In this work we will build a fundamental domain for Deligne–Mostow lattices in
with
2-fold symmetry, which completes the list of Deligne–Mostow lattices in dimension 2.
These lattices were introduced by Mostow, (1980; 1986) and Deligne and
Mostow (1986) using monodromy of hypergeometric functions and have been
reinterpreted by Thurston (1998) as automorphisms on a sphere with cone
singularities. Following his approach, Parker (2006), Boadi and Parker (2015) and
Pasquinelli (2016) built a fundamental domain for the class of lattices with
3-fold symmetry, i.e., when three of five cone singularities have the same cone
angle. Here we extend this construction to the asymmetric case, where only
two of the five cone points on the sphere have the same cone angle, hence
building a fundamental domain for all remaining Deligne–Mostow lattices in
.
