Title: Fundamental group of a complex ball quotient
Speaker: Tathagata Basak (Iowa State University, Ames, USA)
Date: 04 November 2019
Time: 3 pm
Venue: LH-1, Mathematics Department
Let W be a Weyl group and V be the complexification of its natural
reflection representation. Let H be the 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.