Roelcke non-precompactness, simplicity, and non-amenability of the automorphism group of the Fraïssé limit of finite Heyting algebras are proved among others.

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 automatic continuity property, the small index property, and uncountable cofinality for nonopen subgroups. Then we verify it for the Urysohn space $$, the Lebesgue probability measure algebra MALG, and the Hilbert space $\ell _2 $, thus proving that Iso($$), Aut(MALG), $U\left( {\ell _2 } \right)$, and $O\left( {\ell _2 } \right)$ 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 $$, and $\ell _2 $.

We develop the basics of an analogue of descriptive set theory for functions on a Polish space X. We use this to define a version of the small index property in the context of Polish topometric groups, and show that Polish topometric groups with ample generics have this property. We also extend classical theorems of Effros and Hausdorff to the topometric context.

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.

2000 Mathematical Subject Classification: 03C45.