/^creative ^commor ARS MATHEMATICA CONTEMPORANEA Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 8 (2015) 69-82 Tight orientably-regular polytopes Marston Conder University of Auckland Auckland 1142, New Zealand Gabe Cunningham University ofMassachusetts Boston Boston, Massachusetts 02125, USA Received 9 October 2013, accepted 9 February 2014, published online 7 May 2014 Abstract It is known that every equivelar abstract polytope of type {p1,... ,pn-1} has at least 2p1 • • • pn-1 flags. Polytopes that attain this lower bound are called tight. Here we investigate the conditions under which there is a tight orientably-regular polytope of type {p1,... ,pn-1}. We show that it is necessary and sufficient that whenever pi is odd, both pi-1 and pi+1 (when defined) are even divisors of 2pi. Keywords: Abstract regular polytope, equivelar polytope, flat polytope, tight polytope. Math. Subj. Class.: 51M20, 52B15, 05E18 1 Introduction Abstract polytopes are ranked partially-ordered sets that resemble the face-lattice of a convex polytope in several key ways. Many discrete geometric objects can be viewed as an abstract polytope by considering their face-lattices, but there are also many new kinds of structures that have no immediate geometric analogue. A flag of an abstract polytope is a chain in the poset that contains one element of each rank. In many ways, it is more natural to work with the flags of a polytope rather than the faces themselves. For example, every automorphism (order-preserving bijection) of a polytope is completely determined by its effect on any single flag. Regular polytopes are those for which the automorphism group acts transitively on the set of flags. The automorphism group of a regular polytope is a quotient of some string Coxeter group [p1,... ,pn-1], and conversely, every sufficiently nice quotient of a string E-mail addresses: m.conder@auckland.ac.nz (Marston Conder), gabriel.cunningham@gmail.com (Gabe Cunningham) ©® This work is licensed under http://creativecommons.org/licenses/by/3.0/ 70 Ars Math. Contemp. 8 (2015) 149-162 Coxeter group appears as the automorphism group of a regular polytope. Indeed, it is actually possible to reconstruct a regular polytope from its automorphism group, so that much of the study of regular polytopes is purely group-theoretic. We say a regular polytope P is of type (or has Schlafli symbol) {pi,... ,pn-i} if [pi,... ,pn-i] is the minimal string Coxeter group that covers the automorphism group of P, in a way that pi,... ,pn-i are the orders of the relevant generators. There is an equivalent formulation of this property that is entirely combinatorial, and hence it is possible to define a Schlafli symbol for many non-regular polytopes, including chiral polytopes (see [10]) and other two-orbit polytopes (see [7]). Any polytope with a well-defined Schlafli symbol is said to be equivelar. In [3], the first author determined the smallest regular polytope (by number of flags) in each rank. To begin with, he showed that every regular polytope of type {pi,..., pn - i} has at least 2pi • • • pn-i flags. Polytopes that meet this lower bound are called tight. He then exhibited a family of tight polytopes, one in each rank, of type {4,..., 4}. Using properties of the automorphism groups of regular polytopes, he showed that each polytope was the smallest regular polytope in rank n > 9, and that in smaller ranks, the minimum was also attained by a tight polytope (with type or dual type {3}, {3,4}, {4,3,4}, {3, 6, 3,4}, {4, 3,6,3,4}, {3, 6,3,6,3,4} or {4,3, 6, 3,6,3,4}, respectively). The second author showed in [5] that the bound on the number of flags extended to any equivelar polytope, regardless of regularity. Accordingly, it makes sense to extend the definition of tight polytopes to include any polytope of type {pi,... ,pn-i} with 2pi • • • pn-i flags. An alternate formulation was proved as well, showing that an equivelar polytope is tight if and only if every face is incident with all faces (if any) that are 2 ranks higher. Tightness is a restrictive property, and not every Schlafli symbol is possible for a tight polytope. In order for there to be a tight polytope of type {pi,... ,pn-i}, it is necessary that no two adjacent values pi and pi+i are odd. Theorem 5.1 in [5] shows that this condition is sufficient in rank 3. In higher ranks, the question of sufficiency is still open. Constructing non-regular polytopes in high ranks is difficult. In order to determine which Schlafli symbols are possible for a tight polytope, it is helpful to begin by considering only regular polytopes. If every pi is even, then Theorem 5.3 in [3] and Theorem 6.3 in [5] show that there is a tight regular polytope of type {pi,... ,pn-i}. Also the computational data from [2, 6] led the second author to conjecture that if p is odd and q > 2p, there is no tight regular polyhedron of type {p, q}. Although we are currently unable to prove this conjecture, we can show that for tight orientably-regular polyhedra, if p is odd then q must divide 2p. Moreover, we are able to prove the following generalisation in higher ranks: Theorem 1.1. There is a tight orientably-regular polytope of type {pi,... ,pn-i} if and only if each of the integers pi-i and pi+i (when defined) is an even divisor of 2pi whenever pi is odd, for 1 < i < n. 2 Background Our background information is mostly taken from [8, Chs. 2, 3, 4], with a few small additions. 