The adelic closure of triangle groups

TL;DR

This paper investigates the adelic closure of triangle groups related to billiard trajectories in regular polygons, providing explicit computations of their closures and answering questions about the congruence properties of associated functions.

Contribution

It computes the exact adelic closure of triangle groups inside the profinite completion, addressing McMullen's questions on congruence properties of certain functions.

Findings

Computed the closure of triangle groups in the profinite setting

Answered McMullen's questions on congruence functions

Established explicit descriptions of the adelic closure

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 ${\mathcal{O}}$.