The intersection condition for regular polytopes

The automorphism group of every abstract regular polytope of type { k 1 | ⋯ | k m } is a smooth finite quotient of the corresponding Coxeter group [ k 1 , … , k m ] satisfying a particular condition on subgroups known as the intersection condition (or intersection property), and vice versa. In this paper, we consider the question of how many cases of the intersection condition need to be checked, sometimes under additional hypotheses about the finite quotient group or the type of the polytope. We also apply this to the construction of a family of polytopal regular maps of every possible genus, and an infinite family of locally spherical regular 4-polytopes of type { 3 | 5 | 3 } , with all but finitely many alternating groups as automorphism groups.