Presentations for cusped arithmetic hyperbolic lattices

TL;DR

This paper introduces a general computational method to derive presentations of cusped arithmetic hyperbolic lattices, with applications to specific Picard modular groups and quaternion hyperbolic lattices, supported by computer assistance.

Contribution

It provides a novel, general approach to compute presentations of cusped arithmetic hyperbolic lattices using horoball covers and Macbeath's theorem, with explicit examples.

Findings

Computed presentations for Picard modular groups ${ m PU}(2,1, ext{O}_d)$ for d=1,3,7

Derived presentation for quaternion hyperbolic lattice with Hurwitz integers

Method is implemented with computer assistance

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 ${\rm PU}(2,1,\mathcal{O}_d)$ for $d=1,3,7$ and the quaternion hyperbolic lattice ${\rm PU}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$. The implementation of the method for these groups is computer-assisted.