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Liftable automorphisms of right-angled Artin groups

  • Sangrok Oh , Donggyun Seo EMAIL logo and Philippe Tranchida
Published/Copyright: December 3, 2024
Journal of Group Theory
From the journal Journal of Group Theory Volume 28 Issue 3

Abstract

Given a regular covering map φ : Λ → Γ of graphs, we investigate the subgroup LAut ⁡ ( φ ) of the automorphism group Aut ⁡ ( A Γ ) of the right-angled Artin group A Γ . This subgroup comprises all automorphisms that can be lifted to automorphisms of A Λ . We first show that LAut ⁡ ( φ ) is generated by a finite subset of Laurence’s elementary automorphisms. For the subgroup FAut ⁡ ( φ ) of Aut ⁡ ( A Λ ) that consists of lifts of automorphisms in LAut ⁡ ( φ ) , there exists a natural homomorphism FAut ⁡ ( φ ) → LAut ⁡ ( φ ) induced by 𝜑. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup IA ⁡ ( A Λ ) and deduce a short exact sequence reminiscent of results from the Birman–Hilden theory for surfaces.

Funding source: Eusko Jaurlaritza

Award Identifier / Grant number: IT1483-22

Funding source: National Research Foundation of Korea

Award Identifier / Grant number: 2021R1C1C200593811

Funding source: Samsung Science and Technology Foundation

Award Identifier / Grant number: SSTF-BA1702-01

Funding statement: The first author is supported by the Basque Government grant IT1483-22. The first and second authors are supported by the National Research Foundation of Korea (NRF) grant No. 2021R1C1C200593811 from the Korea government (MSIT). The third author is partially supported by the Samsung Science and Technology Foundation under Project Number SSTF-BA1702-01.

Acknowledgements

We would like to thank Sang-hyun Kim, Hyungryul Baik, Thomas Koberda, Junseok Kim, Richard Wade and Dan Margalit for useful comments.

  1. Communicated by: Adrian Ioana

References

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Received: 2023-12-02
Revised: 2024-08-22
Published Online: 2024-12-03
Published in Print: 2025-05-01

© 2024 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Rigid stabilizers and local prosolubility for boundary-transitive actions on tree
  3. Twisted conjugacy and separability
  4. Mixed identities, hereditarily separated actions and oscillation
  5. Decomposition of Thompson group representations arising from Cuntz algebras
  6. Liftable automorphisms of right-angled Artin groups
  7. Uncountable groups in which permutability is a transitive relation
  8. On semiabelian groups
  9. Critical classes of power graphs and reconstruction of directed power graphs
  10. Weights for 𝜋-partial characters of 𝜋-separable groups
https://doi.org/10.1515/jgth-2023-0277

Articles in the same Issue

  1. Frontmatter
  2. Rigid stabilizers and local prosolubility for boundary-transitive actions on tree
  3. Twisted conjugacy and separability
  4. Mixed identities, hereditarily separated actions and oscillation
  5. Decomposition of Thompson group representations arising from Cuntz algebras
  6. Liftable automorphisms of right-angled Artin groups
  7. Uncountable groups in which permutability is a transitive relation
  8. On semiabelian groups
  9. Critical classes of power graphs and reconstruction of directed power graphs
  10. Weights for 𝜋-partial characters of 𝜋-separable groups
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