Picard modular groups generated by complex reflections

TL;DR

This paper demonstrates that certain Picard modular groups and quaternionic hyperbolic lattices are generated by complex or quaternionic reflections, revealing new generating sets and subgroup structures.

Contribution

It shows that specific Picard modular groups are generated by complex reflections and identifies index 4 subgroups generated by complex reflections.

Findings

Picard modular groups for d=1,3,7 are generated by complex reflections.

Quaternionic hyperbolic lattice with Hurwitz integers is generated by quaternionic reflections.

Groups for d=2,11 have index 4 subgroups generated by complex reflections.

Abstract

In this short note we use the presentations found in \cite{MP} and \cite{Po} to show that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=1,3,7$ (respectively the quaternion hyperbolic lattice ${\rm PSp}(2,1,\mathcal{H})$ with entries in the Hurwitz integer ring $\mathcal{H}$) are generated by complex (resp. quaternionic) reflections, and that the Picard modular groups ${\rm PU}(2,1,\mathcal{O}_d)$ with $d=2,11$ have an index 4 subgroup generated by complex reflections.