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From braid groups to mapping class groups. (English) Zbl 1527.20054

F. Castel [Geometric representations of the braid groups. Paris: Société Mathématique de France (SMF) (2016; Zbl 1441.20001)] gave a complete classification of homomorphisms from the Artin braid group \(B_n\) to the pure mapping class group PMod\((S_{g,p})\) of the surface \(S_{g,p}\) of genus \(g\) with \(p\) punctures, for \(n \ge 6\) and \(g \le \frac{n}{2}\).
Using totally symmetric sets and totally symmetric multicurves as the main technical tools, the authors of this paper give a new proof of the result by Castel [loc. cit.] and also extend the classification to the case of \(g < n-2\) but in a reduced range of \(n \ge 23\). The main theorem in this paper states the following:
(Main theorem) For \(k \ge 13\), \(n=2k\) or \(n=2k+1\) and \(g \le 2k-3\), any homomorphism \(\rho : B_n \to \mathrm{PMod}(S_{g,p})\) is either trivial, standard or negative-standard homomorphism, up to transvection.
As a corollary to the above classification theorem, the authors partially recover a result by J. Aramayona and J. Souto [Geom. Topol. 16, No. 4, 2285–2341 (2012; Zbl 1262.57003)] as follows:
(Corollary) For \(g \ge 23\) and \(h < 2g\), any homomorphism \(\rho : \mathrm{PMod}(S_{g,p}) \to \mathrm{PMod}(S_{h,q})\) is either trivial or induced by an embedding.

MSC:

20F36 Braid groups; Artin groups
20F65 Geometric group theory
57K20 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.)
57M07 Topological methods in group theory
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References:

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