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 $\kappa$ and $\delta$ on the cusps $c$ of the corresponding triangle group $\Delta_n$ inside $\mathrm{SL}_2({\mathcal{O}})$, where ${\mathcal{O}} = \mathbf{Z}[\zeta_n+ \zeta^{-1}_n]$. 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 $\Delta_n \subset \mathrm{SL}_2({\mathcal{O}})$ inside $\mathrm{SL}_2(\widehat{{\mathcal{O}}})$, where $\widehat{{\mathcal{O}}}$ is the profinite completion of ${\mathcal{O}}$.