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Conjugacy classes and automorphisms of twin groups

  • Tushar Kanta Naik , Neha Nanda und Mahender Singh EMAIL logo
Veröffentlicht/Copyright: 16. Juli 2020

Abstract

The twin group Tn is a right-angled Coxeter group generated by n-1 involutions, and the pure twin group PTn is the kernel of the natural surjection from Tn onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conjugacy classes of involutions in Tn, which, quite interestingly, is related to the well-known Fibonacci sequence. We also derive a recursive formula for the number of z-classes of involutions in Tn. We give a new proof of the structure of Aut(Tn) for n3, and show that Tn is isomorphic to a subgroup of Aut(PTn) for n4. Finally, we construct a representation of Tn to Aut(Fn) for n2.


Communicated by Manfred Droste


Funding statement: Mahender Singh is supported by the Swarna Jayanti Fellowship grants DST/SJF/MSA-02/2018-19 and SB/SJF/2019-20/04.

Acknowledgements

The authors are grateful to Valeriy Bardakov for his interest in this work and for his many useful comments. Tushar Kanta Naik and Neha Nanda thank IISER Mohali for the Post Doctoral and the PhD Research Fellowships, respectively.

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Received: 2019-11-19
Revised: 2020-03-24
Published Online: 2020-07-16
Published in Print: 2020-09-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Heruntergeladen am 29.9.2025 von https://www.degruyterbrill.com/document/doi/10.1515/forum-2019-0321/html
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