2.1 Definition of a polytope Let P be a ranked partially-ordered set, the elements of which are called faces, and suppose that the faces of P range in rank from -1 to n. We call each face of rank j a j-face, and we M. Conder and G. Cunningham: Tight orientably-regular polytopes 71 say that two faces are incident if they are comparable. We also call the 0-faces, 1-faces and (n - 1)-faces the vertices, edges and facets of P, respectively. A flag is a maximal chain in P. We say that two flags are adjacent if they differ in exactly one face, and that they are j-adjacent if they differ only in their j-faces. If F and G are faces of P such that F < G, then the section G/F consists of those faces H such that F < H < G. If F is a j-face and G is a k-face, then we say that the rank of the section G/F is k—j — 1. If removing G and F from the Hasse diagram of G/F leaves us with a connected graph, then we say that G/F is connected. That is, for any two faces H and H' in G/F (other than F and G themselves), there is a sequence of faces H = Ho, Hi,..., Hk = H' such that F < Hj < G for 0 < i < k and the faces H¿_ i and H¿ are incident for 1 < i < k. By convention, we also define all sections of rank at most 1 to be connected. We say that P is an (abstract) polytope of rank n, or briefly, an n-polytope, if it satisfies the following four properties: (a) There is a unique greatest face Fn of rank n, and a unique least face F_ i of rank — 1. (b) Each flag has n+2 faces. (c) Every section is connected. (d) Every section of rank 1 is a diamond — that is, whenever F is a (j — 1)-face and G is a (j + 1)-face for some j, with F < G, there are exactly two j-faces H with F < H < G. Condition (d) is known as the diamond condition. Note that this condition ensures that for 0 < j < n, every flag $ has a unique j-adjacent flag, which we denote by . In ranks — 1, 0, and 1, there is a unique polytope up to isomorphism. Abstract polytopes of rank 2 are also called abstract polygons, and for each 2 < p < to, there is a unique abstract polygon with p vertices and p edges, denoted by {p}. If F is a j-face and G is a k-face of a polytope with F < G, then the section G/F itself is a (k — j — 1)-polytope. We may identify a face F with the section F/F_ i, and call the section Fn/F the co-face at F. The co-face at a vertex F0 is also called a vertex-figure at Fo. If P is an n-polytope, F is an (i — 2)-face of P, and G is an (i + 1)-face of P with F < G, then the section G/F is an abstract polygon. If it happens that for 1 < i < n, each such section is (isomorphic to) the same polygon {p¿}, no matter which (i — 2)-face F and incident (i + 1)-face G we choose, then we say that P has Schlafli symbol {pi,... ,pn-i}, or that P is of type {pi,... ,pn-1}. Also when this happens, we say that P is equivelar. All sections of an equivelar polytope are themselves equivelar polytopes. In particular, if P is an equivelar polytope of type {p1,... ,pn-1}, then all its facets are equivelar polytopes of type {p1,... ,pn_2}, and all its vertex-figures are equivelar polytopes of type {p2 , . . . ,pn_1}. Next, let P and Q be two polytopes of the same rank. A surjective function 7 : P ^ Q is called a covering if it preserves incidence of faces, ranks of faces, and adjacency of flags. If there exists such a covering 7 : P ^ Q, then we say that P covers Q. The dual of a polytope P is the polytope obtained by reversing the partial order. If P is an equivelar polytope of type {p1,... ,pn-1}, then the dual of P is an equivelar polytope of type {p„_1,... ,p1}. 72 Ars Math. Contemp. 8 (2015) 149-162 2.2 Regularity For polytopes P and Q, an isomorphism from P to Q is an incidence- and rank-preserving bijection. By connectedness and the diamond condition, every polytope isomorphism is uniquely determined by its effect on a given flag. An isomorphism from P to itself is an automorphism of P, and the group of all automorphisms of P is denoted by r(P). We will denote the identity automorphism by e. We say that P is regular if the natural action of r(P) on the flags of P is transitive (and hence regular, in the sense of being sharply-transitive). For convex polytopes, this definition is equivalent to any of the usual definitions of regularity. Now let P be any regular polytope, and choose a flag which we call a base flag. Then the automorphism group r(P) is generated by the abstract reflections p0,..., pn-1, where pi