 3-manifold T_ψthat fibers over the circle with monodromy ψ: S -> S, the monodromy $ψ$ determines an isometry of Teichmuller space with its Weil-Petersson metric whose translation distance ||ψ||_WP is positive. We show there is a constant K >= 1 depending only on the topology of S so that the v…

Paper calculates distances between strata in Teichmüller space, proving a constant separation.

The paper studies gradients of geodesic-length functions and systoles on Teichmüller spaces.

A summary introduction of the Weil-Petersson metric space geometry is presented. Teichmueller space and its augmentation are described in terms of Fenchel-Nielsen coordinates. Formulas for the gradients and Hessians of geodesic-length functions are presented. Applications are considered. A description of the Weil-Peter…

A recent preprint of S. Kojima and G. McShane [KM] observes a beautiful explicit connection between Teichmüller translation distance and hyperbolic volume. It relies on a key estimate which we supply here: using geometric inflexibility of hyperbolic 3-manifolds, we show that for $S$ a closed surface, and $ψ\in \text{Mo…

Study of circle homeomorphisms with square summable diamond shears.

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 …

New compactification of Teichmüller space via renormalized volume.

We give an overview of the proof for Mirzakhani's volume recursion for the Weil-Petersson volumes of the moduli spaces of genus $g$ hyperbolic surfaces with $n$ labeled geodesic boundary components, and her application of this recursion to Witten's conjecture and the study of simple geodesic length spectrum growth rate…

The Weil-Petersson metric for the moduli space of Riemann surfaces has negative sectional curvature. Surfaces represented in the complement of a compact set in the moduli space have short geodesics. At such surfaces the Weil-Petersson metric is approximately a product metric. An almost product metric has sections with …

This research extends topological recursion to hyperbolic surfaces with tight boundaries and conical defects.

Study on stable translation lengths of surface homeomorphisms and their approximations.

A brief history of the investigation of the Weil-Petersson curvature and a summary of Teichmüller theory are provided. A report is presented on the program to describe an intrinsic geometry with the Weil-Petersson metric and geodesic-length functions. Formulas for the metric, covariant derivative and formulas for the c…

Computes volumes of metric maps on surfaces, linking to Weil-Petersson volumes.

In this paper, we study the asymptotic geometry of Teichmuller space of Riemann surfaces and give bounds on the Weil-Petersson sectional curvature of Teichmuller space, in terms of the length of the shortest geodesic on the surface. This will also imply that the sectional curvature is not pinched from above or below by…

We consider the Riemann moduli space $\mathcal M_γ$ of conformal structures on a compact surface of genus $γ>1$ together with its Weil-Petersson metric $g_{\mathrm{WP}}$. Our main result is that $g_{\mathrm{WP}}$ admits a complete polyhomogeneous expansion in powers of the lengths of the short geodesics up to the singu…

Extremal length is an important conformal invariant on Riemann surface. It is closely related to the geometry of Teichmuller metric on Teichmuller space. By identifying extremal length functions with energy of harmonic maps from Riemann surfaces to $\mathbb{R}$-trees, we study the second variation of extremal length fu…

Study critical exponents on hyperbolic surfaces with long boundaries using Weil-Petersson measures.

The study of random surfaces reveals asymptotic lengths of separating geodesics.

In a family of compact, canonically polarized, complex manifolds the first variation of the lengths of closed geodesics is computed. As an application, we show the coincidence of the Fenchel-Nielsen and Weil-Petersson symplectic forms on the Teichmueller spaces of compact Riemann surfaces in a purely geometric way. The…

Exact asymptotic value of Weil-Petersson volumes computed for large genus surfaces.

A bijection proves a polynomial volume for genus-0 hyperbolic surfaces with boundaries.

We prove Poisson approximation results for the bottom part of the length spectrum of a random closed hyperbolic surface of large genus. Here, a random hyperbolic surface is a surface picked at random using the Weil-Petersson volume form on the corresponding moduli space. As an application of our result, we compute the …

We show that an Anosov map has a geodesic axis on the curve graph of a torus. The direct corollary of our result is the stable translation length of an Anosov map on the curve graph is always a positive integer. As the proof is constructive, we also provide an algorithm to calculate the exact translation length for any…

New findings on translation lengths in Teichmüller and curve graphs for pseudo-Anosovs.

An expansion is developed for the Weil-Petersson Riemann curvature tensor in the thin region of the Teichmüller and moduli spaces. The tensor is evaluated on the gradients of geodesic-lengths for disjoint geodesics. A precise lower bound for sectional curvature in terms of the surface systole is presented. The curvatur…

New lattice extensions of Schottky groups in hyperbolic space.

The dynamics of representations into PSL_d(R) are studied for surfaces of genus at least 3.

Study of large-$n$ asymptotics for Weil-Petersson volumes of hyperbolic surfaces with cusps.

The study examines translation lengths of pseudo-Anosov maps on curve graphs.

Random walks on hyperbolic spaces show linear growth in translation lengths.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

Defines quantum intersection number on pants decompositions and relates it to hyperbolic geometry.

A 2008 general overview on Weil-Petersson geometry is offered. A preliminary plan for the subsequent CBMS lectures at Central Connecticut State University is included. Mirzakhani's solution of Witten-Kontsevich is not included - this work essentially requires its own lectures. Lectures on Mirzakhani's Witten-Kontsevich…

Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at the boundary of S perpendicularly are coordinates on the Teichmueller space T(S). We compute the Weil-Petersson Poisson structure on T(S) in this system of coordinates and we prove that i…

We establish bounds on the minimal asymptotic pseudo-Anosov translation lengths on the complex of curves of orientable surfaces. In particular, for a closed surface with genus $g \geqslant 2$, we show that there are positive constants $a_1 < a_2$ such that the minimal translation length is bounded below and above by $a…

Study on hyperbolic surfaces' volumes, proving asymptotic expansion for high genus.

The paper studies translation lengths on sphere complexes and related cones.

Neural machine translation is a relatively new approach to statistical machine translation based purely on neural networks. The neural machine translation models often consist of an encoder and a decoder. The encoder extracts a fixed-length representation from a variable-length input sentence, and the decoder generates…

Maps with a single face converge to hyperbolic surfaces in large genus.

New bound for group action length without diameter restriction.

Study on finiteness property of right-angled Artin groups actions on extension graphs.

The paper calculates the volume growth of hyperbolic surfaces with short geodesics.