Geometry and Topology Seminar

Let B = CH^13 be 13-dimensional complex hyperbolic space (a complex ball). There is an arithmetic group Γ in PU(13) acting on B generated by order 3 Hermitian isometries s_i called triflections. Basak and Allcock have studied the geometry of X = Γ \ B in detail; it is intimately connected with the finite projective plane P^2F_3. A conjecture of Allcock states that if one replaces relations s_i^3=1 in Γ with s_i^2=1, the resulting group is the Bimonster---the wreath product of the monster with Z_2. A resolution of this conjecture likely leads to a resolution of the "Hirzebruch prize question": The existence of a compact 12-complex dimensional manifold with certain topological invariants and an action of the monster. Such a manifold would lead in a known way to a new, geometric proof of monstrous moonshine.

I will discuss three moduli spaces, which might (depending on the status of computations at the time of the talk) be isomorphic to ball quotients of dimensions 13, 7, 4 relating to the projective planes P^2F_3, P^2F_2, "P^2F_1" = {3 points}. The corresponding finite groups, gotten by replacing 3, 4, 6 with 2, should be the bimonster, an orthogonal group O_8(2) of a quadratic form on F_2^8, and the symmetric group S_6 respectively. Should these examples work out, they will produce many surprising things: For instance, a formula for the order of the monster group in terms of Hurwitz numbers. This talk is highly speculative and represents joint with Peter Smillie and Francois Greer.