Consequences of the existence of ample generics and automorphism groups of homogeneous metric structures

الأصل · EN

We define a simple criterion for a homogeneous, complete metric structure X that implies that the 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 U, the Lebesgue probability measure algebra MALG, and the Hilbert space ℓ₂, thus proving that Iso(U), Aut(MALG), U(ℓ₂), and O(ℓ₂) share these properties. We also formulate a condition for X which implies that every homomorphism of Aut(X) into a separable group K with a left-invariant, complete metric, is trivial, and we verify it for U, and ℓ₂.