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### Abstract

Let $\varphi $ be a $\mathbb{Z}/2\mathbb{Z}$-spin structure on a closed oriented surface ${\Sigma}_{g}$ of genus $g\ge 4$. We determine a generating set of the stabilizer of $\varphi $ in the mapping class group of ${\Sigma}_{g}$ consisting of Dehn twists about an explicit collection of $2g+1$ curves on ${\Sigma}_{g}$. If $g=3$ then we determine a generating set of the stabilizer of an odd $\mathbb{Z}/4\mathbb{Z}$-spin structure consisting of Dehn twists about a collection of $6$ curves.

### References

[BCG + 18] Compactification of strata of abelian differentials, Duke Math. J., Volume 167 (2018) no. 12, pp. 2347-2416 | MR | Zbl

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

[Cor89] 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

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

[FM12] A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, 2012 | Zbl

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

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

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

[HJ89] A generalization of winding number functions on surfaces, Proc. Lond. Math. Soc., Volume 58 (1989) no. 2, pp. 366-386 | DOI | MR | Zbl

[KZ03] Connected components of the moduli space of Abelian differentials with prescribed singularities, Invent. Math., Volume 153 (2003) no. 3, pp. 631-678 | DOI | MR | Zbl

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

[LM14] The fine structure of the moduli space of abelian differentials in genus 3, Geom. Dedicata, Volume 169 (2014), pp. 109-128 | DOI | MR | Zbl

[Mat00] 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 | DOI | MR | Zbl

[MS06] 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, Cambridge University Press (2006), pp. 209-218 | Zbl

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

[PV96] Groupe de monodromie géométrique des singularités simples, Math. Ann., Volume 306 (1996) no. 2, pp. 231-245 | DOI | Zbl

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

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

[Wal10] 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 | DOI | MR | Zbl

[Wol10] Families of Riemann surfaces and Weil–Petersson geometry, CBMS Regional Conference Series in Mathematics, 113, American Mathematical Society, 2010 | DOI | Zbl