Locally Toroidal Polytopes and Modular Linear Groups

Lo cally T oroidal P olytop es and Mo dular Linear Groups B. Monson ∗ Universit y of New Brunsw ic k F rederict on, New Brunsw ic k, C anada E3 B 5A3 and Egon Sc hulte † Northea stern Univ ersity Boston, Massach ussetts, USA, 0 2 115 No vem b er 16 , 202 1 Abstract When the standard repr e senta tion of a crystallographic Co xeter group G (with string diagram) is redu ce d m odulo th e in teger d ≥ 2, one obtains a finite group G d whic h is often the automorphism group of an abstract regular p o lytop e. Building on earlier w ork in the case that d is an o dd prime, w e here d ev elop metho ds to h a nd le comp osite mo duli and completely describ e the corresp onding mo d ular p ol ytop es when G is of spherical or Euclidean t yp e. Using a mo dular v arian t of the quotien t criterion, w e then d escrib e the lo cally toroidal p olytop e s p ro vided by ou r constru ct ion, most of whic h are new. Key W ords: locally toroidal p olytop es, abstract regular p olytopes AMS Sub ject Classification (2000 ): Primary: 51M20. Secondary: 20F55. 1 In tr o duction Our fascination with the regula r p olytop es is due not only t o their visual app eal and c harm, but also to the fact that their symmetry g r o ups app ear in suc h v a rie d and unexp ected places. In a recen t series of papers, f o r example, the authors establis hed the basic mac hinery needed to describe a large class of polytop es whose a ut o morphis m groups t ypically ha v e small index in some finite orthogonal group (see [8, 9, 10]). Indeed , in our analysis there w e often had to exploit quite subtle prop erties of the orthogonal group O ( n, p, ǫ ) on an n -dimensional ve ctor space ov er Z p , where p is a n o dd prime. Here we tak e a bit of a detour and consider instead ∗ Suppo rted by NSERC of Ca nada Grant # 4818 † Suppo rted by NSA-gra n ts H98 230-05-1- 0027 and H98 230-07-1- 0 005 1 the p ossibilities released by more generally w orking o v er the ring Z d , with any mo dulus d ≥ 2. (The rank 4 p olytop es describ ed in [11, 12] inv olv e an a na logous excurs ion into the domains of Gaussian and Eisenstein in tegers; and, of course, the related idea o f constructing the automorphism group o f a regular map b y mo dular reduction is natural and we ll established; see [13], for example.) Our main goal is to extend previous results on lo c al ly tor oidal p olytop es , as pro vided b y our construction [10, § 4]. T o that end, in Sections 2 and 3 w e describ e the mo dular reduction of a crystallographic Co xeter group G with string diagram. In Sections 4 and 5 w e completely describe what happ ens when G is of spherical or Euclidean type. Finally , after prov ing a useful quotien t criterion (Theorem 6.1), w e discuss in Section 7 v arious new families o f lo cally toro ida l p olytop es , mainly in ranks 5 and 6. 2 Abstract re gular p olytop es and Coxeter groups Let us b egin with a brief review of some k ey prop erties of abstract regular p olytop es , referring to [6] for details. An (abstr act) n -p olytop e P is a par t ia lly ordered set with a strictly monotone rank function ha ving range {− 1 , 0 , . . . , n } . An elemen t F ∈ P with rank( F ) = j is called a j - f a c e ; t ypically F j will indicate a j -face; P has a unique least face F − 1 and unique greatest face F n . Eac h maximal c ha in or flag in P m ust con ta in n + 2 faces. Next, P m ust satisfy a homo geneity pr op erty : whenev er F < G with rank( F ) = j − 1 and rank( G ) = j + 1, there are exactly t w o j -faces H with F < H < G , just as happ ens fo r conv ex n -p olytop es. It follows that for 0 ≤ j ≤ n − 1 a nd an y flag Φ, there exists a unique adjac ent flag j Φ, differing fro m Φ in just the rank j face. With this notion of