Abstract regular polytopes and Y-shaped presentations for subgroups of the Monster sporadic simple group

9 October 2018
Dima Pasechnik

Abstract regular polytopes are finite quotients of Coxeter complexes
with string diagram, satisfying a natural intersection property, see
e.g. [MMS2002]. They arise in a number of geometric and group-theoretic
contexts. The first class of such objects, beyond the
well-understood examples coming from finite and affine Coxeter groups,
are locally toroidal cases, e.g.  extensions of quotients of the affine
F_4 complex [3,3,4,3].  In 1996 P.McMullen & E.Schulte constructed a
number of examples of locally toroidal abstract regular polytopes of
type [3,3,4,3,3], and conjectured completeness of their list. We
construct counterexamples to the conjecture using a Y-shaped
presentation for a subgroup of the Monster, and discuss various
related questions.