Presentations for cusped arithmetic hyperbolic lattices

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