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Conjugacy classes and automorphisms of twin groups

Conjugacy classes and automorphisms of twin groups AbstractThe twin group Tn{T_{n}}is a right-angled Coxeter group generated by n-1{n-1}involutions, and the pure twin group PTn{\mathrm{PT}_{n}}is the kernel of the natural surjection from Tn{T_{n}}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{T_{n}}, 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{T_{n}}.We give a new proof of the structure of Aut⁡(Tn){\operatorname{Aut}(T_{n})}for n≥3{n\geq 3}, and show that Tn{T_{n}}is isomorphic to a subgroup of Aut⁡(PTn){\operatorname{Aut}(\mathrm{PT}_{n})}for n≥4{n\geq 4}.Finally, we construct a representation of Tn{T_{n}}to Aut⁡(Fn){\operatorname{Aut}(F_{n})}for n≥2{n\geq 2}. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Conjugacy classes and automorphisms of twin groups

Forum Mathematicum , Volume 32 (5): 14 – Sep 1, 2020

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References (19)

Publisher
de Gruyter
Copyright
© 2020 Walter de Gruyter GmbH, Berlin/Boston
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/forum-2019-0321
Publisher site
See Article on Publisher Site

Abstract

AbstractThe twin group Tn{T_{n}}is a right-angled Coxeter group generated by n-1{n-1}involutions, and the pure twin group PTn{\mathrm{PT}_{n}}is the kernel of the natural surjection from Tn{T_{n}}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{T_{n}}, 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{T_{n}}.We give a new proof of the structure of Aut⁡(Tn){\operatorname{Aut}(T_{n})}for n≥3{n\geq 3}, and show that Tn{T_{n}}is isomorphic to a subgroup of Aut⁡(PTn){\operatorname{Aut}(\mathrm{PT}_{n})}for n≥4{n\geq 4}.Finally, we construct a representation of Tn{T_{n}}to Aut⁡(Fn){\operatorname{Aut}(F_{n})}for n≥2{n\geq 2}.

Journal

Forum Mathematicumde Gruyter

Published: Sep 1, 2020

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