Isometry group of the Poincare half-plane ($\mathrm{Isom}({\mathbb{H}}^2)$)

Every orientation-preserving isometry of the Poincare half-plane model lies in $PS{L}_2(\mathbb{R})$. Every orientation-reversing isometry is the composition of one of these with the hyperbolic reflection $z\mapsto -\overline{z}$ in the imaginary axis. Equivalently, the full isometry group consists of real Möbius and anti-Möbius transformations, modulo nonzero scalar matrices.