Geodesic and orthogeodesic identities on hyperbolic surfaces

Orthospectra of Geodesic Laminations and Dilogarithm Identities on Moduli Space

Given a measured lamination on a finite area hyperbolic surface we consider a natural measure Mon the real line obtained by taking the push-forward of the volume measure of the unit tangent bundle of the surface under an intersection function associated with the lamination. We show that the measure M gives summation identities for the Rogers dilogarithm function on the moduli space of a surface.

New Identities for small hyperbolic surfaces

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt the argument from Luo and Tan to give an identity for hyperbolic tori with one geodesic boundary or cusp in terms of dilogarithm functions on the set of lengths of simple closed geodesics on the torus. As a corollary, we are also able to express the Luo-Tan identity as a sum over …

放缩双曲曲面与最优Lipschitz映射的形状

Deflating hyperbolic surfaces and the shapes of optimal Lipschitz maps

本文研究双曲曲面间Lipschitz极小映射在测度叶状结构补集上的性质，引入放缩映射作为阻碍，借助正交测地叶状结构光滑版构造新最优映射族，证明该阻碍是本质唯一的。

Lengths of geodesics on non-orientable hyperbolic surfaces

We give an identity involving sums of functions of lengths of simple closed geodesics, known as a McShane identity, on any non-orientable hyperbolic surface with boundary which generalises Mirzakhani's identities on orientable hyperbolic surfaces with boundary.

McShane's identity for classical Schottky Groups

Greg McShane introduced a remarkable identity for the lengths of simple closed geodesics on cusped hyperbolic surfaces. This was subsequently generalized by the authors to hyperbolic cone-surfaces, possibly with cusps and/or geodesic boundary. In this paper, we generalize the identity further to the case of classical Schottky groups. As a consequence, we obtain some surprising new identities in the case of fuchsian Schottky groups. For classical Schottky groups of rank 2, we also give generalizations of the Weierstrass identities, first given by McShane.

A survey of length series identities for surfaces, % 3-manifolds and representation varieties

We survey some of our recent results on length series identities for hyperbolic (cone) surfaces, possibly with cusps and/or boundary geodesics; classical Schottky groups; representations/characters of the one-holed torus group to $SL(2, \mathbf C)$; and hyperbolic 3 manifolds obtained by hyperbolic Dehn surgery on punctured torus bundles over the circle. These can be regarded as generalizations and variations of McShane's identity for cusped hyperbolic surfaces, which has found some striking applications in the recent work of Mirzakhani. We discuss some of the methods and techniques used to ob…

Generalizations of McShane's identity to hyperbolic cone-surfaces

We generalize McShane's identity for the length series of simple closed geodesics on a cusped hyperbolic surface to hyperbolic cone-surfaces (with all cone angles $\le π$), possibly with cusps and/or geodesic boundary. In particular, by applying the generalized identity to the orbifolds obtained from taking the quotient of the one-holed torus by its elliptic involution, and the closed genus two surface by its hyper-elliptic involution, we obtain generalizations of the Weierstrass identities for the one-holed torus, and identities for the genus two surface, also obtained by McShane using differ…

McShane identities for Higher Teichmüller theory and the Goncharov-Shen potential

We derive generalizations of McShane's identity for higher ranked surface group representations by studying a family of mapping class group invariant functions introduced by Goncharov and Shen which generalize the notion of horocycle lengths. In particular, we obtain McShane-type identities for finite-area cusped convex real projective surfaces by generalizing the Birman--Series geodesic scarcity theorem. More generally, we establish McShane-type identities for positive surface group representations with loxodromic boundary monodromy, as well as McShane-type inequalities for general rank posit…

Moments of the boundary hitting function for the geodesic flow on a hyperbolic manifold

In this paper we consider finite volume hyperbolic manifolds X with non-empty totally geodesic boundary. We consider the distribution of the times for the geodesic flow to hit the boundary and derive a formula for the moments of the associated random variable in terms of the orthospectrum. We show that the the first two moments correspond to two cases of known identities for the orthospectrum. We further obtain an explicit formula in terms of the trilogarithm functions for the average time for the geodesic flow to hit the boundary in the surface case, using the third moment.

The skein algebra of arcs and links and the decorated Teichmüller space

We define an associative algebra AS_h(S) generated by framed arcs and links over a punctured surface S which is a quantization of the Poisson algebra C(S) of arcs and curves on S. We then construct a Poisson algebra homomorphism from C(S) to the space of smooth functions on the decorated Teichmuller space endowed with the Weil-Petersson Poisson structure. The construction relies on a collection of geodesic lengths identities in hyperbolic geometry which generalize Penner's Ptolemy relation, the trace identities and Wolpert's cosine formula. As a consequence, we derive an explicit formula for t…

SL₂(ℝ)特征簇富克斯轨迹上Atiyah-Bott-Goldman辛结构的迹函数泊松括号

Poisson bracket of trace functions on the Fuchsian locus and Wolpert's formulas

该研究针对双曲曲面上莫比乌斯变换乘积的迹，开发了基于SL₂(ℝ)矩阵归一化的系统计算方法，结合Goldman的辛形式描述计算相关迹函数的泊松括号，应用中得到Wolpert的余弦与正弦公式。

Parametrizing Hitchin components

We construct a geometric, real analytic parametrization of the Hitchin component Hit_n(S) of the PSL_n(R)-character variety R_{PSL_n(R)}(S) of a closed surface S. The approach is explicit and constructive. In essence, our parametrization is an extension of Thurston's shear coordinates for the Teichmueller space of a closed surface, combined with Fock-Goncharov's coordinates for the moduli space of positive framed local systems of a punctured surface. More precisely, given a maximal geodesic lamination λin S with finitely many leaves, we introduce two types of invariants for elements of the Hit…

具有锥点的紧超几何曲面模空间体积的递推公式

Recursion formula for the volumes of moduli spaces of compact hyperbolic surfaces with cone points

本文通过广义McShane恒等式推导出具有锥点的紧超几何曲面模空间体积的递推公式，推广了Mirzakhani的结果。