On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus

Bartolini, Gabriel; Izquierdo, Milagros

doi:10.1090/S0002-9939-2011-10881-5

On the connectedness of the branch locus of the moduli space of Riemann surfaces of low genus
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by Gabriel Bartolini and Milagros Izquierdo PDF

Proc. Amer. Math. Soc. 140 (2012), 35-45 Request permission

Abstract:

Let $g$ be an integer $\ge 3$ and let $\mathcal {B}_g = \{X\in \mathcal {M}_g | Aut(X)\neq 1_d \}$, where $\mathcal {M}_g$ denotes the moduli space of compact Riemann surfaces of genus $g$. Using uniformization of Riemann surfaces by Fuchsian groups and the equisymmetric stratification of the branch locus of the moduli space, we prove that the subloci corresponding to Riemann surfaces with automorphism groups isomorphic to cyclic groups of order 2 and 3 belong to the same connected component. We also prove the connectedness of $\mathcal {B}_g$ for $g=5,6,7$ and $8$ with the exception of the isolated points given by Kulkarni.

References

Gabriel Bartolini, Antonio F. Costa, Milagros Izquierdo, and Ana M. Porto, On the connectedness of the branch locus of the moduli space of Riemann surfaces, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM 104 (2010), no. 1, 81–86 (English, with English and Spanish summaries). MR 2666443, DOI 10.5052/RACSAM.2010.08
S. Allen Broughton, The equisymmetric stratification of the moduli space and the Krull dimension of mapping class groups, Topology Appl. 37 (1990), no. 2, 101–113. MR 1080344, DOI 10.1016/0166-8641(90)90055-7
S. Allen Broughton, Classifying finite group actions on surfaces of low genus, J. Pure Appl. Algebra 69 (1991), no. 3, 233–270. MR 1090743, DOI 10.1016/0022-4049(91)90021-S
S. Allen Broughton and Aaron Wootton, Finite abelian subgroups of the mapping class group, Algebr. Geom. Topol. 7 (2007), 1651–1697. MR 2366175, DOI 10.2140/agt.2007.7.1651
E. Bujalance, F. J. Cirre, J. M. Gamboa, and G. Gromadzki, On symmetries of compact Riemann surfaces with cyclic groups of automorphisms, J. Algebra 301 (2006), no. 1, 82–95. MR 2230321, DOI 10.1016/j.jalgebra.2006.03.019
E. Bujalance, F. J. Cirre, M. D. E. Conder, On full automorphism groups of Riemann surfaces, preprint.
José A. Bujalance, Antonio F. Costa, and Ana M. Porto, On the connectedness of the locus of real elliptic-hyperelliptic Riemann surfaces, Internat. J. Math. 20 (2009), no. 8, 1069–1080. MR 2554734, DOI 10.1142/S0129167X09005650
José A. Bujalance, Antonio F. Costa, and Ana M. Porto, On the topological types of symmetries of elliptic-hyperelliptic Riemann surfaces, Israel J. Math. 140 (2004), 145–155. MR 2054842, DOI 10.1007/BF02786630
José A. Bujalance, Antonio F. Costa, and Ana M. Porto, Topological types of symmetries of elliptic-hyperelliptic Riemann surfaces and an application to moduli spaces, RACSAM. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. 97 (2003), no. 1, 69–72 (English, with English and Spanish summaries). MR 2036744
A. Costa, M. Izquierdo, On the connectedness of the branch locus of the moduli space of Riemann surfaces of genus 4, Glasgow Math. J. 52 (2010), 401-408.
Antonio F. Costa and Milagros Izquierdo, On the connectedness of the locus of real Riemann surfaces, Ann. Acad. Sci. Fenn. Math. 27 (2002), no. 2, 341–356. MR 1922193
A. F. Costa and M. Izquierdo, On the locus of real algebraic curves, Atti Sem. Mat. Fis. Univ. Modena 49 (2001), no. suppl., 91–107. Dedicated to the memory of Professor M. Pezzana (Italian). MR 1881092
Antonio F. Costa and Milagros Izquierdo, Maximal order of automorphisms of trigonal Riemann surfaces, J. Algebra 323 (2010), no. 1, 27–31. MR 2564826, DOI 10.1016/j.jalgebra.2009.09.041
A. Costa, M. Izquierdo, On the existence of components of dimension one in the branch loci of moduli spaces of Riemann surfaces, preprint.
Antonio F. Costa and Hugo Parlier, Applications of a theorem of Singerman about Fuchsian groups, Arch. Math. (Basel) 91 (2008), no. 6, 536–543. MR 2465872, DOI 10.1007/s00013-008-2817-3
Ernesto Girondo and Gabino González-Diez, On the structure of the Whittaker sublocus of the moduli space of algebraic curves, Proc. Roy. Soc. Edinburgh Sect. A 136 (2006), no. 2, 337–346. MR 2218156, DOI 10.1017/S0308210500004583
W. J. Harvey, On branch loci in Teichmüller space, Trans. Amer. Math. Soc. 153 (1971), 387–399. MR 297994, DOI 10.1090/S0002-9947-1971-0297994-0
Hideyuki Kimura, Classification of automorphism groups, up to topological equivalence, of compact Riemann surfaces of genus 4, J. Algebra 264 (2003), no. 1, 26–54. MR 1980684, DOI 10.1016/S0021-8693(03)00138-8
Akikazu Kuribayashi and Hideyuki Kimura, Automorphism groups of compact Riemann surfaces of genus five, J. Algebra 134 (1990), no. 1, 80–103. MR 1068416, DOI 10.1016/0021-8693(90)90212-7
Ravi S. Kulkarni, Isolated points in the branch locus of the moduli space of compact Riemann surfaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 16 (1991), no. 1, 71–81. MR 1127697, DOI 10.5186/aasfm.1991.1614
Subhashis Nag, The complex analytic theory of Teichmüller spaces, Canadian Mathematical Society Series of Monographs and Advanced Texts, John Wiley & Sons, Inc., New York, 1988. A Wiley-Interscience Publication. MR 927291
David Singerman, Subgroups of Fuschian groups and finite permutation groups, Bull. London Math. Soc. 2 (1970), 319–323. MR 281805, DOI 10.1112/blms/2.3.319
David Singerman, Finitely maximal Fuchsian groups, J. London Math. Soc. (2) 6 (1972), 29–38. MR 322165, DOI 10.1112/jlms/s2-6.1.29