Construction of chiral polytopes of large rank with alternating or symmetric automorphism group

Chiral polytopes are highly symmetric abstract polytopes that have maximal rotational symmetry but no reflections. Until recently, very few examples of chiral d-polytopes with rank d≥5 were known. In this paper we extend a construction for chiral 4-polytopes (with tetrahedral facets) to produce two infinite families of chiral d-polytopes with simplex facets, for every integer d≥5. Specifically, we show that for every d≥5, there exists a chiral d-polytope with automorphism group isomorphic to the alternating group An and a chiral d-polytope with automorphism group isomorphic to the symmetric group Sn, for all but finitely many n. In each case the facets are simplices, and the vertex-figures are chiral. Our extended construction can also be modified to produce four families of regular d-polytopes with simplices as facets, and automorphism groups isomorphic to An, Sn, An×C2 and Sn×C2 for all but finitely many n.