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

TL;DR

This paper constructs fundamental domains for Deligne-Mostow lattices with three fold symmetry, linking hypergeometric monodromy and cone metrics on the sphere, and generalizes previous work on these complex hyperbolic lattices.

Contribution

It provides a new construction of fundamental domains for a class of Deligne-Mostow lattices with three fold symmetry, expanding the geometric understanding of these lattices.

Findings

Constructed fundamental domains for all such lattices.

Unified interpretation of previous lattice fundamental domains.

Extended the geometric framework for Deligne-Mostow lattices.

Abstract

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