 
Summary: Complex hyperbolic geometry and the monster simple group
(conjectural)
Daniel Allcock
I explained the ideas and coincidences which led me to conjecture in [2] that a group
closely related to the monster simple group is got from the orbifold fundamental
group of a certain 13dimensional complexanalytic variety by adjoining a certain
relation.
Conway conjectured [6] that a group he called the bimonster is generated by 16
involutions satisfying certain braid and commutation relations (the ones specified
by the Y555 diagram), together with one extra relation w10
= 1. The bimonster
is (M × M):2, where M is the monster simple group. (Conway was working with
the bimonster rather than the monster because it made working with a subgroup
1
2 (S5 ×S12) of M more convenient.) Ivanov [8] and Norton [9] proved this. I found
the Y555 diagram appearing in my work in complex hyperbolic reflection groups
[1], so naturally I wondered if there was a connection. For me, it appeared because
one of my reflection groups contains 16 triflections (order 3 complex reflections)
satisfying exactly the same commutation and braid relations.
How can one compare two groups, similar except with generators of different
