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

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, $l(\gamma )=2\mathrm{arccosh}(t(\gamma )/2)$ 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 $S{L}_2(\mathbb{Z})$ (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 Teichmüller orbit.

Comments welcome! (v3: Updated to give more context, add references, fix a few typos and the height function.)

References in corpus (1)

• Multiplicities of simple closed geodesics and hypersurfaces in Teichmüller space