Presentations for quaternionic $S$-unit groups

TL;DR

This paper provides the first explicit presentations of projective S-unit groups in quaternion algebras, acting on products of Bruhat--Tits trees, with computational tools included.

Contribution

It introduces explicit presentations for S-unit groups in quaternion algebras over Q, including computational methods and discussions on the congruence subgroup problem.

Findings

First explicit presentations of S-unit groups in quaternion algebras.

Groups act irreducibly and cocompactly on products of Bruhat--Tits trees.

Provides computational code for arbitrary finite sets S.

Abstract

The purpose of this paper is to give presentations for projective $S$-unit groups of the Hurwitz order in Hamilton's quaternions over the rational field $\mathbb{Q}$. To our knowledge, this provides the first explicit presentations of an $S$-arithmetic lattice in a semisimple Lie group with $S$ large. In particular, we give presentations for groups acting irreducibly and cocompactly on a product of Bruhat--Tits trees. We also include some discussion and experimentation related to the congruence subgroup problem, which is open when $S$ contains at least two odd primes. In the appendix, we provide code that allows the reader to compute presentations for an arbitrary finite set $S$.