Constructions for chiral polytopes

An abstract polytope of rank n is said to be chiral if its automorphismgroup has two orbits on flags, with adjacent flags lying indifferent orbits. In this paper, we describe a method for constructingfinite chiral n-polytopes, by seeking particular normal subgroupsof the orientation-preserving subgroup of an n-generator Coxetergroup (having the property that the subgroup is not normalizedby any reflection and is therefore not normal in the full Coxetergroup). This technique is used to identify the smallest examplesof chiral 3- and 4-polytopes, in both the self-dual and non-self-dualcases, and then to give the first known examples of finite chiral5-polytopes, again in both the self-dual and non-self-dual cases.