HYPERTOPES WITH TETRAHEDRAL DIAGRAM

In this paper we construct an infinite family of hypertopes of rank four having the complete graph K4 as diagram. Their group of rotational symmetries is isomorphic to PSL(2, q). It turns out some elements of this family are regular hypertopes and some are chiral. Moreover, we show that the chiral ones have both improper and proper correlations simultaneously.


Introduction
Abstract polytopes generalize (the face lattice of) convex polytopes to combinatorial structures.The main interest of the theory of abstract polytopes has been the study of their symmetries.Hence, highly symmetric polytopes (in particular those regular and chiral), together with their automorphism groups, are the most studied ones.A polytope can be regarded as a thin residually connected geometry with linear diagram.The concept of hypertope was introduced in [2] and generalizes the concept of a polytope by dropping the linear condition on the diagram.This generalization was made in such a way that the concept of chirality can be extended to hypertopes.As it is the case with maps and polytopes, we are interested in understanding chiral hypertopes, and finding examples of them can be the first step to achieve this goal.
In the 1970's Branko Grünbaum considered rank 4 polytopes that are locally toroidal, meaning that all their facets and vertex figures are either spherical or toroidal, and not all of them are spherical.In [5] an almost complete answer to Grünbaum's problem, the classification of rank 4 regular locally toroidal polytopes, is given.In this paper we follow [2] and generalize the concept of a polytope to that of a hypertope, and study some interesting locally toroidal ones.
In [3] some examples of chiral hypertopes of rank 4 with certain diagrams are given, satisfying that their residues of rank 3 are either spherical or toroidal.In this paper we continue with the study of locally toroidal 4-hypertopes, in the sense that their residues of rank 3 are toroidal.Using P SL (2, q) as their rotational subgroups, we found an infinite family of hypertopes that contains both regular and chiral hypertopes.Moreover, the chiral ones have the very interesting property that they admit both proper and improper correlations, a feature that is known to be impossible for polytopes (see [4,Lemma 3.1]).
We construct these hypertopes as coset geometries Γ = (G; {G 0 , G 1 , G 2 , G 3 }) where G is P SL(2, q) with q = p or q = p 2 for certain primes p.In these geometries two maximal parabolic subgroups are alternating groups A 4 and the other two are E q : C 3 .Therefore 3 must divide q − 1 for our construction to work, and there is a third root of unity e in the field of order q.In the case q = p 2 we require that e is not in the subfield of order p, which is equivalent to requiring that p ≡ 2 mod 3. We get an infinite family of regular hypertopes when q = p 2 and an infinite family of chiral hypertopes when q = p.More precisely in the case q = p 2 two residues are hypermaps of type (3,3,3) (2,0) and the other two are hypermaps of type (3,3,3) (p,0) ; in the case q = p two residues are hypermaps (3,3,3) (2,0) and the other two are both chiral hypermaps of types (3,3,3) (1,e) and (3,3,3) (e,1) , respectively.
The paper is organised as follows.In Section 2, we give the definitions and notation needed to understand this paper.In Section 3, we construct P SL(2, q), with q ∈ {p, p 2 } and p a prime, as a C + -group.In Section 4, we show that the C + -groups obtained in the previous section give hypertopes.In Section 5 we show that the hypertopes obtained are either regular or chiral.In Section 6, we give a geometric description of the hypertopes we constructed, in terms of objects of the projective line P G(1, q).

Preliminaries
2.1.Incidence geometries.Following [1], an incidence system Γ := (X, * , t, I) is a 4-tuple such that • X is a set whose elements are called the elements of Γ; • I is a set whose elements are called the types of Γ; • t : X → I is a type function, associating to each element x ∈ X a type t(x) ∈ I; • * is a binary relation on X called incidence, that is reflexive, symmetric and such that for all x, y ∈ X, if x * y and t(x) = t(y) then x = y.The incidence graph of Γ is the graph whose vertex set is X and where two vertices are joined provided the corresponding elements of Γ are incident.A flag is a clique of the incidence graph of Γ.The type of a flag F is {t(x) : x ∈ F }. A chamber is a flag of type I.
