Two infinite families of chiral polytopes of type {4,4,4} with solvable automorphism groups. (English) Zbl 07286501
Summary: We construct two infinite families of locally toroidal chiral polytopes of type \(\{4, 4, 4 \}\), with \(1024 m^2\) and \(2048 m^2\) 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 \(2^n\) where \(n\) can be any integer greater than 9. (On the other hand, no chiral polytopes of type \([4, 4, 4]\) exist for \(n \leq 9\).) In particular, our two families give a partial answer to a problem proposed by Schulte and Weiss in [21].
MSC:
| 52B15 | Symmetry properties of polytopes |
| 20B25 | Finite automorphism groups of algebraic, geometric, or combinatorial structures |
Software:
MagmaReferences:
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