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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.

An orientation-preserving isometry fixing every point of ${\ell }_{+}$ fixes more than two points in the upper half-plane and hence, as a Möbius transformation, is the identity. The mapfixes ${\ell }_{+}$ pointwise and preserves the hyperbolic metric; it is the hyperbolic reflection in ${\ell }_{+}$. Consequently the pointwise stabilizer is exactly

For any isometry $F$, use part (a) to choose $g\in PS{L}_2(\mathbb{R})$ that agrees with $F$ on one pointed line and its tangent direction. Then $g^{-1}F$ fixes that line pointwise, so it is either the identity or its reflection. Hence the Isometry group of the Poincare half-plane consists ofandwith real coefficients, modulo multiplication of all four coefficients by a nonzero scalar.

The Poincare half-plane model iswith the orientation inherited from the complex plane. Forone hasThese identities show directly that ${\gamma }^{\ast }d{s}^2=d{s}^2$. The map is holomorphic with nonzero derivative, so it preserves orientation. The matrices $I$ and $-I$ induce the same map, giving the action of PSL2(R).

Conversely, let $F$ be an orientation-preserving isometry. The PSL2(R) action is transitive on $\mathbb{H}$, so compose $F$ with an element taking $F(i)$ back to $i$. The resulting isometry fixes $i$ and acts on $T_i\mathbb{H}$ by an orientation-preserving orthogonal map, hence a rotation. The stabilizer of $i$ in PSL2(R),realizes every such tangent rotation. An isometry is determined by its value and differential at one point because it preserves geodesics and the exponential map. Thus the composed isometry belongs to PSL2(R), and so does $F$.

The map $\tau (z)=-\overline{z}$ is an orientation-reversing isometry. Composing any orientation-reversing isometry with $\tau$ gives an orientation-preserving one, so

A hyperbolic line is a vertical Euclidean line or a semicircle orthogonal to the real axis. Its hyperbolic reflection ${\sigma }_{\ell }$ is the unique orientation-reversing isometry that fixes every point of $\ell$. If $\ell ,{\ell }^{\prime }$ meet at angle $\theta$, thenis the hyperbolic rotation about $A$ through angle $2\theta$. The generators satisfyIf $\theta /\pi =p/q$ in lowest terms, then $\rho$ has order $q$ and the generated group is the finite dihedral group of order $2q$. If $\theta /\pi$ is irrational, $\rho$ has infinite order and the generated group is the infinite dihedral group. In the degenerate case $\ell ={\ell }^{\prime }$, the group has order two.