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An infinite presentation for the mapping class group of a non-orientable surface with boundary. (English) Zbl 1493.57007

Let \(N_{g,n}\) be a non-orientable surface of genus \(g\geq 1\) with \(n\geq 0\) boundary components. There are various results on finding a (finite or infinite) presentation for the mapping class group \(\mathcal{M}(N_{g,n})\) of a non-orientable surface (as it is in the orientable surface case), but the results cover only the \(n\in \{0,1\}\) cases (for the orientable case the presentations are for \(g\geq 1\) and \(n\geq 0\)).
In the paper under review, firstly an explicit finite presentation for \(\mathcal{M}(N_{g,n})\) for \(g\geq 1\) and \(n\geq 2\) is given. To do this, the authors use the forgetful exact sequence and the group obtained from the previously known presentations such as [M. Stukow, J. Pure Appl. Algebra 218, No. 12, 2226–2239 (2014; Zbl 1301.57015)]. Then, using Gervais’ method applied to the orientable surface case in [S. Gervais, Trans. Am. Math. Soc. 348, No. 8, 3097–3132 (1996; Zbl 0861.57023)], an infinite presentation for \(\mathcal{M}(N_{g,n})\) for \(g\geq 1\) and \(n\geq 2\) is given. In this infinite presentation, the generators consist of all Dehn twists and all crosscap pushing maps (also known as crosscap slide or Y-homeomorphism when one pushes the Möbius band along a two-sided simple closed curve). The infinite presentation given in the paper under review is a generalization of the presentation given by the second author for the cases \(n\in \{0,1\}\) in [G. Omori, Algebr. Geom. Topol. 17, No. 1, 419–437 (2017; Zbl 1357.57005) ].

MSC:

57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57M05 Fundamental group, presentations, free differential calculus
57M07 Topological methods in group theory
57S05 Topological properties of groups of homeomorphisms or diffeomorphisms
20F05 Generators, relations, and presentations of groups
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References:

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