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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1$\begingroup$ Every torsion free discrete group $G$ acts properly and discontinuousy on the unit sphere of $L^2(G)$, no? $\endgroup$– Mariano Suárez-ÁlvarezCommented May 13, 2013 at 8:55
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$\begingroup$ 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. $\endgroup$– Daniele ZuddasCommented May 13, 2013 at 9:22
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$\begingroup$ I am Thinking mainly about Isometric actions. Thanks for the comments. $\endgroup$– Nicolas BoergerCommented May 13, 2013 at 10:23
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