Complex hyperbolic lattices

Resumen:

Complex hyperbolic space is the still mysterious sibling of real hyperbolic space. In particular, it is the only symmetric space of non-compact type where the relation between arithmeticity and lattices is not completely understood. 

In my talk, I will talk about lattices in the group PU(n,1) of holomorphic isometries of complex hyperbolic space with a particular focus on the non-arithmetic case.

I will give an overview of the known constructions and explain several ways of constructing the better understood class of Deligne-Mostow lattices. 

I will then talk about some future directions of research, which we hope will bring to new constructions of non-arithmetic lattices.

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