maps $ to the unique flag $ that is ¿-adjacent to These generators satisfy p2 = e for all i, and (pipj)2 = e for all i and j such that |i - j| > 2. Every regular polytope is equivelar, and if its Schlafli symbol is {p1,... ,pn-1}, then the order of each pi-1pi is pi. Note that if P is a regular polytope of type {p1;... ,pn-1}, then r(P) is a quotient of the string Coxeter group [p1,... ,pn-1], which is the abstract group generated by n elements x0,..., xn-1 subject to the defining relations x2 = 1, (xi-1xi)pi = 1, and (xixj)2 = 1 whenever |i — j| > 2. Next, if r is any group generated by elements p0,..., pn-1, we define T/ = (pi | i G I} for each subset I of the index set {0,1,..., n — 1}. If r is the automorphism group r(P) of a regular polytope P, then these subgroups satisfy the following condition, known as the intersection condition: r/ n Tj = r/nj for all I, J C {0,1,...,n — 1}. (2.1) More generally, if r is any group generated by elements p0,..., pn-1 of order 2 such that (pipj)2 = 1 whenever |i — j| > 2, then we say that r is a string group generated by involutions, and abbreviate this to say that r is an sggi. If the sggi r also satisfies the intersection condition (2.1) given above, then we call r a string C-group. There is a natural way of building a regular polytope P(r) from a string C-group r such that r(P(r)) = r and P(r(P)) ^ P. In particular, the i-faces of P(r) are taken to be the cosets of the subgroup ri = (pj | j = i}, with incidence of faces ri^ and rj ^ given by ri^ < rj^ if and only if i < j and ri^ n rj^ = 0. This construction is also easily applied when r is any sggi (not necessarily a string C-group), but in that case, the resulting poset is not always a polytope. The following theory from [8] helps us determine when a given sggi is a string C-group: Proposition 2.1. Let r = (p0,..., pn-1} be an sggi, and A = (A0,..., An-1} a string C-group. If there is a homomorphism n : r ^ A sending each ai to Ai, and if n is one-to-one on the subgroup (p0,..., pn-2} or the subgroup (p1,..., pn-1}, then r is a string C-group. Proposition 2.2. Let r = (p0,..., pn-1} be an sggi. If both (p0,..., pn-2} and (p1,..., pn-1} are string C-groups, and (po,..., pn-2} n (p1,..., pn-1} C (p1,..., pn-2}, then r is a string C-group. M. Conder and G. Cunningham: Tight orientably-regular polytopes 73 Given a regular n-polytope P with automorphism group r = (p0,..., pn-1}, we define the abstract rotations a1,... ,an-1 by setting ai = pi-1pi for 1 < i < n. Then the subgroup (u1,..., an-1) of r(P) is denoted by r+(P), and called the rotation subgroup of P. The index of r+(P) in r(P) is at most 2, and when the index is exactly 2, then we say that P is orientably-regular. Otherwise, if r+ (P) = r(P), then we say that P is non-orientably-regular. (This notation comes from the study of regular maps.) A regular polytope P is orientably-regular if and only if r(P) has a presentation in terms of the generators p0,..., pn-1 such that all of the relators have even length. Note that every section of an orientably-regular polytope is itself orientably-regular. 2.3 Flat and tight polytopes The theory of abstract polytopes accommodates certain degeneracies not present in the study of convex polytopes. For example, the face-poset of a convex polytope is a lattice (which means that any two elements have a unique supremum and infimum), but this need not be the case with abstract polytopes. The simplest abstract polytope that is not a lattice is the digon {2}, in which both edges are incident with both vertices. This type of degeneracy can be generalised as follows. If P is an n-polytope, and 0 < k < m < n, then we say that P is (k, m)-flat if every one of its k-faces is incident with every one of its m-faces. If P has rank n and is (0, n - 1)-flat, then we also say simply that P is a flat polytope. Note that if P is (k, m)-flat, then P must also be (i,j)-flat whenever 0 < i < k < m < j < n. In particular, if P is (k, m) -flat, then it is also flat. We will also need the following, taken from [8, Lemma 4E3]: Proposition 2.3. Let P be an n-polytope, and let 0 < k < m < i < n. If each i-face of P is (k, m)-flat, then P is also (k, m)-flat. Similarly, if 0 < i < k < m < n and each co-i-face of P is (k-i-1,m-i-1)-flat, then P is (k, m)-flat. It is easy to see that the converse is also true. In other words, if P is (k, m)-flat, then for i > m each i-face of P is (k, m)-flat, and for j < k each co-j-face of P is (k — j — 1, m— j — 1)-flat. Next, we consider tightness. An equivelar polytope P of type {p1,... ,pn-1} has at least 2p1 • • • pn-1 flags, by [5, Proposition 3.3]. Whenever P has exactly this number of flags, we say that P is tight. It is clear that P is tight if and only if its dual is tight, and that in a tight polytope, every section of rank 3 or more is tight. Also we will need the following, taken from [5, Theorem 4.4]: Theorem 2.4. Let n > 3 and let P be an