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.