No uncountable Polish group can be a right-angled Artin group

by Paolini and Shelah. [PaSh:1112]
Axioms Topical Collection ``Topological Groups'', 2017
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) <= 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 [Sh:744] and ``Polish Group Topologies'' by S. Solecki, where this is proved for free and free abelian uncountable groups.

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