Geometriae Dedicata

, Volume 203, Issue 1, pp 135–154 | Cite as

Structural aspects of twin and pure twin groups

  • Valeriy Bardakov
  • Mahender Singh
  • Andrei VesninEmail author
Original Paper


The twin group \(T_n\) is a Coxeter group generated by \(n-1\) involutions and the pure twin group \(PT_n\) is the kernel of the natural surjection of \(T_n\) onto the symmetric group on n letters. In this paper, we investigate structural aspects of twin and pure twin groups. We prove that the twin group \(T_n\) decomposes into a free product with amalgamation for \(n>4\). It is shown that the pure twin group \(PT_n\) is free for \(n=3,4\), and not free for \(n\ge 6\). We determine a generating set for \(PT_n\), and give an upper bound for its rank. We also construct a natural faithful representation of \(T_4\) into \(\text {Aut}(F_7)\). In the end, we propose virtual and welded analogues of these groups and some directions for future work.


Coxeter group Doodle Eilenberg–Maclane space Free group Hyperbolic plane Pure twin group Twin group 

Mathematics Subject Classification (2010)

57M27 57M25 



After this paper was submitted to the journal and uploaded on the arxiv, we were informed by Harshman and Knapp about their preprint [7]. They refer twin and pure twin groups as triad and pure triad groups, respectively, and explore interesting relations of these groups with three-body hard-core interactions in one dimension. Bardakov and Vesnin are supported by the Russian Science Foundation Grant 16-41-02006. Singh is supported by the DST-RSF Grant INT/RUS/RSF/P-2 and SERB MATRICS Grant MTR/2017/000018.


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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Novosibirsk State University and Agrarian universityNovosibirskRussia
  3. 3.Department of Mathematical SciencesIndian Institute of Science Education and Research (IISER) MohaliS. A. S. NagarIndia
  4. 4.Tomsk State UniversityTomskRussia

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