The adelic closure of triangle groups

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 \operatorname{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 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 \operatorname{SL}_{2}({\mathcal{O}}) inside \operatorname{SL}_{2}(\hat{{\mathcal{O}}}) , where \hat{{\mathcal{O}}} is the profinite completion of  {\mathcal{O}} .