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**Chapter 4 Abelian covers of chiral polytopes**

In this chapter, we introduce a new method for constructing chiral polytopes, as covers of a given ‘base’ example, with a covering group that is abelian, and sometimes cyclic. This method is similar to the ‘mixing’ approach of [2], in that it produces chiral polytopes of the same rank as the given one, but with different type and larger automorphism group. On the other hand, it can produce an infinite family of chiral polytopes from a given one, with the sizes of members of the family growing linearly with one (or more) of the parameters making up its ‘type’ (Schläfli symbol). The constructions of these families of chiral polytopes and their groups can be easily described. We illustrate and apply this method in the construction of several new infinite families of chiral polytopes of ranks 3 to 6.

**Cyclic covers of chiral polytopes**

We now introduce an approach for constructing covers of chiral polytopes with cyclic covering group. The main idea is to take a chiral d-polytope P of type {k_{1},k_{2},…,k_{d} _{−1}} and construct an infinite family {Q^{(n)} : n = 1,2,3,…} of chiral polytopes of the same rank d, such that each member Q^{(n)} of this family has almost the same type as P, with just one of the k_{i} replaced by nk_{i}. For example, below we will exhibit such a family of chiral 4-polytopes having type {3n,6,9} for every positive integer n, and covering a particular chiral 4-polytope of type {3,6,9}.

Our approach is based on the following key theorem:

Theorem 4.1.1 Let U be a group generated by d − 1 elements x_{1},x_{2},…,x_{d−1}, with the property that for some ℓ ∈ {1,2,…,d −1}, the following hold :

(a) (x_{i}x_{i+1} …x _{j})^{2} = 1 for 1 ≤ i < j < d,

x_{i} has finite order k_{i} ≥ 3 for all i ̸= ℓ, while x_{ℓ} has infinite order, x_{ℓ}^{k}^{ℓ} generates a cyclic normal subgroup N of U, for some integer k_{ℓ} ≥ 3, the intersection of N with the subgroup generated by all the x_{i} other than x_{ℓ} is trivial, the images of the generators x_{1},x_{2},…,x_{d−1} in the factor group U/N = U/ x_{ℓ}^{k}^{ℓ} satisfy the intersection condition (2.4), and make U/N the automorphism group of a chiral d-polytope P of type {k_{1},k_{2},…,k_{d−1}}.

Then for every positive integer n, the factor group U^{(n)} = U/ x_{ℓ}^{nk}^{ℓ} is the automorphism group of a chiral d-polytope Q^{(n)} of type {k_{1},…,k_{ℓ−1},nk_{ℓ},k_{ℓ+1},…,k_{d−1}}, covering the chiral polytope Q^{(1)} ^{∼} P. Moreover, if P is flat, then so is Q^{(n)} for all n, and if P is tight,

= then so is Q^{(n)} for all n.

Proof. First, note that N^{(n)} = x_{ℓ}^{nk}^{ℓ} is the only subgroup of index n in the cyclic group N = x_{ℓ}^{k}^{ℓ} , and so is characteristic in N and hence normal in U. Also N^{(n)} intersects trivially the subgroup generated by {x_{i} | i ̸= ℓ}, by (b), and therefore the images of x_{1},…,x_{ℓ−1},x_{ℓ},

intersection condition (2.4). To do this, let I = {1,2,…i} and J = { j, j + 1,…,k} where 1 ≤ i < k and 2 ≤ j ≤ k < d, and then define A = x¯_{r} : r ∈ I and B = x¯_{s} : s ∈ J and C = x¯_{t} : t ∈ I ∩J , and also U = U^{(n)} = U/N^{(n)} and N = N/N^{(n)}. The intersection condition (2.4) requires A ∩ B = C, but as usual, it is easy to see that A ∩ B contains C, and hence all that we need to do is prove the reverse inclusion. Next, if we let N = N/N^{(n)}, then satisfy the intersection condition (2.4), and therefore A ∩ B ⊆ CN. Now if ℓ ∈ I ∩ J, then N = x¯_{ℓ}^{k}^{ℓ} ⊆ x¯_{ℓ} ⊆ x¯_{t} : t ∈ I ∩ J = C, so A ∩ B ⊆ CN = C; while on the other hand, if ̸ I∩J, then by (d) either x_{r} : r ∈ I or x_{s} : s ∈ J intersects N trivially, so A ∩N = 0/ or

B ∩N = 0,/ and therefore (A ∩B) ∩N = 0,/ so A ∩B ⊆ C.

