A note on the isometry group of the hyperbolic plane

Abstract

This note concerns $G$, the connected component of the identity of the isometries of the hyperbolic plane. $G$ operates on the closed unit disk. Here we characterize $G$ both as a Lie group among the centerless, connected, real semisimple Lie groups of noncompact type as having no two dimensional abelian Lie subgroups, then as a transformation group: any two nontrivial isometries of $G$ commute if and only if they have the same fixed point set, this property being particular to $G$ among all isometry groups of irreducible symmetric spaces of noncompact type. From this it follows that any subgroup of $G$ with a nontrivial center is abelian. In particular, each nilpotent subgroup of $G$ is abelian. Finally, we uniquely characterize SO$(2,1)$ among the centerless, connected, real simple Lie groups of noncompact type with this property.