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arXiv · 1701.03021

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

Gianluca Paolini·Saharon Shelah
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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 $lg(x) \leq lg(x^k)$; (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.

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Gianluca Paolini, Saharon Shelah. 2017-01-08. No Uncountable Polish Group Can be a Right-Angled Artin Group. https://arxiv.org/abs/1701.03021

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