An infinite product on the Teichm\"{u}ller space of the once-punctured torus

TL;DR

This paper establishes an infinite product identity involving lengths of simple closed geodesics on the once-punctured torus, connecting geometric, algebraic, and Teichmüller theory aspects with elementary proofs.

Contribution

It introduces a new infinite product identity on the Teichmüller space of the once-punctured torus, linking geodesic lengths, traces, and orbit structures under SL(2,Z), with elementary proofs.

Findings

Proves a novel infinite product identity for geodesics on the once-punctured torus.

Provides an elementary proof of McShane's identity in the same setting.

Connects geometric lengths with algebraic traces through explicit formulas.

Abstract

We prove the identity $$ \prod_{\gamma}\left(\frac{e^{l(\gamma)}+1}{e^{l(\gamma)}-1}\right)^{2h}=\exp\left(\frac{l_1+l_2+l_3}{2}\right), $$ (or $$ \prod_{\gamma}\left(\frac{t(\gamma)^2}{t(\gamma)^2-4}\right)^h=\frac{t_1+\sqrt{t_1^2-4}}{2}\cdot\frac{t_2+\sqrt{t_2^2-4}}{2}\cdot\frac{t_3+\sqrt{t_3^2-4}}{2} $$ in trace coordinates), where the product is over all simple closed geodesics on the once-punctured torus, $l(\gamma)=2\operatorname{arccosh}(t(\gamma)/2)$ is the length of the geodesic, and $l_i$ ($t_i$) are the lengths (traces) of any triple of simple geodesics $\{\gamma_i\}$ intersecting at a single point. The exponent $h=h(\gamma;\{\gamma_i\})$ is a positive integer "height" which increases as we move away from the chosen triple $\{\gamma_i\}$ in its orbit under $SL_2(\mathbb{Z})$ (see Figure 1 for the "definition by picture"). For comparison, a short proof of McShane's identity $$ \sum_{\gamma}\frac{1}{1+e^{l(\gamma)}}=\frac{1}{2}=\sum_{\gamma}\frac{1-\sqrt{1-4/t(\gamma)^2}}{2} $$ in the same spirit is given in an appendix. Both proofs are elementary and proceed by "integrating" around the chosen triple $\{\gamma_i\}$ in its Teichm\"{u}ller orbit.