We define a criterion for a homogeneous, complete metric structure \(X\) that implies that the automorphism group \({\rm Aut}(X)\) 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. We verify it for the Urysohn space \(\mathbb{U}\), the Lebesgue probability measure algebra \({\rm MALG}\), and the Hilbert space \(\ell_2\), thus proving that \({\rm Iso}(\mathbb{U})\), \({\rm Aut}({\rm MALG})\), \(U(\ell_2)\), and \(O(\ell_2)\) share these properties. We also formulate a condition for \(X\) that implies that every homomorphism of \({\rm Aut}(X)\) into a separable group \(K\) with a left-invariant, complete metric, is trivial, and we verify it for \(\mathbb{U}\), and \(\ell_2\).
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