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Axioms 2017, 6(2), 13; doi:10.3390/axioms6020013

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

1
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
Received: 28 March 2017 / Revised: 20 April 2017 / Accepted: 4 May 2017 / Published: 11 May 2017
(This article belongs to the Collection Topological Groups)
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Abstract

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
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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

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