An element x is incident to a flag F and we write x * F for that, when x is incident to all elements of F .An incidence system Γ is a geometry or incidence geometry if every flag of Γ is contained in a chamber.The rank of Γ is the number of types of Γ, namely the cardinality of I.
Let Γ := (X, * , t, I) be an incidence geometry and F a flag of Γ.The residue of F in Γ is the incidence geometry Γ F := (X F , * F , t F , I F ) where • t F and * F are the restrictions of t and * to X F and I F .An incidence system Γ is connected if its incidence graph is connected; Γ is residually connected when each residue of rank at least two of Γ (including itself) has a connected incidence graph; Γ is called thin (resp.firm) when every residue of rank one of Γ contains exactly two (resp.at least two) elements.As in [2], we say that a hypertope is a thin incidence geometry which is residually connected.
Note that α induces a bijection on I.When t(x) = i we say that x is an element of type i, or equivalently, that x is an i-element.The set of automorphisms of Γ is a group denoted by Aut(Γ).
An automorphism α of Γ is called type preserving when for each x ∈ X, t(α(x)) = t(x) (i.e.α maps each element on an element of the same type).The set of type-preserving automorphisms of Γ is a group denoted by Aut I (Γ) and obviously Aut I (Γ) ≤ Aut(Γ).
A correlation is a non-type-preserving automorphism, that is an element of Aut(Γ) \ Aut I (Γ).A duality is correlation that induces an involutory permutation on I.
An incidence geometry Γ is chamber-transitive if Aut I (Γ) is transitive on all chambers of Γ.Finally, an incidence geometry Γ is regular if Aut I (Γ) acts regularly on the chambers, that is, the action is semi-regular (free) and transitive.
Observe that chamber-transitivity implies flag-transitivity, that is, for each J ⊆ I, there is a unique orbits on the flags of type J under the action of Aut(Γ).
Sometimes, when Aut(Γ) is not transitive on the chambers of Γ, it has two orbits.We are also interested in those hypertopes having two orbits with an extra condition.Two chambers C and C of a thin incidence geometry of rank r are called i-adjacent if C and C differ only in their i-elements.We then denote C by C i .Let Γ(X, * , t, I) be a thin incidence geometry.We say that Γ is chiral if Aut I (Γ) has two orbits on the chambers of Γ such that any two adjacent chambers lie in distinct orbits.Moreover, if Γ is residually connected, we call Γ a chiral hypertope.
When Γ is a chiral hypertope, if Aut(Γ) = Aut I (Γ), correlations may either interchange the two orbits or preserve them.A correlation that interchanges the two orbits is said to be improper and a correlation that preserves them is said to be proper.
The following proposition shows how to construct an incident geometry starting from a group.Proposition 2.1.(Tits Algorithm, 1956) [7] Let n be a positive integer and I := {1, . . ., n}.Let G be a group together with a family of subgroups (G i ) i∈I , X the set consisting of all cosets G i g with g ∈ G and i ∈ I, and t : X → I defined by t(G i g) = i.Define an incidence relation * on X × X by : Then the 4-tuple Γ := (X, * , t, I) is an incidence system having a chamber.Moreover, the group G acts by right multiplication as an automorphism group on Γ.Finally, the group G is transitive on the flags of rank less than 3.
When a geometry Γ is constructed using the proposition above, we denote it by Γ(G; (G i ) i∈I ) and call it a coset geometry.The subgroups (G i ) i∈I are called the maximal parabolic subgroups.
If the pair (G + , R) satisfies the following condition called the intersection property IP + , we say that (G + , R) is a C + -group.
for all J, K ⊆ I, with |J|, |K| ≥ 2. The following construction produces an incidence system from a C + -group.