equivelar n-polytope. Then P is tight if and only if it is (i, i + 2)-flatfor 0 < i < n — 3. Later in this paper we will build polytopes inductively, and for that, the following approach is useful. We say that the regular n-polytope P has the flat amalgamation property (or FAP) with respect to its k-faces, if adding the relations pi = e for i > k to r(P) yields a presentation for (p0,..., pk-1). Similarly, we say that P has the FAP with respect to its co-k-faces if adding the relations pi = e for i < k yields a presentation for + ^ . . . , pn-1}. We will also use the following, taken from [8, Theorem 4F9]: Theorem 2.5. Suppose m,n > 2, and 0 < k < m — 2 where k > m — n. Let P1 be a regular m-polytope, and let P2 be a regular n-polytope such that the co-k-faces of P1 are 74 Ars Math. Contemp. 8 (2015) 149-162 isomorphic to the (m — k — l)-faces of P2. Also suppose that Pi has the FAP with respect to its co-k-faces, and that P2 has the FAP with respect to its (m — k-l)-faces. Then there exists a regular (k+n+l)-polytope P such that P is (k, m)-flat, and the m-faces of P are isomorphic to Pi, while the co-k-faces of P are isomorphic to P2. Furthermore, P has the FAP with respect to its m-faces and its co-k-faces. 3 Tight orientably-regular polyhedra We now consider the values of p and q for which there is a tight orientably-regular polyhedron of type {p, q}. By [5, Proposition 3.5], there are no tight polyhedra of type {p, q} when p and q are both odd. Also by [3, Theorem 5.3] and [5, Theorem 6.3], if p and q are both even then there exists a tight orientably-regular polyhedron whose automorphism group is the quotient of the Coxeter group [p, q] obtained by adding the extra relation (x0xix2xi )2 = 1. (Note that this is the group of the polyhedron {p, q | 2} described in [8, p. 196]; see [9] for more information on this and related polyhedra.) Indeed, for even p and q, there can often be multiple (non-isomorphic) tight polyhedra; for example, there are two of type {4, 8} that are non-isomorphic (see [2]). When p is odd and q is even (or vice-versa), the situation is more complicated. Evidence from [2] and [6] led the second author to conjecture that there are no tight regular polyhedra of type {p, q} if p is odd and q > 2p (see [5]). We will show that this is true in the orientably-regular case. In fact, we will prove something stronger, namely that q must divide 2p. We start by showing that if p is odd and q is an even divisor of 2p, then there is a tight orientably-regular polyhedron of type {p, q}. To do this, we define r(p, q) as the group (P0,P1,P21 Po2,Pi2,P22, (poPi)p, (PiP2)q, (P0P2)2, (P0P1P2P1P2)2), which is obtainable by adding one extra relator to the Coxeter group [p, q]. (Note that this is the group of the polyhedron {p, q} ,2 described in [8, p. 196].) Theorem 3.1. Let p > 3 be odd, and let q be an even divisor of 2p. Then there is a tight orientably-regular polyhedron P of type {p, q} such that r(P) = r(p, q). Proof. Let r(p, q) = (p0, pi, p2). In light of the construction in Section 2.2, all we need to do is show that r(p, q) is a string C-group of order 2pq, in which the order of p0pi is p and the order of pi p2 is q. First, note that the element w = (pip2)2 generates a cyclic normal subgroup N of r(p, q), since each of pi and p2 conjugates w to its inverse, and the extra relation (p0pip2pip2)2 = 1 implies that p0 does the same. Factoring out N gives quotient r(p, 2), in which the extra relation (p0pip2pip2)2 = 1 is redundant. In fact r(p, 2) is isomorphic to the string Coxeter group [p, 2], which is an extension of the dihedral group of order 2p, and has order 4p. In particular, r(p, q) covers r(p, 2) = [p, 2], and it follows that p0pi has order p (rather than some proper divisor of p). Also the cover from r(p, q) to r(p, 2) is one-to-one on (p0, pi), and so by Proposition 2.1, we find that r(p, q) is a string C-group. Next, we observe that the dihedral group Dq = (yi, y21 yi2, y22, (yiy2 )q) is a quotient of r(p, q), via an epimorphism taking pi ^ yi, p2 ^ y2 and p0 ^ (yiy2)p-iyi. (Note that the defining relations for r(p, q) are satisfied by their images in Dq, since (yiy2)p-iyi has order 2, the order of (yiy2)p-i divides p (as p — 1 is even and q divides 2p), the order of (yiy2)p-iyiy2 = (yiy2)p divides 2 (as q divides 2p), and (yiy2)p-iy2yiy2 has order 2.) In particular, the image of pip2 is yiy2, which has order q, and hence pip2 has order q. M. Conder and G. Cunningham: Tight orientably-regular polytopes 75 Finally, |r(p,q)| = |r(p,q)/N||N| = |r(p,2)||N| = 4p(q/2) = 2pq, since N = We will show that in fact, the only tight orientably-regular polyhedra of type {p, q} with p odd are those given in Theorem 3.1. We proceed with the help of a simple lemma. Lemma 3.2. Let P be an orientably-regular polyhedron of type {p, q}, with p odd, and with automorphism group r(P) generated by the reflections P0,Pi,P2. If w = (pip2)2 = a22 