## An extension of the classification of high rank regular polytopes

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- by Maria Elisa Fernandes, Dimitri Leemans and Mark Mixer PDF
- Trans. Amer. Math. Soc.
**370**(2018), 8833-8857 Request permission

## Abstract:

Up to isomorphism and duality, there are exactly two nondegenerate abstract regular polytopes of rank greater than $n-3$ (one of rank $n-1$ and one of rank $n-2$) with automorphism groups that are transitive permutation groups of degree $n\geq 7$. In this paper we extend this classification of high rank regular polytopes to include the ranks $n-3$ and $n-4$. The result is, up to isomorphism and duality, there are exactly seven abstract regular polytopes of rank $n-3$ for each $n\geq 9$, and there are nine abstract regular polytopes of rank $n-4$ for each $n \geq 11$. Moreover, we show that if a transitive permutation group $\Gamma$ of degree $n \geq 11$ is the automorphism group of an abstract regular polytope of rank at least $n-4$, then $\Gamma \cong S_n$.## References

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## Additional Information

**Maria Elisa Fernandes**- Affiliation: Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
- MR Author ID: 916550
- ORCID: 0000-0001-7386-4254
- Email: maria.elisa@ua.pt
**Dimitri Leemans**- Affiliation: Université Libre de Bruxelles, Département de Mathématique, C.P.216 - Algèbre et Combinatoire, Boulevard du Triomphe, 1050 Brussels, Belgium
- MR Author ID: 613090
- ORCID: 0000-0002-4439-502X
- Email: dleemans@ulb.ac.be
**Mark Mixer**- Affiliation: Department of Applied Mathematics, Wentworth Institute of Technology, Boston, Massachusetts 02115
- MR Author ID: 954212
- Email: mixerm@wit.edu
- Received by editor(s): September 5, 2017
- Received by editor(s) in revised form: October 4, 2017
- Published electronically: September 13, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 8833-8857 - MSC (2010): Primary 52B11, 20D06
- DOI: https://doi.org/10.1090/tran/7425
- MathSciNet review: 3864397