It is convenient to represent (G + , R) by a graph B with r vertices which we call the B-diagram of (G + , R).The vertex set of B is the set {α 0 , . . ., α r−1 }.The edges {α i , α j } of this graph are labelled by o(α We take the convention of dropping an edge if its label is 2 and of not writing the label if it is 3. Vertices of B are represented by small circles.Finally, we sometimes attach to each vertex α i the corresponding subgroup G i defined in Construction 2.1.
Observe that, thanks to Proposition 2.1 that ensures that G + is transitive on the flags of rank less than 3, we not only know the number of elements of type i for every i, that is the index of G i in G + , but also the number of elements of type j incident to a given element of type i, that is the index of The coset geometry Γ(G + , R) gives an incidence system using Proposition 2.1.In what follows we prove that any such coset geometry has a connected incidence graph if its rank is at least 3. Proposition 2.2.If |R| ≥ 2, then Γ(G + , R) has a connected incidence graph.
Proof.As for any g ∈ G + , {G i g | i ∈ I} is a set of mutually incident elements of Γ, it is sufficient to prove that G i and G i g are in the same connected component of the incidence graph (of Γ) for every g ∈ G + and every i ∈ I.
Although the incidence geometry Γ has a connected incidence graph, it need not be residually connected.Moreover, Γ might not be a thin geometry, and hence Γ need not be a hypertope.Furthermore, in general Γ might not be transitive on flags of rank 3, and it might have many orbits of chambers.In the construction we shall give in the next section, the geometry that we obtain will in fact be a hypertope and have only two orbits of chambers.The following theorem will help us to decide if a hypertope is regular of chiral.
R) be the coset geometry associated to (G + , R) using Construction 2.1.If Γ is a hypertope and G + has two orbits on the set of chambers of Γ, then Γ is chiral if and only if there is no automorphism of G + that inverts all the elements of R. Otherwise, there exists an automorphism σ ∈ Aut(G + ) that inverts all the elements of R and the group G + extended by σ is regular on Γ.
The rotation subgroup G of the automorphism group of a rank three toroidal hypermap is as follows for some integers a and b. (2.1) x, y This hypermap is denoted by (3, 3, 3) (a,b) .The above presentation readily shows that such a hypermap has a B-diagram that is a triangle with no numbers on the edges.
2.4.P SL(2, q) acting on the projective line.For a prime power q = p n let GF (q) * be the multiplicative group of the Galois field GF (q) on q elements.A primitive element of GF (q) is a generator of the multiplicative group GF (q) * .For any positive integer k < q, a solution of x k = 1 which is not a solution of x j = 1 for any j ∈ {1, . . ., k − 1} is called a kth primitive root of unity.If i is a kth primitive root of unity, then k divides q − 1 and Let V be the 2-dimensional vector space GF (q) 2 over GF (q).Consider the relation ∼ in V \ {(0, 0)} defined as follows.
The projective line P G(1, q) is the set of equivalence classes V \ {(0, 0)}/ ∼.The elements of the projective line [x 0 , x 1 ] can be identified with their non-homogeneous coordinates by the following bijection where i is a (q − 1)th root of unity.
Consider the special linear group SL(2, q) = {A ∈ GL(2, q) | det(A) = 1} and denote by Id its identity matrix.As P SL(2, q) = SL(2, q)/Z where Z = {±Id}, the elements of P SL(2, q) can be seen as unordered pairs ±A with A ∈ SL(2, q).Observe that in characteristic 2, |Z| = 1.For convenience, we shall denote P SL(2, q) by G + .An element of G + will be given by one of its two representative elements in SL(2, q).Consequently equalities are to be taken modulo Z = {±Id}.
As −Az = Az , ϕ is well-defined and gives an action of G + on P G(1, q).
Lemma 2.4.[6, Lemma 5.3] Let q be a power of a prime.Then SL(2, q) is generated by the elementary matrices of the form 3. A C + -group for P SL(2, q) Let q ∈ {p, p 2 } for a prime number p with 3 being a divisor of q − 1, and let e ∈ GF (q) be a third primitive root of unity.In the case q = p 2 we also assume that 3 is not a divisor of p − 1 so that In this section we prove that P SL(2, q) is a C + -group with the B-diagram of Figure 1 (where the labels of the corners are the groups corresponding to each rank 3 residue).
