Connected components of strata of abelian differentials over Teichmüller space. (English) Zbl 1447.57018
The author studies connected components of the strata of holomorphic abelian differentials on marked Riemann surfaces with prescribed degrees of zeros. He proves that there are several such connected components, which are distinguished by roots of the cotangent bundle of the surface. He also characterizes the images of the fundamental groups of strata inside of the mapping class group. The main techniques he uses are those of \(r\)-spin structures (\(r\)-th roots of the tangent bundle), mod \(r\) winding numbers and an analogue of the Euclidean algorithm which holds for mapping class groups and systems of simple closed curves.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 57K20 | 2-dimensional topology (including mapping class groups of surfaces, Teichmüller theory, curve complexes, etc.) |
| 32G15 | Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) |
| 30F30 | Differentials on Riemann surfaces |
| 57R15 | Specialized structures on manifolds (spin manifolds, framed manifolds, etc.) |
References:
| [1] | D. Abramovich and T. Jarvis, Moduli of twisted spin curves,Proc. Am. Math. Soc.,131 (2002), no. 3, 685-699.Zbl 1037.14008 MR 1937405 · Zbl 1037.14008 |
| [2] | C. Arf, Untersuchungen über quadratische Formen in Körpern der Charakteristik 2. I,J. Reine Angew. Math.,183(1941), 148-167.Zbl 0025.01403 MR 8069 · JFM 67.0055.02 |
| [3] | M. F. Atiyah, Riemann surfaces and spin structures,Ann. Sci. École Norm. Sup. (4),4 (1971), 47-62.Zbl 0212.56402 MR 286136 · Zbl 0212.56402 |
| [4] | A. Avila, C. Matheus, and J.-C. Yoccoz, Zorich conjecture for hyperelliptic Rauzy-Veech groups,Math. Ann.,370(2018), no. 1-2, 785-809.Zbl 1381.05088 MR 3747501 · Zbl 1381.05088 |
| [5] | A. Avila and M. Viana, Simplicity of Lyapunov spectra: Proof of the Zorich-Kontsevich conjecture,Acta Math.,198(2007), no. 1, 1-56.Zbl 1143.37001 MR 2316268 · Zbl 1143.37001 |
| [6] | C. Boissy, Connected components of the strata of the moduli space of meromorphic differentials,Comment. Math. Helv.,90(2015), no. 2, 255-286.Zbl 1323.30060 MR 3351745 · Zbl 1323.30060 |
| [7] | T. Brendle and D. Margalit, Point pushing, homology, and the hyperelliptic involution, Michigan Math. J.,62(2013), no. 3, 451-473.Zbl 1279.57013 MR 3102525 · Zbl 1279.57013 |
| [8] | A. Calderon and N. Salter, Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space,preprint, 2019.arXiv:1906.03515 |
| [9] | A. Calderon and N. Salter, Framed mapping class groups and the monodromy of strata of Abelian differentials,preprint, 2020.arXiv:2002.02472 |
| [10] | L. Caporaso, C. Casagrande, and M. Cornalba, Moduli of roots of line bundles on curves, Trans. Amer. Math. Soc.,359(2007), no. 8, 3733-3768.Zbl 1140.14022 MR 2302513 · Zbl 1140.14022 |
| [11] | D. Chen and M. Möller, Quadratic differentials in low genus: exceptional and nonvarying strata,Ann. Sci. Éc. Norm. Supér. (4),47(2014), no. 2, 309-369.Zbl 1395.14008 MR 3215925 · Zbl 1395.14008 |
| [12] | D. R. J. Chillingworth, Winding numbers on surfaces. I,Math. Ann.,196(1972), 218-249. Zbl 0221.57001 MR 300304 · Zbl 0221.57001 |
| [13] | D. R. J. Chillingworth, Winding numbers on surfaces. II,Math. Ann.,199(1972), 131-153. Zbl 0231.57002 MR 321091 · Zbl 0221.57001 |