Hence the intersection condition (2.4) is satisfied, making U^{(n)} the orientation-preserving subgroup of the automorphism group of a chiral or directly regular d-polytope Q^{(n)} of type {k_{1},…,k_{ℓ−1},nk_{ℓ},k_{ℓ+1},…,k_{d−1}}.

In fact, Q^{(n)} is chiral. For suppose the contrary, namely that Q^{(n)} is directly regular.

Then there exists an automorphism θ of the group U = U^{(n)} that takes (x¯_{1},x¯_{2},x¯_{3},…,x¯_{d−1}) to (x¯_{1}^{−1},x¯_{1}x¯_{2}^{−1}x¯_{1}^{−1},x¯_{3},…,x¯_{d−1}). Now if ℓ ̸= 2, then θ takes x¯_{ℓ} to x¯_{ℓ}^{±1} and so preserves x¯_{ℓ} = N, while on the other hand if ℓ = 2, then θ takes x¯_{ℓ} to x¯_{1}x¯_{2}^{−1}x¯_{1}^{−1} = x¯_{1}x¯_{ℓ}^{−1}x¯_{1}^{−1}, which generates N (because N is cyclic and normal in U), and so again θ preserves N. But then θ induces an analogous automorphism of U/N = U/N = U^{(1)}, making the polytope Q^{(1)} = P reflexible, a contradiction.

Finally, we consider flatness and tightness. If P is flat, then U^{(1)} = U/N is expressible as the product of the images of x_{1},x_{2},…,x_{d−2} and x_{2},x_{3},…,x_{d−1} . When we move from P to its cover Q^{(n)}, the analogues of these two subgroups are x¯_{1},x¯_{2},…,x¯_{d−2} and x¯_{2},x¯_{3},…,x¯_{d−1} . At least one of these contains x¯_{ℓ} and hence contains N, so their product must be U^{(n)}. Thus Q^{(n)} is also flat. Also if P is tight, then |U^{(1)}| = |Γ(P)| = k_{1}k_{2} …k_{d−1}, and so |Γ(Q^{(n)})| = |U^{(n)}| = |U/N^{(n)}| = |U/N^{(1)}||N^{(1)}/N^{(n)}| = |U^{(1)}|n = nk_{1}k_{2} …k_{d−1}, which is the product of the entries of the Schläfli type {k_{1},…,k_{ℓ−1},nk_{ℓ},k_{ℓ+1},…,k_{d−1}} of Q^{(n)}, and so Q^{(n)} is tight as well.

As our first application of this theorem, we have the following:

Example 4.1.2 An infinite family of chiral 4-polytopes of type {3n,6,9}

To construct this family, take U as the group with presentation u,v,w | (uv)^{2} = (vw)^{2} = (uvw)^{2} = v^{6} = w^{9} = (v^{−1}u^{2})^{2} = [w,u^{3}] = vw^{2}v^{−3}w^{−1} = 1 .

Note that (v^{−1}u^{2})^{2} = 1 can be rewritten as 1 = v^{−1}u^{3}u^{−1}v^{−1}u^{2} = v^{−1}u^{3}vuu^{2} = v^{−1}u^{3}vu^{3}, which implies that v^{−1}u^{3}v = u^{−3}, and it follows that the cyclic subgroup N generated by u^{3} is centralised by u and w and normalised by v. Adding the relation u^{3} = 1 gives the quotient U/N, which by a relatively easy calculation in MAGMA [1] is a group of order 486, and is the automorphism group of the mirror image P of the chiral 4-polytope of type {3,6,9} with 486 automorphisms listed at [11]. Moreover, the Reidemeister-Schreier process [33, 36], implemented via the Rewrite command in MAGMA, shows that the subgroup N is infinite cyclic. Hence the hypotheses (a), (b), (c) and (e) in the above theorem are satisfied, for (x_{1},x_{2},x_{3}) = (u,v,w).

But also v and w satisfy the relations (vw)^{2} = v^{6} = w^{9} = vw^{2}v^{−3}w^{−1} = 1, which by another MAGMA computation define a group of order 54. Moreover, in the factor group U/N of order 486, the image of the subgroup generated by v and w has order 54, and has trivial intersection with the image of the cyclic subgroup generated by u. (Indeed U/N is the complementary product of the images of u and v,w,u^{−1}wu , which have orders 3 and 162 respectively.) Then since the order of the subgroup generated by v and w in U cannot be greater than 54, it follows that the intersection u ∩ v,w is trivial in U, and therefore x_{1}^{3} ∩ x_{2},x_{3} = u^{3} ∩ v,w = N ∩ v,w is trivial as well, so (d) holds too.

