All Alternating Groups $A_n$ with $n\geq12$ Have Polytopes of Rank $\lfloor\frac{n-1}{2}\rfloor$

Abstract

Using the correspondence between abstract regular polytopes and string C-groups, in a recent paper [M. E. Fernandes, D. Leemans, and M. Mixer, J. Combin. Theory Ser. A, 119 (2012), pp. 42–56], we constructed an abstract regular polytope of rank r, for each $r \geq 3$, with automorphism group isomorphic to $A_{2r+3}$ when r is odd, and $A_{2r+1}$ when r is even. In this paper, the remaining cases are completed. It is shown that every group $A_n$, with n sufficiently large, acts on at least one abstract regular polytope of rank $\lfloor\frac{n-1}{2}\rfloor$. We conjecture that this is the highest possible rank for $n\geq 12$.