Mostow’s lattices and cone metrics on the sphere

Abstract

In [8] Mostow constructed a family of lattices in PU(2, 1), the holomorphic isometry group of complex hyperbolic 2-space. These groups are special cases of the lattices constructed by Deligne and Mostow [2] using monodromy of hypergeometric functions. Thurston [12] reinterpreted the work of Deligne and Mostow in terms of cone metrics on the sphere. In this paper we use Thurston's point of view to give a direct construction of fundamental domains for Mostow's lattices. Our approach is a direct generalisation of Parker's construction for Livné's lattices [10]. The details may be found in Boadi's PhD thesis [1]. We note that Deraux, Falbel and Paupert [3] have also constructed fundamental domains for Mostow's groups. Our groups are different from the ones considered in the main part of [3].

Moduli Spaces of Flat Tori with Prescribed Holonomy

“…Among the papers deeply relying on it about the theory of conical flat structures on the Riemann sphere, one can mention [Web93], [Par06], [GLL11], [BP15] and [Pas16] where some particular cases are considered in detail. The recent paper [McM17] deserves to be mentioned as well: in it, the author gives a more detailed treatment of the notion of cone-manifold than in [Thu98] and obtains a nice version of the Gauß-Bonnet theorem for complex hyperbolic cone-manifolds that he eventually uses to compute the volumes of the Picard/Deligne-Mostow/Thurston's moduli spaces.…”