Structural aspects of twin and pure twin groups

Valeriy Bardakov, Mahender Singh, Andrei Vesnin

Research output: Contribution to journalArticlepeer-review

7 Citations (Scopus)


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≥ 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 Aut (F 7 ). In the end, we propose virtual and welded analogues of these groups and some directions for future work.

Original languageEnglish
Pages (from-to)135-154
Number of pages20
JournalGeometriae Dedicata
Issue number1
Publication statusPublished - 1 Dec 2019


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


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