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Abstract:
Any Coxeter group G with string diagram is
the automorphism group of an abstract regular polytope (typically infinite).
When G is crystallographic, its standard real representation is easily reduced
modulo an odd prime p, thus giving a finite representation in some
finite orthogonal space V over Fp. The finite
group need not be polytopal;
and whether or not it is depends in an intricate way on the geometry of V.
The talk presents recent work with Barry Monson, in which we describe
this construction in considerable generality and study in depth the
interplay between the geometric properties of the polytope (if it exist)
and the algebraic structure of the overlying finite orthogonal group.
As a byproduct, we obtain many new maps on surfaces and even more interesting
polytopes of higher rank.
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