Abstract
We generalise to the genus one case several results of Thurston concerning moduli spaces of flat Euclidean structures with conical singularities on the two dimensional sphere. More precisely, we study moduli spaces of flat tori with n cone points and a prescribed holonomy \({\rho}\). In his paper `Flat Surfaces’ Veech has established that under some assumptions on the cone angles, such a moduli space \({\mathcal{F}_{[\rho]} \subset \mathcal{M}_{1,n}}\) carries a natural geometric structure modeled on the complex hyperbolic space \({{\mathbb C}{\mathbb{H}}^{n-1}}\) which is not metrically complete. Using surgeries for flat surfaces, we prove that the metric completion \({\overline{\mathcal{F}_{[\rho]}}}\) is obtained by adjoining to \({ \mathcal{F}_{[\rho]}}\) certain strata that are themselves moduli spaces of flat surfaces of genus 0 or 1, obtained as degenerations of the flat tori whose moduli space is \({ \mathcal{F}_{[\rho]}}\). We show that the \({{\mathbb C}{\mathbb{H}}^{n-1}}\)-structure of \({ \mathcal{F}_{[\rho]}}\) extends to a complex hyperbolic cone-manifold structure of finite volume on \({ \overline{\mathcal{F}_{[\rho]}}}\) and we compute the cone angles associated to the different strata of codimension 1. Finally, we address the question of whether or not the holonomy of Veech’s \({{\mathbb C}{\mathbb{H}}^{n-1}}\)-structure on \({\mathcal{F}_{[\rho]}}\) has a discrete image in \({ {\rm Aut}({\mathbb C}{\mathbb{H}}^{n-1})={\rm PU}(1,n-1)}\). We outline a general strategy to find moduli spaces \({\mathcal{F}_{[\rho]}}\) whose \({{\mathbb C}{\mathbb{H}}^{n-1}}\)-holonomy gives rise to lattices in \({{\rm PU}(1,n-1)}\) and eventually we give a finite list of \({\mathcal{F}_{[\rho]}}\) ’s whose holonomy is a complex hyperbolic arithmetic lattice.
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E. Arbarello, M. Cornalba, and P.A. Griffiths. Geometry of algebraic curves. Volume II,volume 268 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Heidelberg, (2011). With a contribution by Joseph Daniel Harris.
M. R. Bridson and A. Haefliger. Metric spaces of non-positive curvature, volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, (1999).
Boadi R., Parker J.: Mostow’s lattices and cone metrics on the sphere. Adv. Geom., 15(1), 27–53 (2015)
P. Deligne and G. D. Mostow. Monodromy of hypergeometric functions and nonlattice integral monodromy. Inst. Hautes Études Sci. Publ. Math., (63) (1986), 5–89
P. Deligne and G. D. Mostow. Commensurabilities among lattices in \({{\rm PU}(1,n)}\), volume 132 of Annals of Mathematics Studies. Princeton University Press, Princeton, NJ, (1993).
M. Deraux, J. Parker, and J. Paupert.s New non-arithmetic complex hyperbolic lattices. Invent. Math., 203(3) (2016), 681–771
F. Diamond and J. Shurman. A first course in modular forms, volume 228 of Graduate Texts in Mathematics. Springer-Verlag, New York, (2005).
A. Eskin, H. Masur, and A. Zorich. Moduli spaces of abelian differentials: the principal boundary, counting problems, and the Siegel-Veech constants. Publ. Math. Inst. Hautes Études Sci., (97) (2003), 61–179
González A., López-López J.: Shapes of tetrahedra with prescribed cone angles. Conform. Geom. Dyn., 15, 50–63 (2011)
W. M. Goldman. Complex hyperbolic geometry. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, (1999). Oxford Science Publications.
S. Ghazouani and L. Pirio. Moduli spaces of flat tori and elliptic hypergeometric functions. Preprint arXiv:1605.02356.
Goldman W. M., Parker J. R.: Dirichlet polyhedra for dihedral groups acting on complex hyperbolic space. J. Geom. Anal., 2(6), 517–554 (1992)
M. Gromov. Metric structures for Riemannian and non-Riemannian spaces, volume 152 of Progress in Mathematics. Birkhäuser Boston, Inc., Boston, MA, 1999. Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated from the French by Sean Michael Bates.
