The adelic closure of triangle groups

Abstract

Motivated by questions arising from billiard trajectories in the regular $n$-gon, McMullen defined a pair of functions $\hat{I}º$ and $\hat{I}´$ on the cusps $c$ of the corresponding triangle group $\hat{I}{”}_n$ inside ${\mathrm{SL}}_2(\mathcal{O})$, where $\mathcal{O}=\mathbf{Z}[\hat{I}{\P }_n+\hat{I}{\P }_n^{-1}]$. McMullen asks for which $n$ these functions are congruence, that is, when they only depend on the image of the cusp $c\in {\mathbf{P}}^1(\mathcal{O})$ in ${\mathbf{P}}^1(\mathcal{O}/d)$ for some integer $d$. In this note, we answer McMullen's questions. We obtain our results by computing the exact closure of $\hat{I}{”}_n\subset {\mathrm{SL}}_2(\mathcal{O})$ inside ${\mathrm{SL}}_2(\hat{\mathcal{O}})$, where $\hat{\mathcal{O}}$ is the profinite completion of $\mathcal{O}$.

This is the (lightly) revised final version; to appear in Groups, Geometry, and Dynamics