Assume a group $G$ acts faithfully by isometries on a separable infinite dimensional Hilbert space $H$ in such a way that the orbits are closed and the quotient $H/G$ is isometric to a finite dimensional Riemannian manifold.

Edit after YCor comment: Is there a natural way to equip $G$ with a topology so that it is an infinite dimensional Lie Group with a Lie Algebra?

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    $\begingroup$ $G$ can be a large discrete group, say acting transitively. $\endgroup$ – YCor Dec 28 '18 at 21:47
  • $\begingroup$ Hi YCor. Could you please describe an example with that property? I haven't been able to construct one. $\endgroup$ – Sergio Zamora Jan 14 '19 at 17:26
  • $\begingroup$ Endow $G=H$ with the discrete topology, and let it act by translations on $H$. So the quotient is a point, and $G$ is a 0-dimensional Lie group. This is not an interesting example, probably additional hypotheses would be needed to discard it, and motivating examples could help then. $\endgroup$ – YCor Jan 14 '19 at 17:36
  • $\begingroup$ Your addition doesn't help resolve YCor's comment, take $H = H_1 \oplus H_1$ and $G = H_1 \oplus H_1^\delta$, the first term $H_1$ with the usual topology and the second with the discrete topology, considered as an additive group. $\endgroup$ – mme Jan 16 '19 at 19:31
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    $\begingroup$ Hello Mike. If one uses the group of all translations, it could be equipped with the pointwise convergence topology and then it becomes homeomorphic to the Hilbert space. $\endgroup$ – Sergio Zamora Jan 16 '19 at 19:40

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