An infinite presentation for the twist subgroup of the mapping class group of a compact non-orientable surface (Q6589769)

For \(g \geq 1\) and \(n \geq 0\), let \(N_{g,n}\) denote the compact non-orientable surface of genus \(g\) with \(n\) boundary components (with the understanding that \(N_{g,0} := N_g\)). Let \(\mathcal{M}(N_{g,n})\) be the mapping class group of \(N_{g,n}\) and let \(\mathcal{T}(N_{g,n})\) be the twist subgroup of \(\mathcal{M}(N_{g,n})\) generated by the Dehn twists in \(\mathcal{M}(N_{g,n})\). \textit{W. B. R. Lickorish} [Proc. Camb. Philos. Soc. 59, 307--317 (1963; Zbl 0115.40801)] showed that for \(g \geq 2\), \(\mathcal{M}(N_g)\) is generated by Dehn twists and crosscap slides (also known as \(Y\)-homeomorphisms). Subsequently, a finite generating set for \(\mathcal{M}(N_g)\) was derived by \textit{D. R. J. Chillingworth} [Proc. Camb. Philos. Soc. 65, 409--430 (1969; Zbl 0172.48801)], which was later extended to the case of \(\mathcal{M}(S_{g,n})\) by Stukow [6].\N\NFor \(g+n >3\) and \(n \leq 1\), \textit{L. Paris} and \textit{B. Szepietowski} [Bull. Soc. Math. Fr. 143, No. 3, 503--566 (2015; Zbl 1419.57005)] gave a finite presentation for \(\mathcal{T}(N_{g,n})\) using Dehn twists and \(g-1\) crosscap transpositions, which was later simplified by \textit{M. Stukow} [J. Pure Appl. Algebra 218, No. 12, 2226--2239 (2014; Zbl 1301.57015)] to a presentation involving Dehn twists and a single crosscap slide. Using this generating set and the fact [\textit{W. B. R. Lickorish}, Proc. Camb. Philos. Soc. 61, 61--64 (1965; Zbl 0131.20802)] that \([\mathcal{M}(N_{g,n}): \mathcal{T}(N_{g,n})] = 2\), \textit{M. Stukow} [Adv. Geom. 10, No. 2, 249--273 (2010; Zbl 1207.57030)] derived a generating set for \(\mathcal{T}(N_{g,n})\) when \(g+n >3\) and \(n \leq 1\). In this paper, using Stukow's generating set and the Birman exact sequence [\textit{J. S. Birman}, Commun. Pure Appl. Math. 22, 213--238 (1969; Zbl 0167.21503)] in the non-orientable setting, an infinite presentation of \(\mathcal{T}(N_{g,n})\) is obtained for \(g \geq 1\) and \(n \geq 0\).