arXiv research

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 conve…

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 S…

Greg McShane introduced a remarkable identity for lengths of simple closed geodesics on the once punctured torus with a complete, finite volume hyperbolic structure. Bowditch later generalized this and gave sufficient conditions for the identity to hold for general type-preserving representations of a free group on two…

In this article, we discuss a probabilistic interpretation of McShane's identity as describing a finite measure on the space of embedded paths though a point.

In this paper we study McShane's identity in real and complex hyperbolic spaces and obtain various generalizations of the identity for representations of surface groups into the isometry groups of rank one symmetric spaces. Our methods unify most of the existing methods used in the existing literature for proving this …

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.

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- finite…

The authors derive a McShane identity for once-punctured super tori. Relying upon earlier work on super Teichmüller theory by the last two-named authors, they further develop the supergeometry of these surfaces and establish asymptotic growth rate of their length spectra.

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 quotien…

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 tha…

We study general representations of the free group on two generators into $SL(2,C)$, and the connection with generalized Markoff maps, following Bowditch. We show that Bowditch's Q-conditions for generalized Markoff maps are sufficient for the generalized McShane identity to hold for the corresponding representations a…

This paper discovers new identities linking geodesic and orthogeodesic lengths on hyperbolic surfaces.

We prove a McShane-type identity - a series, expressed in terms of geodesic lengths, that sums to 2πfor any closed hyperbolic surface with one distinguished point. To do so, we prove a generalized Birman-Series theorem showing that the set of complete geodesics on a hyperbolic surface with large cone angles is sparse.

Sum of Lagrange numbers equals a specific formula.

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 t…

We prove an extension of Basmajian's identity to $n$-Hitchin representations of compact bordered surfaces. For $n=3$, we show that this identity has a geometric interpretation for convex real projective structures analogous to Basmajian's original result. As part of our proof, we demonstrate that, with respect to the L…

In this survey, we discuss four classes of identities due principally to Basmajian, McShane, Bridgeman-Kahn and Luo-Tan on hyperbolic manifolds and provide a unified approach for proving them. We also elucidate on the connections between the various identities.

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.

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 …

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.

Recursion formula derived for moduli spaces of hyperbolic surfaces with cone points.

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 punc…

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 corres…

The paper explores orthogeodesics on hyperbolic surfaces and their integer traces.

The purpose of this note is to announce complete answers to the following questions. (1) For an essential simple loop on a 2-bridge sphere in a 2-bridge link complement, when is it null-homotopic in the link complement? (2) For two distinct essential simple loops on a 2-bridge sphere in a 2-bridge link complement, when…

Estimates for geodesics on hyperbolic tori improve previous bounds.

We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.

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 ov…

Study volumes of Klein surfaces, extending Mirzakhani's recursion.

Following Riley's work, for each 2-bridge link $K(r)$ of slope $r\in\QQ$ and an integer or a half-integer $n$ greater than 1, we introduce the {\it Heckoid orbifold $\orbs(r;n)$} and the {\it Heckoid group $\Hecke(r;n)=π_1(\orbs(r;n))$ of index $n$ for $K(r)$}. When $n$ is an integer, $\orbs(r;n)$ is called an {\it eve…

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 …

We propose a general theory for constructing functorial assignments $Σ\longmapsto Ω_Σ \in E(Σ)$ for a large class of functors $E$ from a certain category of bordered surfaces to a suitable target category of topological vector spaces. The construction proceeds by successive excisions of homotopy classes of embedded pai…

Characterizes slopes for Markov ordering on prime pairs.

We study a new bordification of the decorated Teichmüller space for a multiply punctured surface F by a space of filtered screens on the surface that arises from a natural elaboration of earlier work of McShane-Penner. We identify necessary and sufficient conditions for paths in this space of filtered screens to yield …

The study limits the number of ribbon concordant fibered knots.

This research connects combinatorial Teichmüller space geometry to Weil-Petersson geometry.

New neural network class approximates Hölder functions with optimal error and sample complexity.

This paper shows that only finitely many knots can be ribbon concordant to any given knot.

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

Establishes a correspondence between two mathematical identities.

Quandle homology was defined from rack homology as the quotient by a subcomplex corresponding to the idempotency, for invariance under the type I Reidemeister move. Similar subcomplexes have been considered for various identities of racks and moves on diagrams. We observe common aspects of these identities and subcompl…

The paper studies pseudo-Anosov maps from typical Thurston constructions.

The paper derives curvature identities for 5D and 6D Einstein manifolds.

Global Pestov identity proved on frame bundle and related fibrations.

Proves Bochner's identity on graphs using a new auxiliary graph.

Doodles link to commutator identities in a 2-sphere.

The paper studies harmonic identity maps on Riemannian manifolds.

User identity linkage is a task of recognizing the identities of the same user across different social networks (SN). Previous works tackle this problem via estimating the pairwise similarity between identities from different SN, predicting the label of identity pairs or selecting the most relevant identity pair based …