Consequences of the Existence of Ample Generics and Automorphism Groups of Homogeneous Metric Structures

Abstract

Abstract. We define a 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 non-open subgroups. Then we verify it for the Urysohn space Í, the Lebesgue probability measure algebra MALG, and the Hilbert space ℓ2, thus proving that Iso(Í), 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 Í, and ℓ2.

Automatic Continuity for Isometry Groups

“…It is worth noting that Theorem 1.4 has already been used, in a modified version of Malicki [26], by Kaïchouh [18] to obtain several new automatic continuity results for infinite powers of Polish groups. Theorem 1.4 can be also applied to give a unified treatment of previously known automatic continuity results for automorphism groups of some metric structures.…”

Polish topologies on groups of non-singular transformations

“…Let us now briefly indicate why Aut(Y, λ) has the automatic continuity property when (Y, λ) is a standard σ -finite space (ie, λ is a non-atomic σ -finite infinite measure on the standard Borel space Y ). To do so, we will simply check the criterions given by Sabok [15] and then simplified by Malicki [13]. We won't give full details since the proofs adapt verbatim and we refer the reader to Malicki's paper for definitions of the terms used thereafter.…”

Polish topologies on groups of non-singular transformations

“…Let us now briefly indicate why Aut(Y, λ) has the automatic continuity property by checking the criterions given by Sabok [Sab19] and then simplified by Malicki [Mal16]. We won't give full details since the proofs adapt verbatim and we refer the reader to Malicki's paper for definitions of the terms used thereafter.…”

“…Later on Sabok developed a framework to show automatic continuity for automorphism groups of metric structures [Sab19]. In particular, he got another proof of automatic continuity for Aut(Y, λ), and then Malicki simplified his approach [Mal16]. We first observe that this framework can also be applied when λ is infinite.…”