Presentations for cusped arithmetic hyperbolic lattices

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