No Uncountable Polish Group Can be a Right-Angled Artin Group

We prove that no uncountable Polish group can admit a system of generators whose associated length function satisfies the following conditions: (i) if $0<k<\omega$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y)<k<\omega$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes results from [3] and [5], where this is proved for free and free Abelian uncountable groups.

In a meeting in Durham in 1997, Evans asked if an uncountable free group can be realized as the group of automorphisms of a countable structure. This was settled in the negative by Shelah [3]. Independently, in the context of descriptive set theory, Becher and Kechris [1] asked if an uncountable Polish group can be free. This was also answered negatively by Shelah [4], generalizing the techniques of [3]. Inspired by the question of Becher and Kechris, Solecki [5] proved that no uncountable Polish group can be free abelian. In this paper we give a general framework for these results, proving that no uncountable Polish group can be a right-angled Artin group (see below for a definition). We actually prove more: Theorem 1. Let G = (G, d) be an uncountable Polish group and A a group admitting a system of generators whose associated length function satisfies the following conditions: (i) if 0 < k < ω, then lg(x) lg(x k ); (ii) if lg(y) < k < ω and x k = y, then x = e. Then G is not isomorphic to A, in fact there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A.
Proof. Let ζ = (ζ n ) n<ω ∈ R ω be such that ζ n < 2 −n , for every n < ω, and g = (g n ) n<ω ∈ G ω such that g n = e and d(g n , e) < ζ n , for every n < ω. Let Λ be a set of power b of increasing functions η ∈ ω ω which is unbounded with respect to the partial order of eventual domination. For transparency we also assume that for every η ∈ Λ we have η(0) > 0. For η ∈ Λ, define the following set of equations: Let G * be the subgroup of G generated by {g n : n < ω} ∪ {b η,n : η ∈ Λ, n < ω}. Towards contradiction, suppose that π is an embedding of G * into A, and let S Partially supported by European Research Council grant 338821. No. 1112 on Shelah's publication list.
Definition 2. Given a graph Γ = (E, V ), the right-angled Artin group A(Γ) is the group with presentation V | ab = ba : aEb .
Thus, for Γ a graph with no edges (resp. a complete graph) A(Γ) is a free group (resp. a free abelian group).

Corollary 5. No uncountable Polish group can be a right-angled Artin group.
Proof. By Theorem 1 it suffices to show that for every right-angled Artin group A(Γ) the associated length function lg satisfies conditions (i) and (ii) of the theorem, but by Fact 4 this is clear.
As well known, the automorphism group of a countable structure is naturally endowed with a Polish topology which respects the group structure, hence: Corollary 6. The automorphism group of a countable structure can not be an uncountable right-angled Artin group.