$2$-groups behaving as automorphism groups of regular $3$-polytopes

Abstract

In this paper, we classify regular polytopes with automorphism groups of order $2^n$ and Schläfli types $\{4,2^{n-3}\},\{4,2^{n-4}\}$ and $\{4,2^{n-5}\}$ for $n\ge 10$, therefore giving a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, Periodica Math. Hungarica 53(2006) 231-255].

21 pages