D. Jackson and T. Visentin. An atlas of the smaller maps in orientable and nonorientable surfaces. CRC Press Series on Discrete Mathematics and its Applications. Chapman & Hall/CRC, Boca Raton, FL, (2001).
Kontsevich M., Zorich A.: Connected components of the moduli spaces of Abelian differentials with prescribed singularities. Invent. Math., 153(3), 631–678 (2003)
R. Le Vavasseur. Sur le système d’équations aux dérivées partielles simultanées auxquelles satisfait la série hypergéométrique à deux variables \({{\rm F}_1\left(\alpha, \beta, \beta',\gamma; x, y\right)}\). Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 7(3) (1893), 1–120
Toshiyuki Mano. The Riemann-Wirtinger integral and monodromy-preserving deformation on elliptic curves. Int. Math. Res. Not. IMRN, pages Art. ID rnn110, 19, 2008.
McMullen C. T.: The gauss-bonnet theorem for cone manifolds and volumes of moduli spaces. Amer. J. Math., 139(1), 261–291 (2017)
G. D. Mostow. Generalized Picard lattices arising from half-integral conditions. Inst. Hautes Études Sci. Publ. Math., (63) (1986), 91–106
Mostow G. D.: On discontinuous action of monodromy groups on the complex n-ball. J. Amer. Math. Soc., 1(3), 555–586 (1988)
H. Masur and J. Smillie. Hausdorff dimension of sets of nonergodic measured foliations. Ann. of Math. (2), 134(3) (1991), 455–543
Masur H., Zorich A.: Multiple saddle connections on flat surfaces and the principal boundary of the moduli spaces of quadratic differentials. Geom. Funct. Anal., 18(3), 919–987 (2008)
Parker J. R.: Cone metrics on the sphere and Livné’s lattices. Acta Math., 196(1), 1–64 (2006)
Pasquinelli. I.: Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere. Conform. Geom. Dyn., 20, 235–281 (2016)
Sauter. J.: Isomorphisms among monodromy groups and applications to lattices in \({{\rm PU}(1,2)}\). Pacific J. Math., 146(2), 331–384 (1990)
R. E. Schwartz. Notes on shape of polyhedra. Preprint arXiv:1506.07252
Terada. T.: Problème de Riemann et fonctions automorphes provenant des fonctions hypergéométriques de plusieurs variables. J. Math. Kyoto Univ., 13, 557–578 (1973)
Tholozan. N.: Sur la complétude de certaines variétés pseudo-riemanniennes localement homogènes. Ann. Inst. Fourier., 65(5), 1921–1952 (2015)
W. P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, (1997). Edited by Silvio Levy.
W. P. Thurston. Shapes of polyhedra and triangulations of the sphere. In: The Epstein birthday schrift, volume 1 of Geom. Topol. Monogr., pages 511–549. Geom. Topol. Publ., Coventry, (1998).
M. Troyanov. Les surfaces euclidiennes à singularités coniques. Enseign. Math. (2), 32(1-2) (1986), 79–94
M. Troyanov. On the moduli space of singular Euclidean surfaces. In: Handbook of Teichmüller theory. Vol. I, volume 11 of IRMA Lect. Math. Theor. Phys., pages 507–540. Eur. Math. Soc., Zürich, (2007).
Veech. W.: Flat surfaces. Amer. J. Math., 115(3), 589–689 (1993)
M. Weber. Fundamentalbereiche komplex hyperbolischer Flächen. Bonner Mathematische Schriften [Bonn Mathematical Publications], 254. Universität Bonn, Mathematisches Institut, Bonn, (1993).
J. Wolf. Spaces of constant curvature. AMS Chelsea Publishing, Providence, RI, sixth edition, (2011).
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Ghazouani, S., Pirio, L. Moduli Spaces of Flat Tori with Prescribed Holonomy. Geom. Funct. Anal. 27, 1289–1366 (2017). https://doi.org/10.1007/s00039-017-0426-7
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DOI: https://doi.org/10.1007/s00039-017-0426-7