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

Abstract

We define a simple criterion for a homogeneous, complete metric structure $X$ that implies that the automorphism group $\mbox{Aut}(X)$ 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 $\mbox{U}$, the Lebesgue probability measure algebra $\mbox{MALG}$, and the Hilbert space ${\ell }_2$, thus proving that $\mbox{Iso}(\mbox{U})$, $\mbox{Aut}(\mbox{MALG})$, $U({\ell }_2)$, and $O({\ell }_2)$ share these properties. We also formulate a condition for $X$ which implies that every homomorphism of $\mbox{Aut}(X)$ into a separable group $K$ with a left-invariant, complete metric, is trivial, and we verify it for $\mbox{U}$, and ${\ell }_2$.

arXiv commentarXiv admin note: text overlap with arXiv:1312.5141 by other authors