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Four Infinite Families of Chiral $3$-Polytopes of Type $\{4, 8\}$ with Solvable Automorphism Groups by Hou, Dong-Dong; Zheng, Tian-Tian; Guo, Rui-Rui is a scholarly article available to read on EtoBox.

What is Four Infinite Families of Chiral $3$-Polytopes of Type $\{4, 8\}$ with Solvable Automorphism Groups about?

We construct four infinite families of chiral $3$-polytopes of type $\{4, 8\}$, with $1024m^4$, $2048m^4$, $4096m^4$ and $8192m^4$ 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 $n \geq 10$. (On the other hand, no chiral polytopes of type $\{4, 8\}$ exist for $n \leq 9$.) 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].

Author

Hou, Dong-Dong; Zheng, Tian-Tian; Guo, Rui-Rui

Published

2023

Language

EN