THE TWIST SUBGROUP IS GENERATED BY TWO ELEMENTS

Altunöz T., Pamuk M., Yildiz O.

Tohoku Mathematical Journal, vol.76, no.2, pp.175-197, 2024 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 76 Issue: 2
  • Publication Date: 2024
  • Doi Number: 10.2748/tmj.20221017
  • Journal Name: Tohoku Mathematical Journal
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH
  • Page Numbers: pp.175-197
  • Keywords: commutators, generating sets, Mapping class groups, nonorientable surfaces, torsion, twist subgroup
  • Middle East Technical University Affiliated: Yes

Abstract

We show that the twist subgroup Tg of a nonorientable surface of genus g can be generated by two elements for every odd g ≥ 21 and even g ≥ 50. Using these generators, we can also show that Tg can be generated by two or three commutators depending on g modulo 4. Moreover, we show that Tg can be generated by three elements if g ≥ 8. For this general case, the number of commutator generators is either three or four depending on g modulo 4 again.