| [14] | V. Delecroix, E. Goujard, P. Zograf, and A. Zorich, Masur-Veech volumes of principal strata, intersection numbers and hyperbolic geometry. Talk by Elise Goujard at the Fields Institute (October, 2018). Available at:http://www.fields.utoronto.ca/videoarchive//event/2359/2018 · Zbl 1436.32055 |
| [15] | A. Eskin, S. Filip, and A. Wright, The algebraic hull of the Kontsevich-Zorich cocycle, Ann. of Math. (2),188(2018), no. 1, 281-313.Zbl 1398.32015 MR 3815463 · Zbl 1398.32015 |
| [16] | C. Faber, S. Shadrin, and D. Zvonkine, Tautological relations and ther-spin Witten conjecture,Ann. Sci. Éc. Norm. Supér. (4),43(2010), no. 4, 621-658.Zbl 1203.53090 MR 2722511 · Zbl 1203.53090 |
| [17] | B. Farb and D. Margalit,A primer on mapping class groups, Princeton Mathematical Series, 49, Princeton University Press, Princeton, NJ, 2012.Zbl 1245.57002 MR 2850125 · Zbl 1245.57002 |
| [18] | S. Filip, Zero Lyapunov exponents and monodromy of the Kontsevich-Zorich cocycle, Duke Math. J.,166(2017), no. 4, 657-706.Zbl 1370.37066 MR 3619303 · Zbl 1370.37066 |
| [19] | G. Forni, Deviation of ergodic averages for area-preserving flows on surfaces of higher genus,Ann. of Math. (2),155(2002), no. 1, 1-103.Zbl 1034.37003 MR 1888794 · Zbl 1034.37003 |
| [20] | P. Griffiths and J. Harris,Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978.Zbl 0408.14001 MR 507725 · Zbl 0408.14001 |
| [21] | R. Gutiérrez-Romo, Classification of Rauzy-Veech groups: proof of the Zorich conjecture, Invent. Math.,215(2019), no. 3, 741-778.Zbl 1420.37020 MR 3935031 · Zbl 1420.37020 |
| [22] | U. Hamenstädt, Quotients of the orbifold fundamental group of strata of abelian differentials,preprint, 2018. |
| [23] | S. Humphries, Generators for the mapping class group, inTopology of Low-Dimensional Manifolds (Chelwood Gate, 1977), 44-47, Lecture Notes in Math., 722, Springer, Berlin, 1979.Zbl 0732.57004 MR 547453 · Zbl 0732.57004 |
| [24] | S. Humphries and D. Johnson, A generalization of winding number functions on surfaces, Proc. London Math. Soc.(3),58(1989), no. 2, 366-386.Zbl 0681.57003 MR 977482 · Zbl 0681.57003 |
| [25] | T. Jarvis, Geometry of the moduli of higher spin curves,Internat. J. Math.,11(2000), no. 5, 637-663.Zbl 1094.14504 MR 1780734 · Zbl 1094.14504 |
| [26] | T. Jarvis, T. Kimura, and A. Vaintrob, Moduli spaces of higher spin curves and integrable hierarchies,Compositio Math.,126(2001), no. 2, 157-212.Zbl 1015.14028 MR 1827643 · Zbl 1015.14028 |
| [27] | D. Johnson, Spin structures and quadratic forms on surfaces,J. London Math. Soc. (2),22 (1980), no. 2, 365-373.Zbl 0454.57011 MR 588283 · Zbl 0454.57011 |
| [28] | M. Kontsevich and A. Zorich, Lyapunov exponents and Hodge theory.arXiv:hepth/9701164 · Zbl 1305.32007 |
| [29] | M. Kontsevich and A. Zorich, Connected components of the moduli spaces of Abelian differentials with prescribed singularities,Invent. Math.,153(2003), no. 3, 631-678. Zbl 1087.32010 MR 2000471 · Zbl 1087.32010 |
| [30] | E. Lanneau, Hyperelliptic components of the moduli spaces of quadratic differentials with prescribed singularities,Comment. Math. Helv.,79(2004), no. 3, 471-501.Zbl 1054.32007 MR 2081723 · Zbl 1054.32007 |
| [31] | E. Lanneau, Connected components of the strata of the moduli spaces of quadratic differentials,Ann. Sci. Éc. Norm. Supér. (4),41(2008), no. 1, 1-56.Zbl 1161.30033 MR 2423309 · Zbl 1161.30033 |
