#### Vol. 302, No. 1, 2019

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Fundamental domains and presentations for the Deligne–Mostow lattices with 2-fold symmetry

### Irene Pasquinelli

Vol. 302 (2019), No. 1, 201–247
##### Abstract

In this work we will build a fundamental domain for Deligne–Mostow lattices in $PU\left(2,1\right)$ 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 $PU\left(2,1\right)$.

##### Keywords
complex hyperbolic lattices, discrete subgroups of Lie groups, ball quotients, Deligne–Mostow lattices
##### Mathematical Subject Classification 2010
Primary: 22E40, 32M05, 51M10, 57M50