TY - JOUR
T1 - No uncountable Polish group can be a right-angled Artin group
AU - Paolini, Gianluca
AU - Shelah, Saharon
N1 - Publisher Copyright:
© 2017 by the authors.
PY - 2017/6/1
Y1 - 2017/6/1
N2 - 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 lg(x) ≤ lg(xk); (ii) if lg(y) < k < ω and xk = 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.
AB - 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 lg(x) ≤ lg(xk); (ii) if lg(y) < k < ω and xk = 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.
KW - Descriptive set theory
KW - Polish group topologies
KW - Right-angled Artin groups
UR - http://www.scopus.com/inward/record.url?scp=85029509939&partnerID=8YFLogxK
U2 - 10.3390/axioms6020013
DO - 10.3390/axioms6020013
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AN - SCOPUS:85029509939
SN - 2075-1680
VL - 6
JO - Axioms
JF - Axioms
IS - 2
M1 - 13
ER -