Polytopes of high rank for the alternating groups

Abstract

There is a well-known correspondence between abstract regular polytopes and string C-groups. In this paper, for each d⩾3, a string C-group with d generators, isomorphic to an alternating group of degree n is constructed (for some n⩾9), or equivalently an abstract regular d-polytope, is produced with automorphism group Alt(n). A method that extends the CPR graph of a polytope to a different CPR graph of a larger (or possibly isomorphic) polytope is used to prove that various groups are themselves string C-groups.