Proof.We first observe that, for each i ∈ {2, 3}, {α 1 , α i } is an independent set, which is sufficient to guarantee that it gives a set of generators of a C + -group.Now let us compute the order of the elements that give the B-diagram.It is straightforward to see that α 1 has order 3. To compute the order of the other elements we use the fact that e is a third root of unity and hence e 2 + e + 1 = 0. Therefore the following equalities hold for any c ∈ GF (q).In particular, this shows that both α 2 and α 3 have order 3.In addition, e 0 e e 2 and α −1 (1, 0, −e 2 , ∞) (1, ∞, 0, −e) (1, −e, ∞, 0) (1, 0, −e, ∞) hence these elements have also order 3.
To prove that the groups are isomorphic to A 4 we first observe that for i ∈ {2, 3} 1) we conclude that G i is isomorphic to A 4 , the rotational subgroup of the hypermap of type (3, 3, 3) (2,0) .From the given enumeration of the elements of these two groups we conclude that their intersection is α 1 .
Consider the following subgroups of P SL(2, q): When q = p, H 0 and H 1 are cyclic groups of order p.More precisely, When q = p 2 , H 0 and H 1 are elementary abelian groups of order p 2 .More precisely, since we assumed that 3 does not divide p − 1 and therefore GF (q) = {a + eb : a, b ∈ GF (p)}.
We claim that G 0 ∼ = H 0 : E and G 1 ∼ = H 1 : E .The following equalities prove that H 0 and H 1 are subgroups of G 0 and G 1 , respectively. (3.1) On the other hand, as e 0 0 e 2 and α −1 With this we have that the groups G 1 and G 0 are both isomorphic to E q : E .
Since an element of G 0 is equal to e i 0 c e 2i for some c ∈ GF (q) and i ∈ {0, 1, 2}, while an element of G 1 is equal to e j b 0 e 2j for some b ∈ GF (q) and j ∈ {0, 1, 2}, we have that ) is a C + -group with tetrahedral B-diagram as in Figure 1.
Proof.By Lemmas 3.1 and 3.2 we have that We have that G 0 is a subgroup of the stabilizer of 0 in G + and G 1 is a subgroup of the stabilizer of ∞ in G + .Figure 2 shows that the only elements of G 2 fixing 0 are the identity, α 2  3 α 1 and α The same figure shows that the only elements of G 2 fixing ∞ are the q q q q q q q q q q q q α −1 1 α2 w w w w w w w w w w w w identity, α 3 and α 2 3 , thus G 1 ∩ G 2 = α 3 .Figure 3 shows that the only elements of G 3 fixing 0 are the identity, α 2 2 α 1 and The same figure shows that the only elements of G 3 fixing ∞ are the identity, α 2 and α 2 2 , thus G 1 ∩ G 3 = α 2 .From this we conclude that (G + , {α 1 , α 2 , α 3 }) is a C + -group.Moreover, as G i ∩ G j are cyclic groups of order three, for i, j ∈ {0, 1, 2, 3} with (i = j), the B-diagram is a complete graph K 4 with edges labelled 3.
Lemma 3.4.The group G + is isomorphic to P SL(2, q).Proof.This follows from the description of the subgroups H 0 and H 1 in Lemma 3.2 and Lemma 2.4.

A hypertope for P SL(2, q)
As pointed out in [2], not every coset geometry is a hypertope.In this section we shall see that the coset geometry Γ := Γ(G + , R) := Γ(G + ; (G i ) i∈I ) where I = {0, 1, 2, 3}, G + is isomorphic to P SL(2, q) and the G i 's are the groups defined in Section 3, is a hypertope.Since Γ is a coset geometry, then we only have to show that it is thin and residually connected.
