Projective special linear group:PSL(2,R)

Definition as a matrix group
$$PSL(2,\R)$$ is defined as the member of family::projective special linear group of degree two over the field of real numbers.

In other words, it is defined as $$SL(2,\R)/\{ \pm I \}$$.

Definition as a group of fractional linear transformations
The group $$PSL(2,\R)$$ can be defined as the group of conformal automorphisms of the upper half-plane. In other words, it is the group of maps of the form:

$$z \mapsto \frac{az + b}{cz + d}$$.

where $$a,b,c,d \in \mathbb{C}$$ are such that these maps send the upper half-plane to itself.