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 $W$.
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.