Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space
For g≥5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2], we find that the...
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| Published in | Advances in mathematics (New York. 1965) Vol. 389; p. 107926 |
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| Format | Journal Article |
| Language | English |
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08.10.2021
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| Abstract | For g≥5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2], we find that the non-hyperelliptic components are classified by an invariant known as an r–spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all r–spin stabilizer subgroups of the mapping class group, completing a project begun by the second author in [18]. Some corollaries in flat geometry and toric geometry are obtained from these results. |
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| AbstractList | For g≥5, we give a complete classification of the connected components of strata of abelian differentials over Teichmüller space, establishing an analogue of Kontsevich and Zorich's classification of their components over moduli space. Building on work of the first author [2], we find that the non-hyperelliptic components are classified by an invariant known as an r–spin structure. This is accomplished by computing a certain monodromy group valued in the mapping class group. To do this, we determine explicit finite generating sets for all r–spin stabilizer subgroups of the mapping class group, completing a project begun by the second author in [18]. Some corollaries in flat geometry and toric geometry are obtained from these results. |
| ArticleNumber | 107926 |
| Author | Salter, Nick Calderon, Aaron |
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| Cites_doi | 10.1016/S1874-575X(02)80015-7 10.1016/0040-9383(85)90049-7 10.1090/bull/1513 10.1515/crll.1941.183.148 10.1007/BF01363897 10.1007/BF01428050 10.4171/CMH/491 10.1007/s00222-003-0303-x 10.1007/BF01475756 10.1090/S0002-9939-1986-0840639-5 10.1112/plms/s3-58.2.366 10.1112/jtopol/jtt029 10.1007/s11856-015-1248-7 10.1007/s00222-018-0845-6 10.1007/s10711-013-9845-2 10.1112/jlms/s2-22.2.365 |
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| Keywords | Translation surfaces Higher spin structures Mapping class groups Abelian differentials Strata Monodromy |
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| References | Chillingworth (br0040) 1972; 196 Calderon (br0020) 2020; 95 Kawazumi (br0110) 2017 Sipe (br0200) 1986; 97 Randal-Williams (br0170) 2013; 7 Arf (br0010) 1941; 183 Kontsevich, Zorich (br0120) 2003; 153 Johnson (br0100) 1985; 24 Johnson (br0090) 1980; 22 Johnson (br0080) 1980; 249 Sipe (br0190) 1982; 260 Looijenga, Mondello (br0140) 2014; 169 Masur, Tabachnikov (br0150) 2002; vol. 1A Calderon, Salter (br0030) 2020 Gutiérrez-Romo (br0060) 2018 Lindsey (br0130) 2015; 210 Humphries, Johnson (br0070) 1989; 58 Wright (br0210) 2016; 53 D. Margalit, R. Winarski, The Birman–Hilden theory, preprint, pp. 1–20. Salter (br0180) 2019; 216 Zorich (br0220) 2006 Farb, Margalit (br0050) 2011 Calderon (10.1016/j.aim.2021.107926_br0020) 2020; 95 Lindsey (10.1016/j.aim.2021.107926_br0130) 2015; 210 Randal-Williams (10.1016/j.aim.2021.107926_br0170) 2013; 7 Chillingworth (10.1016/j.aim.2021.107926_br0040) 1972; 196 Salter (10.1016/j.aim.2021.107926_br0180) 2019; 216 Masur (10.1016/j.aim.2021.107926_br0150) 2002; vol. 1A Looijenga (10.1016/j.aim.2021.107926_br0140) 2014; 169 Humphries (10.1016/j.aim.2021.107926_br0070) 1989; 58 Zorich (10.1016/j.aim.2021.107926_br0220) 2006 Sipe (10.1016/j.aim.2021.107926_br0200) 1986; 97 10.1016/j.aim.2021.107926_br0160 Johnson (10.1016/j.aim.2021.107926_br0100) 1985; 24 Sipe (10.1016/j.aim.2021.107926_br0190) 1982; 260 Johnson (10.1016/j.aim.2021.107926_br0090) 1980; 22 Kawazumi (10.1016/j.aim.2021.107926_br0110) Johnson (10.1016/j.aim.2021.107926_br0080) 1980; 249 Arf (10.1016/j.aim.2021.107926_br0010) 1941; 183 Calderon (10.1016/j.aim.2021.107926_br0030) Gutiérrez-Romo (10.1016/j.aim.2021.107926_br0060) 2018 Wright (10.1016/j.aim.2021.107926_br0210) 2016; 53 Farb (10.1016/j.aim.2021.107926_br0050) 2011 Kontsevich (10.1016/j.aim.2021.107926_br0120) 2003; 153 |
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| SubjectTerms | Abelian differentials Higher spin structures Mapping class groups Monodromy Strata Translation surfaces |
| Title | Higher spin mapping class groups and strata of Abelian differentials over Teichmüller space |
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