Mixing and monodromy of abstract polytopes

Monson, Barry; Pellicer, Daniel; Williams, Gordon

doi:10.1090/s0002-9947-2013-05954-5

Cited by 40 publications

(42 citation statements)

References 23 publications

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“…For more details on this, see Sections 3 and 4 of [15]. Thus, for equivelar toroidal maps N = τ /H, M = τ /G, H < G, it follows that if N M is a K-sheeted covering, then the number K of sheets can be easily found from the representation in (2.1) as follows:…”

Section: Maps and Coversmentioning

confidence: 99%

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Minimal covers of equivelar toroidal maps

Drach¹,

Mixer

2014

Ars Math. Contemp.

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Given any equivelar map on the torus, it is natural to consider its covering maps. The most basic of these coverings are finite toroidal maps or infinite tessellations of the Euclidean plane. In this paper, we prove that each equivelar map on the torus has a unique minimal toroidal rotary cover and also a unique minimal toroidal regular cover. That is to say, of all the toroidal rotary (or regular) maps covering a given map, there is a unique smallest. Furthermore, using the Gaussian and Eisenstein integers, we construct these covers explicitly.

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Section: Maps and Coversmentioning

confidence: 99%

“…It follows from Theorem 4.14 of [15], that any equivelar map M is covered by the universal map U of the same type {p, q}, and that M is a quotient of U by some subgroup of the automorphism group of U.…”

Section: Maps and Coversmentioning

confidence: 99%

Minimal covers of equivelar toroidal maps

Drach¹,

Mixer

2014

Ars Math. Contemp.

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“…It is well-known that if K is regular then M on(K) ∼ = Γ(K) (see [9,Theorem 3.9]). On the other hand, if K is chiral then its automorphism group is related with its monodromy group in a slightly different way.…”

Section: Regular and Chiral Polytopesmentioning

confidence: 99%

Chiral extensions of chiral polytopes

Cunningham¹,

Pellicer²

2014

Discrete Mathematics

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“…. , r d−1 of the monodromy group of K satisfy the intersection condition (2.3) then M on(K) is the automorphism group of the minimal regular cover of K. Further details can be found in [26] and [41].…”

Section: Group Structuresmentioning

confidence: 99%

“…It is known that the monodromy group of all polyhedra are string C-groups [41], and therefore every polyhedron has a minimal regular cover. This is not the case for higher rank polytopes.…”

Section: Group Structuresmentioning

confidence: 99%