More on chiral polytopes of type \{4, 4, …, 4\} with~solvable automorphism groups

2.1 Abstract polytopes

An abstract polytope𝒫 of rank 𝑑 ( ≥ − 1 ) is a partially ordered set endowed with a strictly monotone rank function having range { − 1 , … , d } and satisfying the properties mentioned below. The elements of 𝒫 are called faces. Two faces 𝐹 and 𝐺 of 𝒫 are said to be incident if F ≤ G or F ≥ G . Then a flag is a maximal totally ordered subset of 𝒫.

For − 1 ≤ i ≤ d , the elements of 𝒫 of rank 𝑖 are called 𝑖-faces. Also the 0-faces, 1-faces and the ( d − 1 ) -faces are called vertices, edges and facets, respectively. Each flag of 𝒫 contains exactly d + 2 faces, including a unique least face and a unique greatest face. Two flags are said to be adjacent if they differ by just one face, and 𝑗-adjacent if they differ only in their 𝑗-face.

There are two more important properties 𝒫 should satisfy. Firstly, 𝒫 is required to be strongly flag-connected, which says that, for any two flags Φ and Ψ, there exists a sequence Φ = Φ 0 , Φ 1 , … , Φ k − 1 , Φ k = Ψ of flags such that Φ j − 1 is adjacent to Φ j , and Φ ∩ Ψ ⊆ Φ j for 1 ≤ j ≤ k . Moreover, we also require 𝒫 to satisfy the diamond condition, saying that there are exactly two 𝑖-faces H 1 and H 2 of 𝒫 such that F < H j < G ( j ∈ { 1 , 2 } ) for each pair of faces F < G satisfying rank ⁡ ( G ) − rank ⁡ ( F ) = 2 .

If 𝐹 and 𝐺 are faces with F ≤ G , then the set { H ∣ F ≤ H ≤ G } is called a section of 𝒫, denoted by G / F . A section of rank 𝑟 can be called an 𝑟-section of 𝒫. For any face 𝐺 of 𝒫, the section G / F − 1 may be identified with 𝐺 itself; hence, for example, a facet may be viewed as a ( d − 1 ) -face 𝐺, or as the section F = G / F − 1 . Similarly, if 𝐺 is a 𝑗-face, then the section F d / G is sometimes called the co-face at 𝐺. In particular, a co-vertex is called a vertex-figure. Given a vertex 𝐹 and a facet 𝐺 that contains 𝐹, the section G / F is an ( n − 2 ) -polytope, called a medial section of 𝒫; it is both a vertex-figure of the facet 𝐺 and a facet of the vertex-figure at 𝐹.

We say that 𝒫 has Schläfli symbol { k 1 , … , k d − 1 } or 𝒫 is of type { k 1 , … , k d − 1 } if the section G / F is isomorphic to the poset given by the polygon with k i edges, no matter which ( i − 2 ) -face 𝐹 and ( i + 1 ) -face 𝐺 we choose for 1 ≤ i ≤ d − 1 . Also, when this happens, we say 𝒫 is equivelar. Regular and chiral polytopes, defined in the following subsection, are examples of equivelar polytopes. Every equivelar abstract polytope of type { k 1 , … , k d − 1 } has at least 2 ⁢ k 1 ⁢ ⋯ ⁢ k d − 1 flags. Whenever 𝒫 has exactly this number of flags, it is said to be tight.

In this paper, all equivelar polytopes we will consider are assumed to be finite; hence, entries of their Schläfli symbols are necessarily finite.

2.2 Directly regular polytopes, chiral polytopes and their groups

Given two polytopes 𝒫 and 𝒬 of the same rank, an isomorphism from 𝒫 to 𝒬 is an incidence- and rank-preserving bijection on the set of faces. As usual, an automorphism of 𝒫 is an isomorphism from 𝒫 to itself. All automorphisms of 𝒫 form a group denoted by Γ ⁢ ( P ) .

Due to the strong connectedness and the diamond condition, the stabilizer in Γ ⁢ ( P ) of any given flag fixes all flags of 𝒫 and hence is trivial. Thus, the order of Γ ⁢ ( P ) is bounded above by the number of flags of 𝒫. The group Γ ⁢ ( P ) acts transitively (and hence regularly) on the flags of 𝒫 when this bound is attained.

An abstract polytope 𝒫 is said to be regular whenever Γ ⁢ ( P ) acts transitively on the set of all flags of 𝒫. Let 𝒫 be a regular 𝑑-polytope of type { k 1 , … , k d − 1 } . Then its automorphism group Γ ⁢ ( P ) can be generated by a canonical set of involutions ρ 0 , … , ρ d − 1 , where ρ i is the unique automorphism mapping a given base flagΦ to its 𝑖-adjacent flag Φ i for 0 ≤ i < d . Moreover, the generators ρ 0 , … , ρ d − 1 satisfy the relations

and the following intersection condition:

The rotation group Γ + ⁢ ( P ) of 𝒫 is the subgroup of Γ ⁢ ( P ) generated by

and has index at most 2 in Γ ⁢ ( P ) . We say that 𝒫 is directly regular when this index is exactly 2.

