No uncountable Polish group can be a right-angled Artin group
by Paolini and Shelah. [PaSh:1112]
We prove that no uncountable Polish group can admit a system
generators whose associated length function satisfies the following
(i) if 0 < k < omega, then lg(x) <= 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
be an uncountable right-angled Artin group. This generalizes results
from [Sh:744] and ``Polish Group Topologies'' by S. Solecki,
where this is proved for free and free abelian uncountable groups.
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