We observe that Γ has a lot of symmetry, and we shall use such symmetry to show that it is a hypertope.Note that since G + is a C + -group, then G 1 ∩ G 2 ∩ G 3 is trivial.This implies that the action of G + on the flags by right multiplication is free (semi-regular) on the chambers.Moreover, as Γ is a coset geometry, it is transitive on the flags of rank less than 3.We shall show that in fact Γ is transitive on the flags of rank 3. To show this, in the following two lemmas we analyze some incidences among elements of Γ, and some incidences among elements and flags of Γ.
From the B-diagram of G + , since G i with i ∈ {2, 3} is isomorphic to A 4 and G i ∩ G j ∼ = C 3 , we know that each element of type i ∈ {2, 3} is incident to exactly four elements of type j = i.Similarly, since G i with i ∈ {0, 1} is isomorphic to E q : C 3 , we have that each element of type i ∈ {0, 1} is incident to exactly q elements of type j = i.Remark also that, if for some t ∈ T .More precisely we have the following lemma.Lemma 4.1.Let i, j ∈ I with i = j and let g ∈ G + .Then and {i, j, k} = {0, 2, 3}.
Proof.It can be easily checked that all of the cosets in the set on the right side of the equivalences are incident to the respective G j .Suppose that j ∈ {0, 1}.By Lemma 3.2, H j is a subgroup of G j of order q.
In all the cases G i ∩ G j is generated by an element of order 3 which does not belong to H j , therefore all the intersections (G i ∩ G j ) ∩ H j are trivial, and so g = h.
For the cases when j ∈ {2, 3}, by Lemma 3.3 (see also Figure 4 we have the following: Proposition 4.2.Let i, j, k ∈ I be distinct and let g ∈ G + .Then Proof.Obviously the right hand side of the equivalence implies the left one.The proof of the converse will be divided in cases covering all the possibilities for the set {j, k}.Assume that G i g * {G j , G k }.
Case 4. If {j, k} = {2, 3}, then by Lemma 4.1 we have the following: We are now ready to show that G + is transitive on the flags of rank 3.
Proposition 4.3.Let J = {i, j, k} ⊂ {0, 1, 2, 3} with i, j and k all distinct.The action of G + on the flags of type J is transitive.
To show that Γ is a thin geometry, we need to show that every residue of rank 1 has exactly two elements.A residue of rank 1 is the set of elements of Γ that are incident to a flag of type J ⊂ {0, 1, 2, 3}, with |J| = 3.By Proposition 4.3, all residues of rank 1 corresponding to flags of a given type J are isomorphic.Hence, it is enough to show that, for {i, j, k, l} = {0, 1, 2, 3}, the sets {G j g | G j g * {G i , G k , G l }} all have cardinality two.
Proof.Following the above discussion, we need to show that four sets have cardinality two.To do this, we shall show that: It is straightforward to see that the elements on the sets of the right are incident to the flags on the left.
In order to show that the geometry Γ is a hypertope, we need to show that it is residually connected.That is, we need to show that each residue of rank at least two has a connected incidence graph.We already showed that all the residues of rank 3 are hypermaps of type (3, 3, 3) (Lemmas 3.1 and 3.2).Since we know that all such hypermaps are residually connected, as observed in [2], that will conclude the proof.More precisely, we have the following theorem.Proof.As pointed out before, Γ is a geometry, and by Proposition 4.4, it is thin.By Lemmas 3.1 and 3.2, its rank 3 redidues Γ i are toroidal hypertopes isomorphic to the coset geometries Γ(G i , (G i ∩ G j ) j∈I\{i} ).This, together with Proposition 2.2, implies that Γ is residually connected.Therefore Γ is a hypertope.4.1.The residues of rank 3. The residues of rank three of Γ are hypermaps of type (3, 3, 3) (a,b) for some a and b that depend on the type of the residue.To determine the vector (a, b) of the rank 3 residues of Γ we now fix an order for the generators of each G i .Let the first generator of G i be the one with minimal label when i = 0 and let the first generator of G 0 be α −1 1 α 2 .Lemma 4.6.The residues Γ i with i ∈ {2, 3} are regular hypermaps of type (3, 3, 3) (0,2) .When q = p 2 the residues Γ i with i ∈ {0, 1} are regular hypermaps of type (3, 3, 3) (0,p) .When q = p the residue Γ 0 is the chiral hypermap of type (3,3,3) (1,e) while the residue Γ 1 is the chiral hypermap of type (3, 3, 3) (e, 1) , where e is a third root of unity (which is an integer, as q = p).

