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 ${ \mathcal{ O}}$.
Categories: math.NT math.GT