The sum of Lagrange numbers

Abstract

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 - \varphi - \sqrt 2 \end{equation} where $L_n$ is the $n$th Lagrange number and $\varphi=\frac{1+\sqrt5}2$ is the golden ratio.