generates a normal subgroup of r+ (P), then w is central, and q divides 2p. Proof. For simplicity, let x = a1 = p0p1 and y = a2 = p1p2, so that xy = p0p2 and hence xp = yq = (xy)2 = 1, and also w = y2. By hypothesis, (y2) is normal, and so xy2x-1 = y2k for some k. It follows that y2 = xpy2x-p = y2fcp and that y2 = (xy)2y2(xy)-2 = y2k , and therefore 2 = 2k2 = 2kp mod q. Then also 2 = 2k2kp-2 = 2kp-2 mod q, and by induction 2 = 2kp = 2kp-2 = • • • = 2k mod q, sincep is odd. Thus xy2x-1 = y2k = y2, and so w = y2 is central. Moreover, since y2 commutes with x (which has order p) and xy = y-1x-1, we find that y2p = xpy2p = (xy2)p = (y-1x-1y)p = y-1x-py = 1, and so 2p is a multiple of q. □ We also utilise a connection between tight polyhedra and regular Cayley maps, as is explained in [4]. Specifically, suppose that the finite group G is generated by two non-involutory elements x and y such that xy has order 2, and that G can be written as AY where Y = (y) is core-free in G (that is, Y contains no non-trivial normal subgroup of G), and A is a subgroup of G such that A n Y = {1}. Then G is the group r+(M) of orientation-preserving automorphisms group of a regular Cayley map M for the group A. Furthermore, this map M is reflexible if and only if G admits an automorphism taking x ^ xy2 (= y-1x-1y) and y ^ y-1. Theorem 3.3. Let p > 3 be odd. If P is a tight orientably-regular polyhedron of type {p, q}, then q is an even divisor of 2p, and r(P) is isomorphic to r(p, q). Proof. Let G = r+ (P), and let a1 = p0p1 and a2 = p1p2 be its standard generators. Also take F = (a1) and V = ( 4, the group r(p1,... ,pn-1) is the amalgamation of r(p1,... ,pn-2) with r(p2,... ,pn-1) in the obvious way, subject to the extra relation (x0xn-1)2 = 1. Also let P(p1,... ,pn-1) be the poset obtained from r(p1,... ,pn-1), using the construction in Section 2.2. We will show that r(p1,... ,pn-1) is a string C-group of order 2p1p2.. .pn-1, and then since every relator of r(p1,... ,pn-1) has even length, it follows that P(p1,... ,pn-1) is a tight orientably-regular polytope of type {p1,... ,pn-1}. We start by considering the order of r(p 1,..., pn -1). Proposition 4.2. Let pi be even. Then every element of r(p1,... ,pn-1) either commutes with (xi-1xi)2 or inverts it by conjugation. In particular, the square of every element of r(p1,... ,pn-1) commutes with (xi-1xi)2. ri = ;+1) -1) 2 2 if pi and pi+1 are both even, or if pi is odd and pi+1 is even, or if pi is even and pi+1 is odd. M. Conder and G. Cunningham: Tight orientably-regular polytopes 77 Proof. Let w = (xj_ixj)2. If j < i — 3 or j > i + 2, then xj commutes with both xj_i and xj, and so commutes with w. Also it is clear that xi_1 and xj both conjugate w to w-1, so it remains to consider only xi-2 and xj+1. Now since pj is even, the relator rj_1 is either (xj_2xj_1xjxj_1)2 or (xj_2xj_1xjxj_1xj)2. In the first case, xi-2 commutes with xj_1xjxj_1 and hence with (xj_1xjxj_1)xj = w, while in the second case, we have (xj_2w)2 = 1 and so xj_2 conjugates w to w_1. Similarly, the relator r is (xj_1xjxj+1xj)2 or (xj+1xjxj_1xjxj_1)2, and in these two cases we find that xj+1wxj+1 = w or w_1, respectively. Thus every generator xj of r(p1,... ,pn_1) either commutes with (xj_1xj)2 or inverts it by conjugation, and it follows that the same is true for every element of r(p1,... ,pn_1). The rest follows easily. □ Proposition 4.3. Let (p1,... ,pn_1) be an admissible (n — 1)-tuple with the property that for every i strictly between 1 and n—1, eitherpj = 2 orpj_1 = pj+1 = 2. Then y = xj_1xj has order pj for all i, and |r(p1,... ,pn_1)| = 2p1 • • • pn_1. Proof. We use induction on n, together with the observation that if pj = 2, then 1 = (xj_1xj)2, so that xj_1 commutes with xj, and therefore (x0,..., xj_1) centralises (xj ,...,x„_1). First, if p1 = 2, then r(p1,p2,... ,pn_1) = r(2,p2,... ,pn_1) = (xo) xr(p2, . . . , Pn_ 1 ), and so |r(p1, . . . ,p„_1)| = 2|r(p2, . . . ,Pn_1 )| = 4p2 • • • Pn_1 = 2P1P2 ••• Pn_ 1. Otherwise p2 = 2 and r(p1,p2,... ,p„_1) = r(p1,2,p3,... ,p„_0 = r(p1) x r(p3,...,p„_1 ), and therefore |r(p1,... ,Pn_1)| = |r(p1)| |r(p3,... ,Pn_1)| = 2p1 2p3 • • • pn_1 = 2p1p2 • • • pn_1. The claim about the orders of the elements yj = xj_1xj follows easily by induction as well. □ Lemma 4.4. Let qj be the order of xj_1xj in r(p1,... ,pn_1), for 1 < i < n. Then qj = pj whenever pj is odd, and also |r(p1,... ,pn_1)| = 2q1 • • • qn_1, which divides 2p1•• • Pn_1. Proof. Let kj = pj when pj is odd, or 2 when pj is even. Then since kj divides pj for all i, there exists an epimorphism n : r(p1,... ,pn_1) ^ r(k1,..., kn_1). Also the (n— 1)-tuple (k1,..., kn_1) is admissible, and indeed kj_1 and kj+1 are both 2 whenever kj is odd (since pj_1 andpj+1 are both even whenever pj is odd). Thus (k1,..., kn_1) satisfies the hypotheses of Proposition 4.3, and so |r(k1,..., kn_1) | = 2k1 • • • kn_1. Moreover, Proposition 4.3 tells us that when pj is odd, the