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

TL;DR

This paper establishes criteria under which automorphism groups of certain homogeneous metric structures exhibit properties like the small index property and automatic continuity, verified for spaces like the Urysohn space, Lebesgue measure algebra, and Hilbert space.

Contribution

It introduces a new criterion linking homogeneity and metric properties to automorphism group characteristics, and verifies these properties for key classical structures.

Findings

Automorphism groups of Urysohn space, Lebesgue measure algebra, and Hilbert space satisfy ample generics consequences.

The paper proves that these groups have the small index property and automatic continuity.

It also provides conditions under which homomorphisms into separable groups are trivial.

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 $\ell_2$.