Norm-one group of a quaternion order

For an order $\mathcal O$ in a quaternion algebra, its norm-one group is

\mathcal O^1=\{x\in\mathcal O^\times:\operatorname{nrd}(x)=1\},

where $\mathcal O^\times$ denotes its units and $\operatorname{nrd}$ is the reduced norm. Multiplicativity of the reduced norm makes this a subgroup of the unit group.

The matrix case

For $\mathcal O=M_2(\mathcal O_K)$ , the reduced norm is the determinant, hence

\mathcal O^1=\operatorname{SL}_2(\mathcal O_K).

This is the bridge from quaternionic constructions to Bianchi groups.

Complex realization

Under a complex splitting, the norm-one elements map into $\operatorname{SL}_2(\mathbb C)$ . Their image in $\operatorname{PSL}_2(\mathbb C)$ identifies the central elements $1$ and $-1$ .

References

F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, Arithmeticity, discreteness and volume, Transactions of the AMS 349 (1997). Author-hosted paper, §4, definition using O¹.