Four infinite families of chiral 3-polytopes of type \4, 8\ with solvable automorphism groups

2023/07/23 by Dong-Dong Hou, Tiantian Zheng, Hou, Dong-Dong +3 · 1 citation
Mathematics · #20B25 52B15 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry and Number Theory #Combinatorics (math.CO) #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2307.12999

openalex publication_date 2023/07/23 · openalex created_date 2023/07/27 · openalex updated_date 2026/07/28

Abstract

We construct four infinite families of chiral 3-polytopes of type \4, 8\, with 1024m⁴, 2048m⁴, 4096m⁴ and 8192m⁴ 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 ≥ 10. (On the other hand, no chiral polytopes of type \4, 8\ exist for n ≤ 9.) In particular, our families give a partial answer to a problem proposed by Schulte and Weiss in [Problems on polytopes, their groups, and realizations, \em Period. Math. Hungar. 53 (2006), 231-255] and a problem proposed by Pellicer in [Developments and open problems on chiral polytopes, \em Ars Math. Contemp 5 (2012), 333-354].

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