Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere

Pasquinelli, Irene

doi:10.1090/ecgd/299

Deligne-Mostow lattices with three fold symmetry and cone metrics on the sphere
HTML articles powered by AMS MathViewer

by Irene Pasquinelli

Conform. Geom. Dyn. 20 (2016), 235-281

DOI: https://doi.org/10.1090/ecgd/299

Published electronically: July 19, 2016

PDF | Request permission

Abstract:

Deligne and Mostow constructed a class of lattices in $PU(2,1)$ using monodromy of hypergeometric functions. Thurston reinterpreted them in terms of cone metrics on the sphere. In this spirit we construct a fundamental domain for the lattices with three fold symmetry in the list of Deligne and Mostow. This is a generalisation of the works of Boadi and Parker and gives a different interpretation of the fundamental domain constructed by Deraux, Falbel, and Paupert.

References

Richard K. Boadi and John R. Parker, Mostow’s lattices and cone metrics on the sphere, Adv. Geom. 15 (2015), no. 1, 27–53. MR 3300709, DOI 10.1515/advgeom-2014-0022
Martin Deraux, Elisha Falbel, and Julien Paupert, New constructions of fundamental polyhedra in complex hyperbolic space, Acta Math. 194 (2005), no. 2, 155–201. MR 2231340, DOI 10.1007/BF02393220
P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651, DOI 10.1007/BF02831622
Martin Deraux, John R. Parker, and Julien Paupert, New non-arithmetic complex hyperbolic lattices, Invent. Math. 203 (2016), no. 3, 681–771. MR 3461365, DOI 10.1007/s00222-015-0600-1
Elisha Falbel and John R. Parker, The geometry of the Eisenstein-Picard modular group, Duke Math. J. 131 (2006), no. 2, 249–289. MR 2219242, DOI 10.1215/S0012-7094-06-13123-X
William M. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR 1695450
G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876, DOI 10.2140/pjm.1980.86.171
G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106. MR 849652, DOI 10.1007/BF02831623
G. D. Mostow, On discontinuous action of monodromy groups on the complex $n$-ball, J. Amer. Math. Soc. 1 (1988), no. 3, 555–586. MR 932662, DOI 10.1090/S0894-0347-1988-0932662-7
John R. Parker, Cone metrics on the sphere and Livné’s lattices, Acta Math. 196 (2006), no. 1, 1–64. MR 2237205, DOI 10.1007/s11511-006-0001-9
John R. Parker, Complex hyperbolic lattices, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 1–42. MR 2581913, DOI 10.1090/conm/501/09838
John Kurt Sauter Jr., Isomorphisms among monodromy groups and applications to lattices in $\textrm {PU}(1,2)$, Pacific J. Math. 146 (1990), no. 2, 331–384. MR 1078386, DOI 10.2140/pjm.1990.146.331
William P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549. MR 1668340, DOI 10.2140/gtm.1998.1.511