A finite presentation for the twist subgroup of the mapping class group of a nonorientable surface
Let \(N\) be a compact surface with boundary and let \(\mathrm{Diff}(N)\) be the group of all diffeomorphisms of \(N\) which are the identity on each boundary component of \(N.\) The mapping class group \(\mathcal{M}(N)\) of \(N\) is the quotient of \(\mathrm{Diff}(N)\) by the subgroup consisting of maps isotopic to the identity, where the isotopies are assumed to be the identity on each boundary component. If \(N\) is orientable, a well known theorem discovered by Max Dehn affirms that \(\mathcal{M}(N)\) is generated by Dehn twists. In the nonorientable case, Lickorish first observed that Dehn twists do not generate \(\mathcal{M}(N)\) but a subgroup \(\mathcal{T}(N)\) which is of index 2 in \(\mathcal{M}(N).\) In this paper compact nonorienable surfaces \(N_{g,s}\) of genus \(g\) with \(s\) boundary components are considered. \textit{L. Paris} and \textit{B. Szepietowski} [Bull. Soc. Math. Fr. 143, No. 3, 503--566 (2015; Zbl 1419.57005)] obtained an explicit presentation for the mapping class group \(\mathcal{M}(N_{g,s})\) of \(N_{g,s},\) where \(s\in\{0,1\}\) and \(g+s>3.\) Based on this work the author obtains a finite presentation for the subgroup \(\mathcal{T}(N_{g,s})\) of \(\mathcal{M} (N_{g,s}).\)
scientific article; zbMATH DE number 6594591
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