# Proper actions on unitary spheres of a Hilbert space

Free group action of spheres, or products of spheres by finite groups have been studied extensively in the literature, giving in many cases restrictions to the cohomological properties of the acting group. Is there a characterization of groups acting properly and discontinuously on the unitary sphere of a Hilbert space?

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Every torsion free discrete group $G$ acts properly and discontinuousy on the unit sphere of $L^2(G)$, no? –  Mariano Suárez-Alvarez May 13 '13 at 8:55
If you mean only topological actions, then this is the same of actions on Hilbert spaces, because the infinite-dimensional unitary sphere is diffeomorphic with the Hilbert space, see C. Bessaga, "Every infinite-dimensional Hilbert space is diffeomorphic with its unit sphere", Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 14 1966 27-31. –  Daniele Zuddas May 13 '13 at 9:22
I am Thinking mainly about Isometric actions. Thanks for the comments. –  Nicolas Boerger May 13 '13 at 10:23