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Inspired by a recent work of Marcin Sabok, we define a criterionfor a homogeneous metric structure X that implies that its automorphism group Aut(X) satisfies all the main consequences of the existence of ample generics: it has the small index property, the automatic continuity property, and uncountable cofinality for non-open subgroups. Then we verify it for the Urysohn space, the Lebesgue probability measure algebra, and the Hilbert space, regarded as metric structures, thus proving that their automorphism groups share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group with a left-invariant, complete metric, is trivial, and we verify it for the Urysohn space, and the Hilbert space.