Knowlpedia

Definition

Projective special linear group over a ring

The determinant-one matrix group modulo its scalar determinant-one matrices.

For a nonzero commutative ring RR with identity and n≥2n\ge2, use the convention

PSL⁡n(R)=SL⁡n(R)/{uIn:u∈R×, un=1}.\operatorname{PSL}_n(R)=\operatorname{SL}_n(R)/\{uI_n:u\in R^\times,\ u^n=1\}.

The denominator is the central subgroup of scalar matrices in the , so the is defined.

The case needed for Bianchi groups

If R⊆CR\subseteq\mathbb C and n=2n=2, then u2=1u^2=1 forces u=±1u=\pm1. Consequently

PSL⁡2(R)=SL⁡2(R)/{±I}↪PSL⁡2(C).\operatorname{PSL}_2(R)=\operatorname{SL}_2(R)/\{\pm I\} \hookrightarrow\operatorname{PSL}_2(\mathbb C).

The kernel calculation proves the displayed injection.

Convention warning

This is an abstract group quotient. It makes no assertion that forming points of a group-scheme quotient commutes with quotienting groups of points. Over fields it agrees with the existing .