Conversely, if Γ is a group generated by 𝑑 elements ρ 0 , ρ 1 , … , ρ d − 1 that satisfy both conditions (2.1) and (2.2), then a regular polytope 𝒫 can be constructed such that Γ ⁢ ( P ) ≅ Γ (see [16]).

An abstract polytope 𝒫 of rank 𝑑 is said to be chiral if its automorphism group Γ ⁢ ( P ) has two orbits on flags, with any two adjacent flags lying in distinct orbits. Given a base flag Φ, then for each j = 1 , … , d − 1 , there exists an automorphism σ j of 𝒫 mapping Φ to the flag ( Φ j ) j − 1 which differs from Φ in precisely its ( j − 1 ) - and 𝑗-faces. This automorphism σ j is the analogue of the abstract rotation ρ j − 1 ⁢ ρ j in the regular case for each 𝑗. In other words, the polytope 𝒫 admits the analogues of all abstract rotations, it has maximum possible rotational symmetry, but on the other hand, it admits no reflection. It follows that the automorphism group Γ ⁢ ( P ) of 𝒫 can be generated by σ 1 , σ 2 , … , σ d − 1 , and hence Γ + ⁢ ( P ) can be defined as Γ ⁢ ( P ) when 𝒫 is chiral.

Directly regular polytopes and chiral polytopes have many common properties, this is particularly reflected in their rotation groups.

Let 𝒫 be a directly regular or chiral polytope of type { k 1 , … , k d − 1 } , with rotation group Γ + ⁢ ( P ) and canonical generators σ 1 , … , σ d − 1 . Then the elements σ 1 , … , σ d − 1 satisfy the relations

and the following intersection condition (of chiral form) (see [22]):

As in the case of regular polytopes, if Γ is any finite group generated by d − 1 elements σ 1 , … , σ d − 1 that satisfy relation (2.3) and the intersection condition (2.4), then we can construct a directly regular or chiral 𝑑-polytope 𝒫 such that its rotation group Γ + ⁢ ( P ) is isomorphic to Γ by taking as its 𝑗-faces the (right) cosets of the subgroup generated by

and defining incidence by non-empty intersection (see [19, Theorem 1]).

The resulting polytope is denoted by P ⁢ ( σ 1 , … , σ d − 1 ) , and it has the specific type { k 1 , … , k d − 1 } , where k i is the order of σ i . If 𝐹 and 𝐺 are incident faces of 𝒫 of ranks i − 2 and j + 1 with i ≤ j , then the section G / F is isomorphic to the ( j − i + 2 ) -polytope P ⁢ ( σ i , … , σ j ) . Moreover, the polytope P ⁢ ( σ 1 , … , σ d − 1 ) is regular if and only if Γ has an automorphism mapping

respectively, and the polytope must be directly regular when this happens.

We let 𝑊 be the Coxeter group of type [ k 1 , … , k d ] , so it is generated by 𝑑 elements x 0 , x 1 , … , x d − 1 satisfying

and let W + be the orientation-preserving subgroup of 𝑊, which is generated by x i − 1 ⁢ x i for 1 ≤ i < d . Then we notice that (2.1) and (2.3) are exactly the defining relations of 𝑊 and W + , respectively. The latter implies that, for a directly regular or chiral polytope 𝒫 of type { k 1 , … , k d − 1 } , with rotation group Γ + ⁢ ( P ) and canonical generators σ 1 , … , σ d − 1 , there exists an epimorphism φ : W + ↦ Γ + taking x i − 1 ⁢ x i to σ i for 1 ≤ i < d , and hence the kernel of 𝜑 is a normal subgroup 𝐾 of W + .

In particular, if 𝒫 is a chiral polytope with base flag Φ, then the two flag orbits yield two sets of generators which are not conjugate in Γ ⁢ ( P ) , and so there is no automorphism of 𝒫 taking Φ to Φ 0 . Actually, chiral polytopes occur in pairs. For each pair, one is the enantiomorphic form of the other one. In this case, the kernel 𝐾 of 𝜑 is not normal in 𝑊, but is conjugated by any orientation-reversing element (in 𝑊) to another subgroup K c , such that W / K c is the automorphism group of P c which is the enantiomorphic form of 𝒫. Actually, Γ ⁢ ( P ) = Γ ⁢ ( P c ) , but they have different canonical generating sets: a base flag of P c can be chosen such that σ 1 − 1 , σ 1 2 ⁢ σ 2 , σ 3 , σ 4 , … , σ d − 2 , σ d − 1 are the canonical generators for Γ ⁢ ( P c ) , which would be the conjugates of σ 1 , σ 2 , … , σ d − 1 by x 0 (see more in [20, Section 3]).

Therefore, given the rotation subgroup W + of the Coxeter group 𝑊 with type [ k 1 , … , k d − 1 ] , and a normal subgroup 𝐾 of W + , W + / K is the automorphism group of a chiral 𝑑-polytope if and only if 𝐾 is not normal in 𝑊 and W + / K satisfies the intersection condition (2.4) with respect to its generators.