Hence by Theorem 4.1.1 we obtain an infinite family {Q^{(n)} : n = 1,2,3,…} of chiral 4-polytopes of type {3n,6,9}, as indicated earlier.

The ‘base’ polytope P = Q^{(1)} has 9 vertices, 27 edges, 81 2-faces and 9 facets (and 972 flags), with each facet F being a directly regular 3-polytope of type {3,6} having full automorphism group of order 108, and each vertex-figure V being a chiral 3-polytope of type {6,9} having automorphism group of order 54 (isomorphic to v,w ). It follows that Q^{(n)} has 9n vertices, 27n edges, 81 2-faces, 9 facets and 972n flags. Moreover, each facet F^{(n)} of Q^{(n)} is a directly regular 3-polytope of type {3n,6}, with 9n vertices, 27n edges and 18 faces, and full automorphism group of order 108n, while each vertex-figure V ^{(n)} is isomorphic to the same tight chiral 3-polytope of type {6,9} as for P, with 6 vertices, 27 edges and 9 faces, and automorphism group of order 54.

**Abelian coverings**

In this section, the above approach will be extended to produce families of covers over chiral polytopes for which the covering group is abelian but not necessarily cyclic. We use the following:

Theorem 4.2.1 Let U be a group generated by d − 1 elements x_{1},x_{2},…,x_{d−1}, with the property that for some subset L of {1,2,…,d −1}, the following hold :

(x_{i}x_{i+1} …x _{j})^{2} = 1 for 1 ≤ i < j < d,

x_{i} has finite order k_{i} ≥ 3 for all i ̸∈L, while x_{ℓ} has infinite order for all ℓ ∈ L, for all ℓ ∈ L, there exists an integer k_{ℓ} ≥ 3 such that x_{ℓ}^{k}^{ℓ} generates a cyclic normal subgroup N_{ℓ} of U that intersects x_{i} : i ̸= ℓ trivially, the normal subgroup N = x_{ℓ}^{k}^{ℓ} : ℓ ∈ L = ∏_{ℓ}_{∈}_{L} N_{ℓ} intersects x_{i} : i ̸∈L trivially,

the images of the generators x_{1},x_{2},…,x_{d−1} in U/N satisfy the intersection condi-tion (2.4), and make U/N the automorphism group of a chiral d-polytope P of type {k_{1},k_{2},…,k_{d−1}}.

Then for every indexed sequence S_{L} = (n_{ℓ})_{ℓ}_{∈}_{L} of positive integers, the factor group U^{(S}L^{)} = U/ x_{ℓ}^{n}^{ℓ}^{k}^{ℓ} : ℓ ∈ L is the automorphism group of a chiral d-polytope Q^{(S}L^{)} that covers P and has type {s_{1},s_{2},…,s_{d−1}}, where s_{i} = k_{i} for all i ̸∈L and s_{ℓ} = n_{ℓ}k_{ℓ} for all ℓ ∈ L, and the covering group for Q^{(S}^{L}^{)} over P is isomorphic to the abelian group ∏_{ℓ}_{∈}_{L} C_{n}_{ℓ} .

Moreover, if P is flat, then so is Q^{(S}L^{)}, and if P is tight, then so is Q^{(S}L^{)}. Also if P is properly (resp. improperly) self-dual, then Q^{(S}L^{)} is properly (resp. improperly) self-dual if and only if d −ℓ ∈ L and n_{d−ℓ} = n_{ℓ} whenever ℓ ∈ L.

Proof. Most of this follows easily from Theorem 4.1.1, by induction on |L|. Note that N = x_{ℓ}^{k}^{ℓ} : ℓ ∈ L is the product of the normal subgroups N_{ℓ} = x_{ℓ}^{k}^{ℓ} for ℓ ∈ L, which are cyclic and have trivial pairwise intersections, and hence N is abelian. In turn, this implies.

For the final claims about duality, necessity follows from the fact that the type of the dual of an equivelar polytope is the reverse of the given type, while sufficiency can be proved by showing that when d − ℓ ∈ L and n_{d−ℓ} = n_{ℓ} whenever ℓ ∈ L, any duality of P can be extended to a duality of Q^{(S}L^{)}.

