Coxeter-Mostow diagrams and lattices in PU(n,1)

Coxeter diagrams and their corresponding Coxeter groups have a
well known relationship to hyperbolic reflection groups. In the 1980s
Vinberg proved a number of major results about these groups, e.g. in real
hyperbolic spaces of dimension ≥30 there are no hyperbolic reflection
groups of compact type.
In this talk we describe the analagous construction for complex hyperbolic
spaces, Coxeter-Mostow diagrams. Then we will summarise some results about
reflection group lattices in PU(2,1) and present a possible approach to
extend these results into higher dimensions.