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From braid groups to mapping class groups

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Abstract

In this paper, we classify homomorphisms from the braid group on n strands to the mapping class group of a genus g surface. In particular, we show that when \(g<n-2\), all representations are either cyclic or standard. Our result is sharp in the sense that when \(g=n-2\), a generalization of the hyperelliptic representation appears, which is not cyclic or standard. This gives a classification of surface bundles over the configuration space of the complex plane. As a corollary, we partially recover the result of Aramayona–Souto (Geom Topol 16(4):2285–2341, 2012), which classifies homomorphisms between mapping class groups, with a slight improvement.

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Acknowledgements

We acknowledge the financial support of the National Science Foundation via Grant Nos. DMS-2005409 and the Caltech SURF program through Prof. Vladimir Markovic’s McArthur chair account and the Mark Reinecke SURF Fellowship. This paper started as a joint paper of Lei Chen with Kevin Kordek and Dan Margalit, and transformed into a SURF program project at Caltech. The authors would like to thank Kevin and Dan for all the helpful conversations regarding this paper. The authors would also like to thank the anonymous referee for giving very useful advice to the current paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland College Park, 4176 Campus Dr, College Park, MD, 20742, USA

    Lei Chen

  2. Department of Mathematics, University of Texas Austin, 2515 Speedway, Austin, TX, 78712, USA

    Aru Mukherjea

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  1. Lei Chen
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  2. Aru Mukherjea
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Correspondence to Aru Mukherjea.

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Chen, L., Mukherjea, A. From braid groups to mapping class groups. Math. Z. 303, 27 (2023). https://doi.org/10.1007/s00209-022-03167-5

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