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

Abstract

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 $lg(x)\le lg(x^k)$; (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.