Tohoku Mathematical Journal
2024
June
SECOND SERIES VOL. 76, NO. 2

Tohoku Math. J.
76 (2024), 175-197

Title THE TWIST SUBGROUP IS GENERATED BY TWO ELEMENTS

Author Tülin Altunöz, Mehmetcik Pamuk and Oğuz Yildiz

(Received January 28, 2022, revised October 12, 2022)
Abstract. We show that the twist subgroup $\mathcal{T}_g$ of a nonorientable surface of genus $g$ can be generated by two elements for every odd $g \geq 21$ and even $g \geq 50$. Using these generators, we can also show that $\mathcal{T}_g$ can be generated by two or three commutators depending on $g$ modulo 4. Moreover, we show that $\mathcal{T}_g$ can be generated by three elements if $g \geq 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo 4 again.

Mathematics Subject Classification. Primary 57M07; Secondary 20F38, 20F05.
Key words and phrases. Mapping class groups, nonorientable surfaces, twist subgroup, torsion, generating sets, commutators.

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