Group metrics for graph products of cyclic groups

Abstract

Abstract. We complement the characterization of the graph products of cyclic groups G(Γ, p) admitting a Polish group topology of [9] with the following result. Let G = G(Γ, p), then the following are equivalent:(i) there is a metric on Γ which induces a separable topology in which E Γ is closed; (ii)

Some Results on Polish Groups

“…By the ℵ 1 -half graph Γ(ℵ 1 ) we mean the graph on vertex set {a α : α < ℵ 1 }∪ {b β : β < ℵ 1 } with edge relation a α E Γ b β if and only if α < β. In the process of characterization of the graph products of cyclic groups embeddable in a Polish group [6], we observed that the commutation relation x −1 y −1 xy = e can never define the ℵ 1 -half graph in a Polish group G. Here we generalize this to: Theorem 1.1. No quantifier-free formula ϕ(x,ȳ) in the language of group theory can define the ℵ 1 -half graph in a Polish group G.…”

Some Results on Polish Groups

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We prove that no quantifier-free formula in the language of group theory can define the ℵ 1 -half graph in a Polish group, thus generalising some results from [6]. We then pose some questions on the space of groups of automorphisms of a given Borel complete class, and observe that this space must contain at least one uncountable group.

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Polish topologies for graph products of groups

“…In this case the situation is substantially more complicated, and the solution of the problem establishes that admits a Polish group topology if and only if it admits a non‐Archimedean Polish group topology if and only if with a countable graph product of cyclic groups and a direct sum of finitely many continuum‐sized vector spaces over a finite field. Concerning Problem , in the authors give a complete solution in the case all the are cyclic, proving that is embeddable into a Polish group if and only if it is embeddable into a non‐Archimedean Polish group if and only if admits a metric which induces a separable topology in which is closed. We hope to conclude this series of studies with an answer to Problem at the same level of generality of this paper.…”