New non-arithmetic complex hyperbolic lattices II

TL;DR

This paper introduces a method to construct fundamental domains for complex hyperbolic triangle groups, expanding the classification of non-arithmetic lattices in PU(2,1) by identifying new commensurability classes.

Contribution

It provides a general procedure for producing fundamental domains for complex hyperbolic triangle groups and identifies new non-arithmetic commensurability classes.

Findings

New fundamental domain construction method for complex hyperbolic triangle groups

Identification of additional non-arithmetic commensurability classes

Total of 22 known non-arithmetic commensurability classes

Abstract

We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups, a class of groups that contains a representative of the commensurability class of every known non-arithmetic lattice in ${\rm PU}(2,1)$. We discuss several commensurability invariants for lattices, and show that some triangle groups yield new commensurability classes, bringing the number of known non-arithmetic commensurability classes to 22.