order of the image of xj in r(k1,..., kn_1) is kj = pj, and so qj = pj; on the other hand, if pj is even, then the order of the image of xj in r(k1,..., kn_1 ) is kj = 2, and so qj is even in that case. Now the kernel of the epimorphism n is the smallest normal subgroup of r(p1,..., pn_1) containing the elements (xj_1xj)2 for those i such that pj is even. By Proposition 4.2, however, the subgroup N generated by these elements is normal in r(p1,..., pn_1), and abelian. Hence in particular, N = ker n, and also by the intersection condition, |N | is the product of the numbers qj/2 over all i for which pj is even. Thus |r(p1, . . . ,Pn_1)| = 2q1 ••• q„_1. □ In order to use Theorem 2.5 to build our tight regular polytopes recursively, we need two more observations. The first concerns the flat amalgamation property (FAP): Proposition 4.5. fp2 is even, then P(p1,..., pn_1 ) has the FAP with respect to its 2-faces, and if pn_2 is even, then P(p1,... ,pn_1) has the FAP with respect to its co-(n — 3)-faces. 78 Ars Math. Contemp. 8 (2015) 149-162 Proof. Let p2 be even, and consider the effect of killing the generators xj of r(pi,..., pn-1), for i > 2 (that is, by adding the relations xj = 1 to the presentation for r(p1,..., pn-1)). Each of the relators r3,..., rn-2 contains only generators xj with i > 2, so becomes redundant, and may be removed. The relator r2 reduces to x2 or x4, while r1 reduces to (x0x1)2, which is equivalent to x2, and hence all of these become redundant too. Thus adding the relations xj = 1 to r(p1,... ,pn-1) has the same effect as adding the relations xj = 1 to the string Coxeter group [p1,... ,pn-1]. It is easy to see that this gives the quotient group with presentation (x0,x1 | x2,x2, (x0x1)pi}, which is the automorphism group of the 2-faces of P(p1,... ,pn-1). Thus P(p1,... ,pn-1) has the FAP with respect to its 2-faces. The second claim can be proved by a dual argument. □ Proposition 4.6. Let P be an equivelar n-polytope with tight m-faces and tight co-k-faces, where m > k + 3. Then P is tight. Proof. Since the m-faces are tight, they are (i, i+2)-flat for 0 < i < m—3, by Theorem 2.4, and then by Proposition 2.3, the polytope P is (i, i+2)-flat for 0 < i < m—3. Similarly, the co-k-faces are (i, i+2)-flat for 0 < i < n—k—4, and P is (i, i + 2)-flat for k+1 < i < n—3. Finally, since m > k + 3, we see that P is (i, i + 2)-flat for 0 < i < n — 3, and again Theorem 2.4 applies, to show that P is tight. □ We can now prove the following. Theorem 4.7. Let (p1,... ,pn-1) be an admissible (n—1)-tuple, with n > 4. Also suppose that pj-1 and pj+1 are both even, for some i (with 2 < i < n—2). IfP(p1,... ,pj) isatight orientably-regular polytope of type {p1,... ,pj}, and P (pj,... ,pn-1) is a tight orientably-regular polytope of type {pj,... ,pn-1}, then P (p1,... ,pn-1) isa tight orientably-regular polytope of type {p1,... ,p„-1}. Proof. Let P1 = P(p1,... ,pj) and P2 = P(pj,... ,pn-1), which by hypothesis are tight orientably-regular polytopes of the appropriate types. Since 1 and pj+1 are even, Proposition 4.5 tells us that P1 has the FAP with respect to its co-(i — 2)-faces, and P2 has the FAP with respect to its 2-faces. Then by Theorem 2.5, there exists a regular polytope P with (i+1)-faces isomorphic to P1 and co-(i—2)-faces isomorphic to P2. Moreover, since P1 and P2 are both tight, Proposition 4.6 implies that P is also tight, and then since P is of type {p1,... ,pn-1}, we find that |r(P)| = 2p1 • • • pn-1. But also the (i + 1)-faces of P are isomorphic to P1, and the co-(i — 2)-faces are isomorphic to P2, and so the standard generators of r(P) must satisfy all the relations of r(p1,... ,pn-1). In particular, r(P) is a quotient of r(p1,... ,pn-1), and so |r(p1,... ,pn-1)| > 2p1 • • • pn-1. On the other hand, |r(p1,... ,pn_1)| < 2p1 ••• p„_1 by Lemma 4.4. Thus |r(p1,... 1)| = 2p1 • • • Pn-1, and hence also r(P) = r(p1,... ,pn-1), and P = P(p1,... ,pn-1). Thus P(p1,... ,pn-1) is a tight polytope of type {p1,... ,pn-1}, and finally, since all the defining relations of r(p1,... ,pn-1) have even length, we find that P(p1,... ,pn-1) is orientably-regular. □ Note that the above theorem helps us deal with a large number of possibilities, once we have enough 'building blocks' in place. We are assuming that the (n — 1)-tuple (p1,..., pn-1) is admissible, so that pj-1 and pj+1 are even divisors of 2pj whenever pj is odd. Now suppose that n > 6. If p2 and are both even, then Theorem 4.7 will apply, and if not, then one of them is odd, say p2, in which case p1 and p3 must both be even, and M. Conder and G. Cunningham: Tight orientably-regular polytopes 79 again Theorem 4.7 will apply. Hence this leaves us with just a few cases to verify, namely admissible (n- 1)-tuples (pi,... ,pn-1) with n = 4 or 5 for which there is no i such that Pi_i and pj+1 are both even. The only such cases are as follows: • n = 4, with p1 odd, p2 even and p3 even, or