 
Summary: A MONSTROUS PROPOSAL
DANIEL ALLCOCK
Dedicated to Domingo Toledo and to John McKay
Abstract. We explain a conjecture relating the monster sim
ple group to an algebraic variety that was discovered in a non
monstrous context.
The purpose of this note is to explain a conjecture I have circulated
privately since 1997. On one level it is purely about group theory and
complex hyperbolic geometry, but if it is true then the most natural
explanation for it would be algebrageometric: a certain ball quotient
would be the moduli space for some sort of objects, the objects admit
ting some sort of marking related to the monster simple group. My
original grounds for the conjecture were flimsy, but in his dissertation,
Tathagata Basak discovered some very suggestive coincidences. The
conjecture is still speculative, but now I am taking it seriously.
In brief, Conway described the bimonster M × M : 2 as being gen
erated by 16 involutions satisfying some braid and commutation re
lations, subject to the additional relation that a certain word w has
order 10. On the other hand, I discovered a certain complex analytic
orbifold X having a nice uniformization by complex hyperbolic 13
