Regular polygonal complexes in space, II

Pellicer, Daniel; Schulte, Egon

doi:10.1090/S0002-9947-2012-05684-4

Regular polygonal complexes in space, II
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Trans. Amer. Math. Soc. 365 (2013), 2031-2061 Request permission

Abstract:

Regular polygonal complexes in euclidean $3$-space $\mathbb {E}^3$ are discrete polyhedra-like structures with finite or infinite polygons as faces and with finite graphs as vertex-figures, such that their symmetry groups are transitive on the flags. The present paper and its predecessor describe a complete classification of regular polygonal complexes in $\mathbb {E}^3$. In Part I we established basic structural results for the symmetry groups, discussed operations on their generators, characterized the complexes with face mirrors as the $2$-skeletons of the regular $4$-apeirotopes in $\mathbb {E}^3$, and fully enumerated the simply flag-transitive complexes with mirror vector $(1,2)$. In this paper, we complete the enumeration of all regular polygonal complexes in $\mathbb {E}^3$ and in particular describe the simply flag-transitive complexes for the remaining mirror vectors. It is found that, up to similarity, there are precisely 25 regular polygonal complexes which are not regular polyhedra, namely 21 simply flag-transitive complexes and $4$ complexes which are $2$-skeletons of regular $4$-apeirotopes in $\mathbb {E}^3$.

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