The symmetries of Γ
As pointed out in the introduction, the hypertopes we have constructed in this paper are highly symmetric.Recall that we have two families: a family of hypertopes Γ (p) = Γ(G + , R) with G + = P SL(2, p) for any prime number p satisfying p ≡ 1 mod 3, and a family of hypertopes Γ (p 2 ) = Γ(G + , R) with G + = P SL(2, p 2 ) for any prime p satisfying p ≡ 2 mod 3, where R = {α 1 , α 2 , α 3 } for both families.In this section we shall show that all hypertopes Γ (p) are chiral, while all hypertopes Γ (p 2 ) are regular.Furthermore, we study the correlations of the hypertopes and find that all the chiral hypertopes Γ (p) have both proper and improper correlations.For any Γ in one of these two families we have the following lemma.which fixes α 1 and interchanges α 2 and α 3 (since p ≡ 2 mod 3 and therefore e p = e 2 in that case).Let ψ be the automorphism of G + given by conjugation with Then ψϕ is an automorphism of G + inverting α 1 , α 2 and α 3 .Hence, by Theorem 2.3 Γ (p 2 ) is a regular hypertope.
For the chiral hypertopes Γ (p) we will now exhibit proper and improper correlations.In order to give a proper correlation of Γ (p) , we remark the following general fact.Given a coset geometry Γ := Γ(G, (G i ) i∈I ), any ϕ ∈ Aut(G) satisfying {ϕ(G i ) | i ∈ I} = {G i | i ∈ I} gives rise to an automorphism of Γ mapping G i g to ϕ(G i g) = ϕ(G i )ϕ(g).In particular, we have the following lemma.gives rise to a proper duality of Γ (p) .
To give an improper correlation of Γ (p) , we extend the assignment Note that A is an involution and that α A . Therefore µ(G 0 ) A = α 1 G 0 and µ(G i ) A = G i for any i ∈ {1, 2, 3}.Then, for any j ∈ I we have showing that µ is well-defined and a bijection.To prove that µ is a correlation we need the following lemma.
Proof.By definition µ sends the chamber {G 0 , G 1 , G 2 , G 3 } to the adjacent chamber {G 0 α −1 1 , G 1 , G 2 , G 3 }.Hence we have only to prove that µ is a correlation.It is enough to prove that for any g ∈ G + and any i, j ∈ I with i > j µ(G i g) * µ(G j ) ⇔ G i g * G j .
According to the definition of µ and Lemma 4.1, we have

Lemma 5 . 1 . 4 . 5 . 2 .
G + acts with two orbits on chambers of Γ with adjacent chambers in different orbits.Proof.This is a consequence of Propositions 4.3 and 4.Lemma Any hypertope Γ (p) is chiral while any hypertope Γ (p 2 ) is regular.Proof.By Lemma 4.6, one residue of Γ (p) is chiral.Hence Γ (p) is itself chiral, indeed by Theorem 2.3 the residues of a regular hypertope must all be regular.For Γ (p 2 ) we have that G + = P SL(2, p 2 ).Then, the Frobenius automorphism of GF (p 2 ) gives rise to the involutory automorphism ϕ : G + → G + , a b c d → a p b p c p d p

Lemma 5 . 3 .
The automorphism η of G + = P SL(2, p) defined by conjugation with the involution bijection of the set of elements of Γ (p) by settingµ(G i g) := µ(G i )g A ,where A