Polish topometric groups

Abstract

Abstract. We define and study the notion of ample metric generics for a Polish topological group, which is a weakening of the notion of ample generics introduced by Kechris and Rosendal in [KR07]. Our work is based on the concept of a Polish topometric group, defined in this article. Using Kechris and Rosendal's work as a guide, we explore consequences of ample metric generics (or, more generally, ample generics for Polish topometric groups). Then we provide examples of Polish groups with ample metric generics, such as the isometry group Iso(U 1 ) of the bounded Urysohn space, the unitary group U (ℓ 2 ) of a separable Hilbert space, and the automorphism group Aut([0, 1], λ) of the Lebesgue measure algebra on [0, 1]. We deduce from this and earlier work of Kittrell and Tsankov that this last group has the automatic continuity property, i.e., any morphism from Aut([0, 1], λ) into a separable topological group is continuous.

Lipschitz functions on topometric spaces

“…General topometric spaces (i.e., non compact) were defined and studied further from an abstract point of view in [2], where the formalism is shown to be useful for the analysis of perturbation structures on type spaces. The same idea was also shown to be useful in the context of (very non compact) Polish groups, which may admit "topometric ample generics" even when no purely topological ample generics need exist, see [4].…”

Automatic Continuity for Isometry Groups

“…The groups of automorphisms of metric structures can be endowed with many topologies and this is the starting point of the analysis of Ben Yaacov, Berenstein and Melleray [5], who consider two types of topologies: the Polish topologies of pointwise convergence and variants of strong (non separable) topologies. We are, however, primarily interested in separable topologies on these groups.…”

“…The group Aut([0, 1], λ) (with the weak topology) is isomorphic to the group of automorphism of the measure algebra and applying Theorem 1.4 to the measure algebra we get a new proof of automatic continuity. After the work of Ben Yaacov, Berenstein and Melleray [5], Tsankov [53] further showed the automatic continuity property for the infinite-dimensional unitary group. Given that the group U (ℓ 2 ) is the automorphism group of the Hilbert space ℓ 2 (or the isometry group of the sphere in ℓ 2 ), we can apply Theorem 1.4 to the Hilbert space and get a new proof of this result.…”

Grey Subsets of Polish Spaces

“…Below, for x ∈ G and C ⊆ G we denote by C x the set x −1 Cx. By [3, Lemma 3.6] we can find a mapping a ∈ 2 → h a ∈ G such that if a, b ∈ 2 are distinct then Notice that [3,Theorem 4.7] follows from Theorem 3.6 via a straightforward reduction to the case where the target group is metrisable, giving a more elegant proof than the one appearing there. Conversely, Theorem 3.6 also follows from a combination of Proposition 3.2 with [3,Theorem 4.7].…”