Presentations for cusped arithmetic hyperbolic lattices

Abstract

We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $\Gamma$, applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. As applications we compute presentations for the Picard modular groups $\mathrm{PU}(2,1,{\mathcal{O}}_d)$ for $d=1,3,7$ and the quaternion hyperbolic lattice $\mathrm{PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. The implementation of the method for these groups is computer-assisted.

35 pages, 5 figures, 9 tables. V2: added relations to presentations for d=3 and 7. V3: corrected a typo in the presentation for the Hurwitz modular group. V4: Made changes in response to a referee report. Found new points, resulting in a few more relations being added to the presentation of the Hurwitz modular group. V5: Computer assisted verification of all presentations

Cited by in corpus (2)

• Hybrid lattices and thin subgroups of Picard modular groups
• Presentations for Singly-Cusped Bianchi Groups