Presentations for quaternionic $S$-unit groups

Ted Chinburg, Holley Friedlander, Sean Howe, Michiel Kosters, Bhairav Singh, Matthew Stover, Ying Zhang, Paul Ziegler

Published 2014-04-24Version 1

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 $S$.