| [32] | E. Looijenga and G. Mondello, The fine structure of the moduli space of abelian differentials in genus 3,Geom. Dedicata,169(2014), no. 1, 109-128.Zbl 1308.14034 MR 3175239 · Zbl 1308.14034 |
| [33] | D. Margalit and R. Winarski, The Birman-Hilden theory.arXiv:1703.03448 |
| [34] | H. Masur, Interval exchange transformations and measured foliations,Ann. of Math. (2), 115(1982), no. 1, 169-200.Zbl 0497.28012 MR 644018 · Zbl 0497.28012 |
| [35] | C. Matheus, What is. . . the Kontsevich-Zorich cocycle? Available at:https: //matheuscmss.wordpress.com/2014/05/23/what-is-the-kontsevichzorich-cocycle/ |
| [36] | M. Mirzakhani, Weil-Petersson volumes and intersection theory on the moduli space of curves,J. Amer. Math. Soc.,20(2007), no. 1, 1-23.Zbl 1120.32008 MR 2257394 · Zbl 1120.32008 |
| [37] | A. Polischuk, Moduli spaces of curves with effectiver-spin structures, inGromov-Witten theory of spin curves and orbifolds, 1-20, Contemp. Math., 403, Amer. Math. Soc., Providence, RI, 2006.Zbl 1112.14028 MR 2234882 |
| [38] | N. Salter, On the monodromy group of the family of smooth plane curves, 2016. arXiv:1610.04920 |
| [39] | N. Salter, Monodromy and vanishing cycles in toric surfaces,Invent. Math.,216(2019), no. 1, 153-213.Zbl 07051046 MR 3935040 · Zbl 1440.57028 |
| [40] | P. L. Sipe, Roots of the canonical bundle of the universal Teichmüller curve and certain subgroups of the mapping class group,Math. Ann.,260(1982), no. 1, 67-92. Zbl 0502.32017 MR 664367 · Zbl 0502.32017 |
| [41] | P. L. Sipe, Some finite quotients of the mapping class group of a surface,Proc. Amer. Math. Soc.,97(1986), no. 3, 515-524.Zbl 0596.57007 MR 840639 · Zbl 0596.57007 |
| [42] | R. Trapp, A linear representation of the mapping class groupMand the theory of winding numbers,Topology Appl.,43(1992), no. 1, 47-64.Zbl 0748.57004 MR 1141372 · Zbl 0748.57004 |
| [43] | W. Veech, Gauss measures for transformations on the space of interval exchange maps, Ann. of Math. (2),115(1982), no. 1, 201-242.Zbl 0486.28014 MR 644019 · Zbl 0486.28014 |
| [44] | K. C. Walker, Fundamental groups of moduli spaces of quadratic differentials, Ph.D. thesis, The University of Chicago, 2007.MR 2710204 |
| [45] | K. C. Walker, Connected components of the strata of quadratic differentials over the Teichmüller space,Geom. Dedicata,142(2009), no. 1, 47-60.Zbl 1182.30070 MR 2545455 · Zbl 1182.30070 |
| [46] | K. C. Walker, Quotient groups of the fundamental groups of certain strata of the moduli space of quadratic differentials,Geom. Topol.,14(2010), no. 2, 1129-1164. Zbl 1213.30076 MR 2651550 · Zbl 1213.30076 |
| [47] | E. Witten, Algebraic geometry associated with matrix models of two-dimensional gravity, inTopological methods in modern mathematics (Stony Brook, NY, 1991), 235-269, Publish or Perish, Houston, TX, 1993.Zbl 0812.14017 MR 1215968 · Zbl 0812.14017 |
| [48] | A. Wright, Cylinder deformations in orbit closures of translation surfaces,Geom. Topol., 19(2015), no. 1, 413-438.Zbl 1318.32021 MR 3318755 · Zbl 1318.32021 |
| [49] | A. Wright, From rational billiards to dynamics on moduli space,Bull. Amer. Math. Soc. (N.S.),53(2016), no. 1, 41-56.Zbl 1353.37076 MR 3403080 · Zbl 1353.37076 |
| [50] | A. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
