Presentations for sisters of one-cusped Picard modular groups
We generalize a method devised by Mark and Paupert for finding group presentations of the single-cusped standard Picard modular groups $\text{PU}(2, 1; \mathcal{O}_d)$, where $\mathcal{O}_d$ is the ring of integers of $\mathbb{Q}(i\sqrt{d})$, to a more general class of lattices in $\text{PU}(2, 1)$. As an application, we give presentations for the so-called \textit{sisters} of the standard Picard modular groups for $d=3, 7,$ and $11$. In particular, this shows that the sister of $\text{PU}(2, 1; \mathcal{O}_7)$ is a perfect group.
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