Past exam of the mathematics course of the University of Cambridge / 2019 / ib / Paper 4 / 15E / b

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.