# Colloquium abstracts

**Tathagata Basak**

Iowa State University, U.S.A.

October 24, 2019

**Fundamental group of a complex ball quotient**: Let W be a Weyl group and V be the complexificaion of its natural reflection representation. Let H be discriminant divisor in (V/W), that is, the image in (V/W) of the hyperplanes fixed by the reflections in W. It is well known that the fundamental group of the discriminant complement ((V/W) - H) is the Artin group described by the Dynkin diagram of W. We want to talk about an example for which an analogous result holds. Here W is an arithmetic lattice in PU(13,1) and V is the unit ball in complex thirteen dimensional vector space. Our main result (joint with Daniel Allcock) describes Coxeter type generators for the fundamental group of the discriminant complement ((V/W) - H). This takes a step towards a conjecture of Allcock relating this fundamental group with the Monster simple group. The example in PU(13,1) is closely related to the Leech lattice. Time permitting, we shall give a second example in PU(9,1) related to the Barnes-Wall lattice for which some similar results hold.