Jø rgensen subgroups of the Picard group

Abstract

Let $G$ be a subgroup of rank two of the Möbius group $\textit{PSL}(2, \mathbb{C})$. The Jørgensen number $J(G)$ of $G$ is defined by \[ J(G) = \inf \{|\tr^{2} A - 4| + \mathopen|\tr[A, B]-2|\colon \langle A, B\rangle= G\}. \] We describ e all subgroups $G$ of the Picard group $\textit{PSL}(2, \mathbb{Z} + i\mathbb{Z})$ with $J(G) = 1$.