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Open AccessArticle

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

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Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israel
2
Department of Mathematics, The State University of New Jersey, Hill Center-Busch Campus, Rutgers, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA
*
Author to whom correspondence should be addressed.
Academic Editor: Sidney A. Morris
Axioms 2017, 6(2), 13; https://doi.org/10.3390/axioms6020013
Received: 28 March 2017 / Revised: 20 April 2017 / Accepted: 4 May 2017 / Published: 11 May 2017
(This article belongs to the Collection Topological Groups)
We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 < k < ω , then l g ( x ) l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. View Full-Text
Keywords: descriptive set theory; polish group topologies; right-angled Artin groups descriptive set theory; polish group topologies; right-angled Artin groups
MDPI and ACS Style

Paolini, G.; Shelah, S. No Uncountable Polish Group Can be a Right-Angled Artin Group. Axioms 2017, 6, 13.

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