Our main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

Keywords:

affine actions, Hilbert spaces, minimal actions, nilpotent groups affine actions, Hilbert spaces, minimal actions, nilpotent groups

MSC Classifications:

22D10, 43A35, 20F69 show english descriptions Unitary representations of locally compact groups
Positive definite functions on groups, semigroups, etc.
Asymptotic properties of groups 22D10 - Unitary representations of locally compact groups
43A35 - Positive definite functions on groups, semigroups, etc.
20F69 - Asymptotic properties of groups

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