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


Authors: Gabriel Bartolini and Milagros Izquierdo
Journal: Proc. Amer. Math. Soc. 140 (2012), 35-45
MSC (2010): Primary 14Hxx, 30F10; Secondary 57M50
DOI: https://doi.org/10.1090/S0002-9939-2011-10881-5
Published electronically: May 9, 2011
MathSciNet review: 2833515
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Abstract: Let $ g$ be an integer $ \ge 3$ and let $ \mathcal{B}_g = \{X\in \mathcal{M}_g \vert 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.


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Additional Information

Gabriel Bartolini
Affiliation: Matematiska institutionen, Linköpings Universitet, 581 83 Linköping, Sweden
Email: gabar@mai.liu.se

Milagros Izquierdo
Affiliation: Matematiska institutionen, Linköpings Universitet, 581 83 Linköping, Sweden
Email: milagros.izquierdo@liu.se

DOI: https://doi.org/10.1090/S0002-9939-2011-10881-5
Keywords: Moduli spaces, Teichmüller modular group, automorphism group
Received by editor(s): December 17, 2009
Received by editor(s) in revised form: November 2, 2010
Published electronically: May 9, 2011
Additional Notes: The second author was partially supported by the Swedish Research Council (VR)
Communicated by: Martin Lorenz
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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