As an application of this more general theorem, we have the following:

Example 4.2.2 An infinite family of chiral 3-polytopes of type {4m,4n}

To construct this family, take U as the group with presentation u,v | (uv)^{2} = (v^{−1}u^{3})^{2} = (u^{−1}v^{3})^{2} = uv^{−1}uv^{−1}u^{2}v^{2}u^{−2}v^{−2} = 1 .

Note that (v^{−1}u^{3})^{2} = 1 can be rewritten as 1 = v^{−1}u^{4}v^{−1}v^{−1}u^{3} = v^{−1}u^{4}vuu^{3} = v^{−1}u^{4}vu^{4}, which implies that v^{−1}u^{4}v = u^{−4}, and hence that the cyclic subgroup generated by u^{4} is centralised by u and normalised by v. Similarly, the relation (u^{−1}v^{3})^{2} = 1 implies that u^{−1}v^{4}u = v^{−4}, and hence that the cyclic subgroup generated by v^{4} is normalised by u and centralised by v. Thus N = u^{4},v^{4} is normal in U. Adding the relations u^{4} = v^{4} = 1 gives the quotient U/N, which by a calculation in MAGMA is a group of order 80, and is the automorphism group of an improperly self-dual chiral 3-polytope of type {4,4} listed in [11]. In particular, the intersection of the images of u and v in U/N is trivial. Moreover, the Reidemeister-Schreier process shows that the subgroup N is free abelian of rank 2 (with just a single defining relation [u^{4},v^{4}] = 1), and it follows that the cyclic subgroups generated by u and v have trivial intersection in U. Hence the hypotheses (a), (b), (c) and (e) in the above theorem are satisfied, for (x_{1},x_{2}) = (u,v). The hypothesis (d) is vacuous in this case.

By Theorem 4.2.1, we obtain for every ordered pair of positive integers m and n a chiral 4-polytope Q^{(m,n)} of type {4m,4n}, with automorphism group of order 80mn. The ‘base’ polytope P = Q^{(1,1)} is the chiral polytope of type {4,4} mentioned above, with 20 vertices, 40 edges, 20 2-faces and automorphism group of order 80, and it follows that the covering polytope Q^{(m,n)} is also improperly self-dual, with 20m vertices, 40mn edges, 20n 2-faces, and 160mn flags.

Before giving more examples, we note that analogues of Theorems 4.1.1 and 4.2.1 can be proved also in finite cases, where the subgroups x_{ℓ}^{k}^{ℓ} are finite of order s_{ℓ} (divisible by k_{ℓ}), and the integer n or n_{ℓ} is restricted to divisors of s_{ℓ}/k_{ℓ}. The proofs are essentially the same, and applications appear in families (2), (14), (19), (22), (23), (26), (27) and (29) to (31) in the next subsection.

**Infinite and finite families**

This section exhibits further applications of Theorems 4.1.1 and 4.2.1, to the construction of new infinite families of chiral polytopes of ranks 3 to 6, and some additional finite families in the rank 6 case.

We use the same notation as above, but for simplicity we write group presentations in the form X |R where X is the generating set and R is the set of defining relators (with a relation of the form w = z written as the relator wz^{−1}. Also we use R(d) as an abbreviation for the set of relators (x_{i}x_{i+1} …x _{j})^{2} for 1 ≤ i < j < d and the implied relators [x_{i},x _{j}] when j −i > 2, and we use the symbols u,v,w,x and y in place of x_{1},x_{2},x_{3},x_{4} and x_{5}.

In each case we give the finitely-presented group U and indicate the relevant normal subgroup N, and then summarise particular properties of the polytopes in the resulting family, but without the kind of detail given in Examples 4.2.2 and 4.1.2.

Chiral 3-polytopes of type {4m, 4n}

U = u,v | R(3),(v^{−1}u^{3})^{2},(u^{−1}v^{3})^{2},uv^{−1}uv^{−1}u^{2}v^{2}u^{−2}v^{−2}

N = u^{4},v^{4} ^{∼} Z⊕Z (free abelian of rank 2), with quotient U/N of order 80

= Q^{(m,n)} has 20m vertices, 40mn edges, 20n 2-faces, and 160mn flags, for all m,n ≥ 1

Q^{(n,n)} is improperly self-dual, for all n ≥ 1.

