McShane's identity for classical Schottky Groups

McShane-type identities for quasifuchsian representations of nonorientable surfaces

We show that Norbury's McShane identity for nonorientable cusped hyperbolic surfaces N generalizes to quasifuchsian representations of pi_1(N) as well as pseudo-Anosov mapping Klein bottles with singular fibers given by N.

超麦山身份

Super McShane identity

本文推导了超环面的麦山身份，并研究了其长度光谱的渐近增长速率。

A variation of McShane's identity for 2-bridge links

We give a variation of McShane's identity, which describes the cusp shape of a hyperbolic 2-bridge link in terms of the complex translation lengths of simple loops on the bridge sphere. We also explicitly determine the set of end invariants of $SL(2,\mathbb{C})$-characters of the once-punctured torus corresponding to the holonomy representations of the complete hyperbolic structures of 2-bridge link complements.

McShane-type Identities for Affine Deformations

We derive an identity for Margulis invariants of affine deformations of a complete orientable one-ended hyperbolic sur- face following the identities of McShane, Mirzakhani and Tan- Wong-Zhang. As a corollary, a deformation of the surface which infinitesimally lengthens all interior simple closed curves must in- finitesimally lengthen the boundary.

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…

高维Teichmüller-Thurston理论中的Basmajian恒等式

Basmajian's identity in higher Teichmüller-Thurston theory

本研究将Basmajian恒等式推广到紧带边曲面的$n$-Hitchin表示，给出$n=3$时的凸实射影结构几何解释，证明相关极限集测度为零，并指出其与Labourie-McShane扩展的紧密联系。

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.

A new identity for SL(2,C)-characters of the once punctured torus group

We obtain new variations of the original McShane identity for those SL(2,C)-representations of the once punctured torus group which satisfy the Bowditch conditions, and also for those fixed up to conjugacy by an Anosov mapping class of the torus and satisfying the relative Bowditch conditions.

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…

The SL(2,C) character variety of the one-holed torus

In this note we announce several results concerning the SL(2,C) character variety ${\mathcal X}$ of the one-holed torus. We give a description of the largest open subset ${\mathcal X}_{BQ}$ of ${\mathcal X}$ on which the mapping class group $Γ$ acts properly discontinuously, in terms of two very simple conditions, and show that a series identity generalizing McShane's identity for the punctured torus holds for all characters in this subset. We also give variations of the McShane-Bowditch identities to characters fixed by an Anosov element of $Γ$ with applications to closed hyperbolic three man…

The sum of Lagrange numbers

Combining McShane's identity on a hyperbolic punctured torus with Schmutz's work on the Markov Uniqueness Conjecture (MUC), we find that MUC is equivalent to the identity \begin{equation} \sum_{n=1}^\infty \, \left( 3- L_n \right) \, = \, 4 - φ- \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $φ=\frac{1+\sqrt5}2$ is the golden ratio.

Simple geodesics and Markoff quads

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corresponding fundamental group to prove: a sharp upper-bound for the length of its shortest geodesic, a McShane identity and the surprising result of non-polynomial growth for the number of simple closed geodesic lengths.

正交积分曲面

Ortho-integral surfaces

本文研究双曲曲面上正交测地线的组合结构，提出计算其迹的递归方法，并证明存在所有正交测地线迹均为整数的曲面，同时发现一种新的根翻转并给出 Basmajian 恒等式的组合证明。

An infinite product on the Teichmüller space of the once-punctured torus

We prove the identity $$ \prod_γ\left(\frac{e^{l(γ)}+1}{e^{l(γ)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$ \prod_γ\left(\frac{t(γ)^2}{t(γ)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2} $$ in trace coordinates), where the product is over all simple closed geodesics on the once-punctured torus, $l(γ)=2\operatorname{arccosh}(t(γ)/2)$ is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of any triple of simple geodesics $\{γ_i\}$ intersecting at a single point. The exponent $h=h(γ;\{γ_i\})$ i…

Geodesics with one self-intersection, and other stories

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension of the Teichmuller space of S. Since closed geodesics with one double point fall into a finite number of orbits under the mapping class group of S, we get the same asympotic estimate for the number of such geodesics of length bounded by L. We also use our (elementary) methods to do a more precise study of geodesics with a single double point on a punctured to…