  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.
