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Equivelar toroids with few flag-orbits. (English) Zbl 07308717
Toroids are generalisations of maps on the torus to higher dimensions. To be more specific, an $$(n+1)$$-toroid is a quotient of a tessellation of the n-dimensional Euclidean space with a lattice group.
The classification of regular and chiral toroids of any rank was completed by P. McMullen and E. Schulte [Adv. Math. 117, No. 1, 17–51 (1996; Zbl 0847.52010)]. In particular, they showed that there exist no chiral toroids of rank greater than 3. Furthermore, M. I. Hartley et al. showed for $$n\ge 2$$, the only Euclidean space form that admits chiral tessellations is the $$n$$-torus and proved this is only possible when $$n = 2$$ (see [Can. J. Math. 51, No. 6, 1230–1239 (1999; Zbl 0965.51008)]). In [Discrete Comput. Geom. 48, No. 4, 1110–1136 (2012; Zbl 1263.51016)], I. Hubard et al. studied properties of the automorphism groups of equivelar toroids and classified all equivelar toroids of ranks 3 and 4 into families according to their symmetry types.
The present nice paper gives a classification of equivelar $$(n+1)$$-toroids with at most $$n$$ orbits. In particular, a classification of 2-orbit $$(n+1)$$-toroids is discussed. Symmetry types of families of equivelar $$(n+1)$$-toroids with at most $$n$$ orbits are also determined.
##### MSC:
 51B15 Laguerre geometries 52C22 Tilings in $$n$$ dimensions (aspects of discrete geometry) 51F15 Reflection groups, reflection geometries 51M20 Polyhedra and polytopes; regular figures, division of spaces
##### Keywords:
symmetries of toroids; maps; polytopes; regular tessellation
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##### References:
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