Groups of rotations of Euclidean and hyperbolic spaces (Q6158243)

The authors ``present a new, and entirely geometric, proof of'' the following theorem, first proved by \textit{P. Fatou} [Fonctions automorphes. (P. Appell, E. Goursat: Théorie des fonctions algébriques et des transcendantes qui s'y rattachent; t. II. 2. ed., revue et augmentée). Paris: Gauthier-Villars (1930; JFM 56.0330.01)], which, they believe, ``is more direct, transparent, and simpler, than any of the earlier proofs'': If \(G\) is a (finite or infinite) group of rotations of three-dimensional hyperbolic space, then the non-trivial rotations in \(G\) either have a common axis or a unique common fixed point.