Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups

Abstract

We construct two infinite families of locally toroidal chiral polytopes of type {4,4,4}, with 1024m2 and 2048m2 automorphisms for every positive integer m, respectively. The automorphism groups of these polytopes are solvable groups, and when m is a power of 2, they provide examples with automorphism groups of order 2n where n can be any integer greater than 9. (On the other hand, no chiral polytopes of type [4,4,4] exist for n≤9.) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21].