dually, p1 even, p2 even and p3 odd, • n = 4, with p1 odd, p2 even and p3 odd, • n = 5, with p1 odd, p2 even, p3 even and p4 odd. We start with the cases where n = 4: Proposition 4.8. If p1 isodd, p2 is an even divisor of 2p1, and p3 > 2, then r(p1,p2,p3) is a string C-group, and P (p1, p2, p3) is a tight orientably-regular polytope of type {p1, q, p3 }, for some even q dividing p2. Proof: Let r = r(p1,p2,p3) = (X0,X1,X2,X3), and let r = r(p1, 2,p3) = (y0,y1,y2, y3), where y = xj is the image of xj under the natural epimorphism n : r(p1,p2,p3) ^ r(p1, 2,p3), for 0 < i < 3. By Proposition 4.3, we know that r(p1,2,p3) = [p1,2,p3]. Also the subgroup (x0, x1, x2) of r covers [p1,2], and this cover is one-to-one on (x0, x1), so (x0,x1 ,x2) is a string C-group, by Proposition 2.1. A similar argument shows that (x1,x2,x3) is a string C-group. Now the intersection of these two string C-groups is (x1, x2), since the intersection of their images in r is (y0, y1, y2) n (y1, ^2,^3) = (^1,^2), and the kernel of n is ((x1x2)2). Hence by Proposition 2.2, r(p1,p2,p3) is a string C-group, and the rest follows easily from Lemma 4.4. □ Lemma 4.9. If pj = 2 for some i, then in r(p1,... ,pn-1) = (x0,..., xn-1), we have (x0,..., xj) = r(p1,... ,pj) and (xj-1,..., x„_1) = r(pj,... ,pn-1). Proof. First consider A = (x0,..., xj). This is obtainable as a quotient of r(p1,... ,pj) by adding extra relations. But also it can be obtained from the group r(p1,... ,pn-1) by killing the unwanted generators xj+1,..., xn-1. (Note that for i +1 < j < n -1, the relation rj = 1 and all relations of the form (p)m = 1 become redundant and may be removed, and the same holds for the relations rj = 1 and rj-1 = 1 since the assumption that pj = 2 implies that [ xj—1, xj] — (xj—1, x j)2 = 1 and hence that x0,..., xj-1 commute with xj,..., xn-1.) It follows that A = r(p1,... ,pj). Also (xj-1,..., xn-1) = r(pj,..., pn-1), by the dual argument. □ Theorem 4.10. If p1 is odd, p2 is an even divisor of 2p1, andp3 is even, then P(p1,p2,p3) is a tight orientably-regular polytope of type {p1 ,p2,p3}. Proof. First, the group r(p1,p2,p3) = (x0, x1, x2,x3) covers r(p1,p2, 2) = (^0,^1,^2, y3), say, since p3 is even. Also by Lemma 4.9 we know that (y0, y1, y2) is isomorphic to r(p1,p2), and hence in particular, y1y2 has order p2. It follows that the order of x1x2 is also p2, and then the conclusion follows from Proposition 4.8. □ Theorem 4.11. If p1 and p3 are odd, and p2 is an even divisor of both 2p1 and 2p3, then P (p1,p2,p3) is a tight orientably-regular polytope of type {p1,p2,p3}. Proof. Under the given assumptions, the group r(p1,p2,p3) is obtained from the string Coxeter group [p1,p2,p3] = (x0,x1,x2,x3) by adding the two extra relations (x0x1x2x1x2)2 = 1 and (x3x2x1x2x1)2 = 1. These imply that the element w = (x1x2)2 80 Ars Math. Contemp. 8 (2015) 149-162 is inverted under conjugation by each of x0 and x3, and hence by all the xj. Now let y = xj-ixj for 1 < i < 3. These elements generate the orientation-preserving subgroup r+(pi,p2,p3), and they all centralise w = y22. It follows that r+(p1 ,p2,p3) has presentation (yi,y2,y31 ypi^3, (yiy2^ (y2y3^ (yiy2y3^ [yi^2], ). We now exhibit a permutation representation of this group on the Cartesian product Zpi x Zp2, by letting each yj induce the permutation n, where (j,k)n2 = (j,k +1) for all (j,k), It is easy to see that n and n2 have orders p1 and p2 respectively (since p2 divides 2pi), and that the order of n3 divides p2 /2 and hence divides p3. It is also easy to verify that they satisfy the other defining relations for r+(p1,p2,p3), and thus we do have a permutation representation. In particular, since n2 has order p2, so does y2 = x1x2, and again the conclusion follows from Proposition 4.8. □ We now handle the remaining case. Theorem 4.12. If p1 and p4 are odd, p2 is an even divisor of 2p1, and p3 is an even divisor of 2p4, then P (p1,p2,p3,p4) is a tight orientably-regular polytope of type {p1,... ,p4}. Proof. Take r = T(pb .. .,p4) = (xo,... ,x4), and A = T(pbp2, 2,p4) = (yo,.. . ,y4). Then r covers A, and this induces a cover from (x0,..., x3) to (y0,..., y3), which is isomorphic to r(p1,p2, 2) by Lemma 4.9. Similarly we have a cover from (x0, x1, x2) to (y0, y1, y2), which is isomorphic to r(p1,p2). But on the other hand, (x0, x1, x2) is a quotient of r(p1,p2), and hence these two groups are isomorphic. In particular, the cover from (x0,..., x3) to r(p1,p2, 2) is one-to-one on the facets, so (x0,..., x3) is a string C-group. By a dual argument, (x1,..., x4) is also a string C-group. Next, let A = r(p1, 2, 2,p4) = (z0,..., z4), and let n be the covering homomorphism from r to A. The kernel of n is the subgroup generated by (x1x2)2 and (x2x3)2, since the defining relations for r = r(p1,... ,p4) imply that these two elements are centralised or inverted under conjugation by each generator xj. In particular, kern C (x1,x2,x3). As also the intersection