Chiral 4-polytopes of type {3, 4m, 4n} for m = 1, 2, 3 or 6

U = u,v,w | R(4),v^{24},u^{3},[u,v^{4}],(w^{−1}v^{3})^{2},[u,w^{4}],(v^{−1}w^{3})^{2},uv^{−1}uw^{2}v^{−1}wuw

N = v^{4},w^{4} ^{∼} Z ⊕Z, with quotient U/N of order 480 _{6}

Q^{(m,n)} has 6 vertices, 60m edges, 80mn 2-faces, 20n facets, and 960mn flags, for all n ≥ 1, whenever m ∈ {1,2,3,6}

Each facet is a directly regular 3-polytope of type {3,4m} with 48m automorphisms

Each vertex-figure is a chiral 3-polytope of type {4m,4n} with 80mn automorphisms, isomorphic to the one of type {4m,4n} in family (1) above when m = 1 or 3.

(3) Chiral 4-polytopes of type {3n, 6, 9}

U = u,v,w | R(4),v^{6},w^{9},(v^{−1}u^{2})^{2},[w,u^{3}],vw^{2}v^{−3}w^{−1}

N = u^{3} ^{∼} Z, with quotient U/N of order 486

=Q^{(n)} has 9n vertices, 27n edges, 81 2-faces, 9 facets, and 972n flags, for all n ≥ 1

Each facet is a directly regular 3-polytope of type {3n,6} with 108n automorphisms

Each vertex-figure is a tight chiral 3-polytope of type {6,9} with 54 automorphisms.

(4) Chiral 4-polytopes of type {4, 3n, 6}

U = u,v,w | R(4),u^{4},w^{6},(u^{−1}v^{2})^{2},(w^{−1}v^{2})^{2},(uwv^{−1}w)^{2}

N = v^{3} ^{∼} Z, with quotient U/N of order 576

= Q^{(n)} has 8 vertices, 48n edges, 72n 2-faces, 24 facets, and 1152n flags, for all n ≥ 1

Each facet is a directly regular 3-polytope of type {4,3n} with 48n automorphisms

Each vertex-figure is a directly regular 3-polytope of type {3n,6} with 144n automor-phisms.

(5) Chiral 4-polytopes of type {3m, 4, 6n}

U = u,v,w | R(4),v^{4},(v^{−1}u^{2})^{2},(v^{−1}w)^{4},u^{−1}wvw^{−1}vuw^{−2}

N = u^{3},w^{6} ^{∼} Z⊕Z, with quotient U/N of order 576

= Q^{(m,n)} has 6m vertices, 48m edges, 96n 2-faces, 24n facets, and 1152mn flags, for all m,n ≥ 1

Each facet is a directly regular 3-polytope of type {3m,4} with 48m automorphisms

Each vertex-figure is a directly regular 3-polytope of type {4,6n} with 192n automor-phisms.

(6) Chiral 4-polytopes of type {3m, 8, 3n}

U = u,v,w | R(4),v^{8},(v^{−1}u^{2})^{2},(v^{−1}w^{2})^{2},(w^{−1}v^{3})^{2},uwvuv^{−1}uwv^{−1}wuwv

N = u^{3},w^{3} ^{∼} Z⊕Z, with quotient U/N of order 576

= Q^{(m,n)} has 12m vertices, 96m edges, 96n 2-faces, 12n facets, and 1152mn flags, for all m,n ≥ 1

Each facet is a directly regular 3-polytope of type {3m,8} with 96m automorphisms

Each vertex-figure is a directly regular 3-polytope of type {8,3n} with 96n automor-phisms

Q^{(n,n)} is properly self-dual, for all n ≥ 1.

**Table of contents**

**List of figures **

**List of tables **

**1 Introduction **

**2 Further background**

2.1 Abstract polytopes

2.2 Regular polytopes

2.3 Chiral polytopes

2.4 Construction of regular and chiral polytopes from groups

2.5 Properties of polytopes

**3 The smallest chiral 6-polytopes **

3.1 Preliminaries

3.2 The smallest known examples of given rank

3.3 Chiral 6-polytopes

3.4 Proof of Theorem 3.1.1

**4 Abelian covers of chiral polytopes **

4.1 Cyclic covers of chiral polytopes

4.2 Abelian coverings

4.3 Infinite and finite families

**5 Constructing chiral polytopes as abelian covers of regular polytopes **

5.1 A general approach to finding chiral abelian covers of regular polytopes

5.2 Chiral polytopes of type {4, 4, . . . , 4} .

5.3 Chiral abelian covers of regular polytopes with smaller covering group

**6 Conclusions and future research **

6.1 Contributions

6.2 Future research

References

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