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AUTOMORPHISM AND OUTER AUTOMORPHISM GROUPS OF RIGHT-ANGLED ARTIN GROUPS ARE NOT RELATIVELY HYPERBOLIC

Published online by Cambridge University Press:  24 January 2022

JUNSEOK KIM
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon34141, South Korea e-mail: jsk8818@kaist.ac.kr
SANGROK OH*
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon34141, South Korea
PHILIPPE TRANCHIDA
Affiliation:
Department of Mathematical Sciences, KAIST, 291 Daehak-ro Yuseong-gu, Daejeon34141, South Korea e-mail: ptranchi@kaist.ac.kr

Abstract

We show that the automorphism groups of right-angled Artin groups whose defining graphs have at least three vertices are not relatively hyperbolic. We then show that the outer automorphism groups are also not relatively hyperbolic, except for a few exceptional cases. In these cases, the outer automorphism groups are virtually isomorphic to either a finite group, an infinite cyclic group or $\mathrm {GL}_2(\mathbb {Z})$ .

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

The first and third authors were partially supported by Samsung Science and Technology Foundation grant No. SSTF-BA1702-01 and the National Research Foundation of Korea (NRF) grant funded by the Korean government (MSIT) (No. 2020R1C1C1A01006912).

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AUTOMORPHISM AND OUTER AUTOMORPHISM GROUPS OF RIGHT-ANGLED ARTIN GROUPS ARE NOT RELATIVELY HYPERBOLIC
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