Geodesic
Simplicial structures and normal forms for mapping class groups and braid groups
Berrick, A J1 ; Hanbury, E2
1Yale-NUS College, 6 College Avenue East #B1-01, Singapore 138614, Singapore
2Department of Mathematics, University of Durham, Science Laboratories, Durham, DH1 3LE, UK
Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3259-3280
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In this paper we show that the mapping class groups of any surface with nonempty boundary form a simplicial group as the number of marked points varies. This extends the simplicial structure on braid groups of surfaces found by Berrick, Cohen, Wong and Wu. We use the simplicial maps to construct compatible normal forms for elements of the braid groups and mapping class groups of surfaces with boundary.

DOI : 10.2140/agt.2014.14.3259
Classification : 20F38, 20E22, 55U10
Keywords: braid group, mapping class group, configuration space, crossed simplicial group, combing, normal form

Berrick, A J  1   ; Hanbury, E  2

1 Yale-NUS College, 6 College Avenue East #B1-01, Singapore 138614, Singapore
2 Department of Mathematics, University of Durham, Science Laboratories, Durham, DH1 3LE, UK 
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Berrick, A J; Hanbury, E. Simplicial structures and normal forms for mapping class groups and braid groups. Algebraic and Geometric Topology, Tome 14 (2014) no. 6, pp. 3259-3280. doi: 10.2140/agt.2014.14.3259

[1] V G Bardakov, R Mikhaillov, V V Vershinin, J Wu, On Cohen braids, preprint

[2] A J Berrick, F R Cohen, Y L Wong, J Wu, Configurations, braids, and homotopy groups, J. Amer. Math. Soc. 19 (2006) 265

[3] A J Berrick, E Hanbury, J Wu, Brunnian subgroups of mapping class groups and braid groups, Proc. Lond. Math. Soc. 107 (2013) 875

[4] A J Berrick, E Hanbury, J Wu, Delta-structures on mapping class groups and braid groups, Trans. Amer. Math. Soc. 366 (2014) 1879

[5] F R Cohen, On combinatorial group theory in homotopy, from: "Homotopy theory and its applications" (editors A Adem, R J Milgram, D C Ravenel), Contemp. Math. 188, Amer. Math. Soc. (1995) 57

[6] C J Earle, A Schatz, Teichmüller theory for surfaces with boundary, J. Differential Geometry 4 (1970) 169

[7] B Farb, D Margalit, A primer on mapping class groups, Princeton Math. Series 49, Princeton Univ. Press (2012)

[8] Z Fiedorowicz, J L Loday, Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991) 57

[9] D Gonçalves, J Guaschi, Braid groups of nonorientable surfaces and the Fadell–Neuwirth short exact sequence, Journal Pure Applied Algebra 214 (2010) 667

[10] J Li, J Wu, Artin braid groups and homotopy groups, Proc. Lond. Math. Soc. 99 (2009) 521

[11] J Wu, A braided simplicial group, Proc. London Math. Soc. 84 (2002) 645

[12] J Wu, On maps from loop suspensions to loop spaces and the shuffle relations on the Cohen groups, Mem. Amer. Math. Soc. 851, Amer. Math. Soc. (2006)

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