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Complex hyperbolic geometry and the monster simple group (conjectural)

Summary: Complex hyperbolic geometry and the monster simple group
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 13-dimensional complex-analytic variety by adjoining a certain
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
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


Source: Allcock, Daniel - Department of Mathematics, University of Texas at Austin


Collections: Mathematics