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

Abstract

Abstract. Deligne and Mostow constructed a class of lattices in P U(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.

New Nonarithmetic Complex Hyperbolic Lattices II

“…Presentations for various of these groups have been given in several places, including [22], [12], [28], [45] for instance. A unified presentation for all Deligne-Mostow groups with three fold symmetry was given in [32]. It is straightforward to check that our presentation is equivalent to hers.…”

Representations of Deligne-Mostow lattices into PGL(3, C) -- Part II

“…A fundamental domain and a presentation for each of these lattices can be found in [Pas16]. There are four types of lattices with 3-fold symmetry.…”

Representations of Deligne-Mostow lattices into PGL(3, C)

“…In this work we will only look at some of the lattices with 3-fold symmetry. For all of those, one can find a construction for a fundamental domain and a presentation in [Pas16]. In principle the 2-dimensional Deligne-Mostow lattices depend on the 5 elements of the ball 5-tuple.…”

“…The main result which allows computations is the explicit presentation of the Deligne-Mostow lattices. It was obtained in several steps culminating in [Pas16], where a unified treatment of all lattices is given. We will use a presentation of each Deligne-Mostow group using two generators easily obtained from the presentations in [Pas16].…”