of (z0,..., z3) and (z1,..., z4) in A is (z1, z2,z3), it follows that intersection of (x0,..., x3) and (x1,..., x4) is (x1, x2, x3). Hence r is a string C-group. Now (x0, x1, x2) = r(p1 ,p2), and by Theorem 3.1 we know the polytope P(p1,p2) has type {p1 ,p2}. Similarly (x2,x3,x4) = r(p3,p4), and P(p3,p4) has type {p3,p4}. It follows that P(p1,p2,p3,p4) is an orientably-regular polytope of type {p1,... ,p4}. In particular, the order of xj_ 1xi is p» (for 1 < i < 4), and so by Lemma 4.4, P(p1, p2, p3, p4) is tight. □ This gives us all the building blocks we need. With the help of Theorem 4.7, we now know that P(p1,... ,pn-1) is a tight orientably-regular polytope of type {p1,... ,pn-1} whenever (p1,... ,pn-1) is admissible, and the proof of Theorem 1.1 is complete. M. Conder and G. Cunningham: Tight orientably-regular polytopes 81 5 Tight non-orientably-regular polytopes We have not yet been able to completely characterise the Schlafli symbols of tight, non-orientably-regular polytopes, but we have made some partial progress. For example, we can easily find an infinite family of tight, non-orientably-regular polyhedra. Theorem 5.1. For every odd positive integer k, there exists a non-orientably-regular tight polyhedron of type {3k, 4}, with automorphism group A(k) having presentation ( P0,Pl,P2 1 Po2,Pl ^ (P0Pl)3fc, (P0P2^ (PlP2)4,P0PlP2P1P0P1P2P1P2 ). Proof. We note that A(1) is the automorphism group of the hemi-octahedron (of type {3,4}), and that A(k) covers A(1), for every k. Hence in each A(k), the order of plp2 is 4 (and not 1 or 2). Next, because the covering is one-to-one on the vertex-figures, it follows from Proposition 2.1 that A(k) is a string C-group. Also A(k) has a relation of odd length, and so it must be the automorphism group of a non-orientably-regular polyhedron. Now let N be the subgroup generated by the involutions p2 and plp2pl. Since their product has order 2, this is a Klein 4-group. Moreover, N is normalised by pl and p2, and also by p0 since p0 centralises p2 and the final relation in the definition of A(k) gives (plp2pl)Po = plp2plp2. It is now easy to see that N is the normal closure of (p2). The quotient A(k)/N is isomorphic to (p0,pl |Po,Pi, (p0pl)3k ), with the final relator for A(k) becoming trivial, and so A(k)/N is dihedral of order 6k. In particular, this shows that p0pl has order 3k (and that |A(k)| = |A(k)/N||N| = 24k). □ The computational data that we have on polytopes with up to 2000 flags (obtained with the help of Magma [1]) suggests that these polyhedra are the only tight non-orientably-regular polyhedra of type {p, q} with p odd. Using Theorem 5.1, it is possible to build tight, non-orientably-regular polytopes in much the same way as we did in Theorem 4.7. In particular, the regular polytope with automorphism group A(k) has the FAP with respect to its 2-faces, and its dual has the FAP with respect to its vertex-figures (co-0-faces). Then by Theorem 2.5, we know there are tight, non-orientably-regular polytopes of type {4,3k, 4} for each odd k, and of type {4, 3k, r} for each odd k and each even r dividing 3k. It is possible to continue in this fashion, building tight non-orientably-regular polytopes of every rank. Finally, just as with orientably-regular polytopes, there are some kinds of Schlafli symbol for which no examples can be constructed using Theorem 2.5. In fact (and in contrast with the situation for orientably-regular polytopes), there seem to be no tight non-orientably-regular polytopes of some of these types at all. For example, there are no tight regular polytopes of type {3,4, r} with r > 3 and with 2000 flags or fewer, but the reason for this is not clear. References [1] W. Bosma, J. Cannon and C. Playoust, The Magma algebra system I, The user language, J. Symbolic Comput. 24 (1997), 235-265. [2] M. Conder, Lists of all regular polytopes with up to 2000 flags, http://www.math. auckland.ac.nz/~conder. [3] M. Conder, The smallest regular polytopes of any given rank, Adv. Math. 236 (2013), 92-110. 82 Ars Math. Contemp. 8 (2015) 149-162 [4] M. D. E. Conder and T. W. Tucker, Regular Cayley maps for cyclic groups, Trans. Amer. Math. Soc. 366 (2014), 3585-3609. [5] G. Cunningham, Minimal equivelar polytopes, Ars Math. Contemp. 7 (2014), 299-315. [6] M. I. Hartley, An atlas of small regular abstract polytopes, Period. Math. Hung. 53 (2006), 149-156. [7] I. Hubard, Two-orbit polyhedra from groups, Eur. J. Comb. 31 (2010), 943-960. [8] P. McMullen and E. Schulte, Abstract regular polytopes, Encyclopedia of Mathematics and its Applications, vol. 92, Cambridge University Press, Cambridge, 2002. [9] E. Schulte, Amalgamation of regular incidence-polytopes, Proc. London Math. Soc. 3 (1988), 303-328. [10] E. Schulte and A. Ivic Weiss, Chiral polytopes, Applied geometry and discrete mathematics, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 4, Amer. Math. Soc., Providence, RI, 1991, pp. 493-516.