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 $0 < k < ω$, then $lg(x) \leq lg(x^k)$; (ii) if $lg(y) < k < ω$ and $x^k = y$, then $x = e$. In particular, the automorphism group of a countable str