Generating the spin mapping class group by Dehn twists
Annales Henri Lebesgue, Volume 4 (2021), pp. 1619-1658.

Metadata

KeywordsSpin mapping class group, Dehn twists, curve systems, group generators

Abstract

Let ϕ be a /2-spin structure on a closed oriented surface Σ g of genus g4. We determine a generating set of the stabilizer of ϕ in the mapping class group of Σ g consisting of Dehn twists about an explicit collection of 2g+1 curves on Σ g . If g=3 then we determine a generating set of the stabilizer of an odd /4-spin structure consisting of Dehn twists about a collection of 6 curves.


References

[BCG + 18] Bainbridge, Matt; Chen, Dawei; Gendron, Quentin; Grushevsky, Samuel; Möller, Martin Compactification of strata of abelian differentials, Duke Math. J., Volume 167 (2018) no. 12, pp. 2347-2416 | MR 3848392 | Zbl 1403.14058

[Cal20] Calderon, Aaron Connected components of strata of abelian differentials over Teichmüller space, Comment. Math. Helv., Volume 95 (2020) no. 2, pp. 361-420 | Article | MR 4115287 | Zbl 1447.57018

[Cor89] Cornalba, Maurizio Moduli of curves and theta characteristics, Proceedings of the first college on Riemann surfaces held in Trieste, Italy, November 9-December 18, 1987, World Scientific, 1989, pp. 560-589 | Zbl 0800.14011

[CS21] Calderon, Aaron; Salter, Nick Higher spin mapping class groups and strata of abelian differentials over Teichmüller space (2021) (https://arxiv.org/abs/1906.03515)

[FM12] Farb, Benson; Margalit, Dan A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012 | Zbl 1245.57002

[Hai95] Hain, Richard M. Torelli groups and geometry of the moduli space of curves, Current topics in complex algebraic geometry (Clemens, Herbert, ed.) (Mathematical Sciences Research Institute Publications), Volume 28, Cambridge University Press, 1995, pp. 97-143 | MR 1397061 | Zbl 0868.14006

[Hir02] Hirose, Susumu On diffeomorphisms over surfaces trivially embedded in the 4-sphere, Algebr. Geom. Topol., Volume 2 (2002), pp. 791-824 | Article | MR 1928177 | Zbl 1022.57016

[Hir05] Hirose, Susumu Surfaces in the complex projective plane and their mapping class groups, Algebr. Geom. Topol., Volume 5 (2005), pp. 577-613 | Article | MR 2153115 | Zbl 1092.57018

[HJ89] Humphries, Stephen P.; Johnson, Dennis A generalization of winding number functions on surfaces, Proc. Lond. Math. Soc., Volume 58 (1989) no. 2, pp. 366-386 | Article | MR 977482 | Zbl 0681.57003

[KZ03] Kontsevich, Maxim; Zorich, Anton Connected components of the moduli space of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | Article | MR 2000471 | Zbl 1087.32010

[Lei04] Leininger, Christopher J. On groups generated by two positive multi-twists: Teichmüller curves and Lehmer’s number, Geom. Topol., Volume 8 (2004), pp. 1301-1359 | Article | MR 2119298 | Zbl 1088.57002

[LM14] Looijenga, Eduard; Mondello, Gabriele The fine structure of the moduli space of abelian differentials in genus 3, Geom. Dedicata, Volume 169 (2014), pp. 109-128 | Article | MR 3175239 | Zbl 1308.14034

[Mat00] Matsumoto, Makoto A presentation of mapping class groups in terms of Artin groups and geometric monodromy of singularities, Math. Ann., Volume 316 (2000) no. 3, pp. 401-418 | Article | MR 1752777 | Zbl 0968.57014

[MS06] Masur, Howard; Schleimer, Saul The pants complex has only one end. Proceedings of the programme “Spaces of Kleinian groups and hyperbolic 3-manifolds”, Cambridge, UK, July 21–August 15, 2003, Spaces of Kleinian groups (London Mathematical Society Lecture Note Series), Volume 329 (2006), pp. 209-218 | Zbl 1099.30037

[Put08] Putman, Andrew A note on connectivity of certain complexes associated to surfaces, Enseign. Math., Volume 54 (2008) no. 3-4, pp. 287-301 | MR 2478089 | Zbl 1182.57004

[PV96] Perron, Bernard; Vannier, Jean-Pierre Groupe de monodromie géométrique des singularités simples, Math. Ann., Volume 306 (1996) no. 2, pp. 231-245 | Article | Zbl 0863.32013

[Sal19] Salter, Nick Monodromy and vanishing cycles in toric surfaces, Invent. Math., Volume 216 (2019) no. 1, pp. 153-213 | Article | MR 3935040 | Zbl 1440.57028

[Wal09] Walker, Katharine C. Connected components of strata of quadratic differentials over Teichmüller space, Geom. Dedicata, Volume 142 (2009), pp. 47-60 | Article | MR 2545455 | Zbl 1182.30070

[Wal10] Walker, Katharine C. Quotient groups of fundamental groups of certain strata of the moduli space of quadratic differentials, Geom. Topol., Volume 14 (2010) no. 2, pp. 1129-1164 | Article | MR 2651550 | Zbl 1213.30076

[Wol10] Wolpert, Scott A. Families of Riemann surfaces and Weil–Petersson geometry, CBMS Regional Conference Series in Mathematics, 113, American Mathematical Society, 2010 | Article | Zbl 1198.30049