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Vertex-transitive graphs with local action the symmetric group on ordered pairs

  • Luke Morgan EMAIL logo
Published/Copyright: October 25, 2022

Abstract

We consider a finite, connected and simple graph Γ that admits a vertex-transitive group of automorphisms 𝐺. Under the assumption that, for all x V ( Γ ) , the local action G x Γ ( x ) is the action of Sym ( n ) on ordered pairs, we show that the group G x [ 3 ] , the point-wise stabiliser of a ball of radius three around 𝑥, is trivial.

Award Identifier / Grant number: EP/R014604/1

Award Identifier / Grant number: P1-0285

Award Identifier / Grant number: J1-1691

Award Identifier / Grant number: N1-0160

Award Identifier / Grant number: J1-2451

Funding statement: This work was supported by EPSRC grant no EP/R014604/1. This work is supported in part by the Slovenian Research Agency (research program P1-0285 and research projects J1-1691, N1-0160, J1-2451).

Acknowledgements

The author would like to thank the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme “Groups, representations and applications: New perspectives” where work on this paper was undertaken. The author also thanks the referee for helpful comments which have improved the manuscript.

  1. Communicated by: Timothy C. Burness

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Received: 2022-03-03
Revised: 2022-09-05
Published Online: 2022-10-25
Published in Print: 2023-05-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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