The adelic closure of triangle groups

Abstract

Motivated by questions arising from the study of billiard trajectories in the regular $n$-gon, McMullen (2024) defined a pair of functions $\kappa$ and $\delta$ on the cusps $c$ of the corresponding triangle group ${\Delta }_n$ inside ${\mathrm{SL}}_2(\mathcal{O})$, where $\mathcal{O}=\mathbf{Z}[{\zeta }_n+{\zeta }_n^{-1}]$. McMullen asks for which $n$ these functions are congruence, that is, when they depend only on the image of the cusp $c\in {\mathbf{P}}^1(\mathcal{O})$ in ${\mathbf{P}}^1(\mathcal{O}/N)$ for some integer $N$. In this note, we answer McMullen’s questions. We obtain our results by computing the exact closure of ${\Delta }_n\subset {\mathrm{SL}}_2(\mathcal{O})$ inside ${\mathrm{SL}}_2(\hat{\mathcal{O}})$, where $\hat{\mathcal{O}}$ is the profinite completion of $\mathcal{O}$.