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

Gianluca Paolini; Saharon Shelah

doi:10.3390/axioms6020013

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

Gianluca Paolini^*
, Saharon Shelah

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

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 lg(x) ≤ lg(x^k); (ii) if lg(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.

Original language	English
Article number	13
Journal	Axioms
Volume	6
Issue number	2
DOIs	https://doi.org/10.3390/axioms6020013
State	Published - 1 Jun 2017

Bibliographical note

Publisher Copyright:
© 2017 by the authors.

Keywords

Descriptive set theory
Polish group topologies
Right-angled Artin groups

Access to Document

10.3390/axioms6020013

Cite this

@article{868b384a986948b19c5b7feb9ad8c9f4,

title = "No uncountable Polish group can be a right-angled Artin group",

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 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.",

keywords = "Descriptive set theory, Polish group topologies, Right-angled Artin groups",

author = "Gianluca Paolini and Saharon Shelah",

note = "Publisher Copyright: {\textcopyright} 2017 by the authors.",

year = "2017",

month = jun,

day = "1",

doi = "10.3390/axioms6020013",

language = "אנגלית",

volume = "6",

journal = "Axioms",

issn = "2075-1680",

publisher = "Multidisciplinary Digital Publishing Institute (MDPI)",

number = "2",

}

TY - JOUR

T1 - No uncountable Polish group can be a right-angled Artin group

AU - Paolini, Gianluca

AU - Shelah, Saharon

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 - https://www.scopus.com/pages/publications/85029509939

U2 - 10.3390/axioms6020013

DO - 10.3390/axioms6020013

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85029509939

SN - 2075-1680

VL - 6

JO - Axioms

JF - Axioms

IS - 2

M1 - 13

ER -

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

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this