Necessary and sufficient conditions for McShane's identity and variations

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.

On the absence of McShane-type identities for the outer space

A remarkable result of McShane states that for a punctured torus with a complete finite volume hyperbolic metric we have \[ \sum_γ \frac{1}{e^{\ell(γ)}+1}={1/2} \] where $γ$ varies over the homotopy classes of essential simple closed curves and $\ell(γ)$ is the length of the geodesic representative of $γ$. We prove that there is no reasonable analogue of McShane's identity for the Culler-Vogtmann outer space of a free group.

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.

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.

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…

McShane-Rivin范数球与简单长度乘数

Stable-norm boundary geometry on once-punctured tori

研究通过McShane-Rivin范数球的正常转估计，证明了每个有限面积完备一次孔洞扭结面X中，长度恰好为L≥2的简单闭曲线数目不超过C_X(logL)^2，并改进了Markoff数的纤维界。