The dual of a polytope is the polytope obtained by reversing its partial order. A polytope 𝒫 can sometimes be self-dual (isomorphic to its dual). In this case, there exists an incidence reversing bijection δ : P ↦ P , which is called a duality of 𝒫. Unlike the case for regular polytopes (where the dualities must preserve the orbit of flags), there are two kinds of self-duality for chiral polytopes. If 𝒫 is chiral with the base flag Φ, then it is said to be properly self-dual if there exists a duality 𝛿 of 𝒫 mapping Φ to a flag Φ δ in the same orbit as Φ under Γ ⁢ ( P ) , or improperly self-dual if the image of Φ under a duality of 𝒫 lies in the other orbit of Γ ⁢ ( P ) .

A polytope is said to be flat whenever all of its facets contain every vertex. For a given directly regular polytope 𝒫 with automorphism group and rotation group

respectively, by [19, Proposition 4E4], we have that 𝒫 is flat if and only if

which is equivalent to

2.3 Coverings

Let 𝒫 and 𝒬 be any two directly regular or chiral polytopes of the same rank, say 𝑑, with Γ + ⁢ ( P ) = ⟨ σ 1 , … , σ d − 1 ⟩ and Γ + ⁢ ( Q ) = ⟨ λ 1 , … , λ d − 1 ⟩ , with respect to their respective base flags Φ and Ψ. From [16, p. 43], we know the polytope 𝒬covers𝒫 if there is a mapping γ : Q ↦ P preserving incidence of faces, ranks of faces and adjacency of flags. If this happens, due to the flag-connectedness of 𝒫, the mapping 𝛾 is necessarily surjective, and we call 𝛾 a covering. On the other hand, we say the group Γ + ⁢ ( Q ) covers Γ + ⁢ ( P ) if there is a well-defined homomorphism α : Γ + ⁢ ( Q ) ↦ Γ + ⁢ ( P ) that maps λ i to σ i for i = 1 , … , d − 1 (an analogue of the definition given in [10]).

Then, due to the correspondence between directly regular or chiral polytopes and their rotation groups, if Γ + ⁢ ( Q ) covers Γ + ⁢ ( P ) , the homomorphism 𝛼 that maps λ i to σ i ( 1 ≤ i ≤ d − 1 ) naturally induces a covering γ : Q ↦ P mapping Ψ to Φ, indicating that 𝒬 covers 𝒫. However, the converse is not true, as there may be a covering from 𝒬 to 𝒫 that maps Ψ to Φ 0 , but not a group homomorphism from Γ + ⁢ ( Q ) to Γ + ⁢ ( P ) mapping τ i to σ i for i = 1 , … , d − 1 . This is the case for the two enantiomorphic forms of the same chiral polytope.

As mentioned, if 𝒫 and 𝒬 are polytopes of the same rank 𝑑 that are either chiral or directly regular, then their rotation groups are both quotients of the orientation-preserving subgroup W + of the rank 𝑑 Coxeter group [ ∞ , ∞ , … , ∞ ] . This group W + is generated by d − 1 elements y 1 , y 2 , … , y d − 1 , subject to the defining relations ( y i ⁢ y i + 1 ⁢ ⋯ ⁢ y j ) 2 = 1 for 1 ≤ i < j ≤ d − 1 . Let 𝐽 and 𝐾 be the corresponding kernels, with Γ + ⁢ ( P ) ≅ W + / J and Γ + ⁢ ( Q ) ≅ W + / K . It is easy to see that if K ≤ J , one has

and there is a natural homomorphism from Γ + ⁢ ( Q ) to Γ + ⁢ ( P ) that maps y i ⁢ K to y i ⁢ J for i = 1 , 2 , … , d − 1 . This homomorphism induces a covering from 𝒬 to 𝒫 mapping the base flag of 𝒬 to the base flag of 𝒫; hence the polytope 𝒬 covers 𝒫. Actually, our construction for chiral covers will be given in this way. In that case, we call the quotient J / K the covering group, or the group of covering transformations. In particular, the polytope 𝒬 is said to be an abelian cover of 𝒫 if J / K is abelian, and a Boolean cover of 𝒫 if J / K is a covering group of exponent 2. There is no doubt that a Boolean cover must be an abelian cover.

2.4 Group theory

In this subsection, we recall some concepts on group theory that we will use later.

Let 𝐺 be a group and let a , b ∈ G . The product a ⁢ b ⁢ a − 1 ⁢ b − 1 is called the commutator of 𝑎 and 𝑏, denoted by [ a , b ] = a ⁢ b ⁢ a − 1 ⁢ b − 1 . The subgroup G ′ of 𝐺, generated by all commutators of elements of 𝐺, is a normal subgroup called the commutator (or derived) subgroup of 𝐺. The quotient group G / G ′ is an abelian group. Moreover, if 𝑁 is a normal subgroup of 𝐺 and G / N is abelian, then G ′ ≤ N .

Assume that G ′ is the commutator subgroup of 𝐺, and let G ′′ be the commutator subgroup of G ′ , etc. If the sequence of subgroups

of 𝐺 contains the trivial subgroup, then the group 𝐺 is called solvable. There are many special solvable groups, e.g. any finite 𝑝-group, abelian group, and group of odd order (see [13]).