adjacency the flags of P form a flag gr aph . The final defining prop ert y of P is that the flag graph for eac h section m ust b e connected, so tha t P is str o n gly flag–c on ne cte d . Recall here that whenev er F ≤ G are faces of ranks j ≤ k in P , then the se c t ion of P determined b y F and G is giv en by G/F := { H ∈ P | F ≤ H ≤ G } . In fact, t his is a ( k − j − 1)- polytop e in its own right. Naturally , the symmetry of P is exhibited b y its automorphi s m gr oup Γ( P ), containing all o rder preserving bijections on P . Henceforth, w e shall consider only r e g u lar p olytop e s P , for whic h Γ( P ) is, b y definition, tra nsitive on flags. Clearly a regular n -p olytop e P must ha v e all sorts of lo cal combinatorial symmetry . In particular, P will b e e quivelar of some t yp e { p 1 , . . . , p n − 1 } , where 2 ≤ p j ≤ ∞ ; this means that f o r each fixed j ∈ { 1 , . . . , n − 1 } and eac h pair of inciden t fa ces F and G in P , with ra nk ( F ) = j − 2 a nd rank( G ) = j + 1, the rank 2 section G/F has the structure of a p j -gon (indep en den t of choice of F < G ). Thus , eac h 2- f ace ( p olygon) o f P is isomorphic t o a p 1 -gon, and in eve ry 3- f ace o f P , eac h 0-face is surrounded by a n a lt e rnating cycle of p 2 edges and p 2 p olygons, etc. T o f ur t her understand the structure o f Γ( P ) when P is regular, w e fix a b ase flag Φ = { F − 1 , F 0 , . . . , F n − 1 , F n } , with r ank ( F j ) = j . F or 0 ≤ j ≤ n − 1, let ρ j b e the (unique) automorphism with ρ j (Φ) = j Φ. If P is regular, then Γ( P ) is g e nerated by ρ 0 , ρ 1 , . . . , ρ n − 1 , whic h are in v olutions satisfying at least the relations ρ 2 j = ( ρ j − 1 ρ j ) p j = ( ρ i ρ j ) 2 = 1 , 0 ≤ i, j ≤ n − 1 , | j − i | ≥ 2 (1) 2 (with j ≥ 1 for ρ j − 1 ρ j ). Also, an interse ction c ondition on standard subgroups holds: h ρ i | i ∈ I i ∩ h ρ i | i ∈ J i = h ρ i | i ∈ I ∩ J i (2) for all I , J ⊆ { 0 , . . . , n − 1 } . In short, Γ( P ) is a v ery particular quotien t of the Co xeter group G = [ p 1 , . . . , p n − 1 ], whose diagram is a string with branche s lab elled p 1 , . . . , p n − 1 . (W e a llo w p j = 2, in whic h case the ‘string’ is disconnected.) Conv ersely , giv en any group Γ = h ρ 0 , . . . , ρ n − 1 i g e nerated b y in v olutions and satisfying (1) a nd (2), one may construct a p olytop e P with Γ( P ) = Γ (see [6, Theorem 2E11]). W e then sa y that Γ( P ) is a string C-gr oup . Since P can b e uniquely reconstructed from Γ( P ), we may therefore shift our fo cus to an appropriate class of groups of particular in terest. Recall that if P 1 and P 2 are regular n -p olytop es with n ≥ 2, then h P 1 , P 2 i denotes the class of a ll regular ( n + 1)-p olytop es whose facets are isomorphic to P 1 and whose v ertex- figures a r e isomorphic to P 2 . If this class is non-empty , then it contains a universal regular ( n + 1) -polytop e, denoted { P 1 , P 2 } , which co v ers an y other p olytop e in the class [6, Th. 4A2]. Let us lo ok more closely at the a bstract Co xeter group G = [ p 1 , . . . , p n − 1 ], which is itself a string C-group with resp ect to the usual generators and whic h ma y w ell b e infinite. The corresp onding p olytope { p 1 , . . . , p n − 1 } := P ( G ) is univ ersal in a more lo cal sense, as described in [6, Th. 3 D5]. No w, lik e an y finitely generated Coxe ter gro up, G can b e iden tified with its image under the standard faithf ul represen ta t io n in real n -space V [4, Cor. 5.4]. Conse quen tly , w e may supp ose G = h r 0 , . . . , r n − 1 i to b e the line ar Coxeter gr oup generated b y certain reflections r j on V . In fact, these reflections leav e in v arian t a symmetric bilinear form x · y on V , so that G is a subgroup of the corresp onding ortho g onal group O ( V ) ⊂ GL ( V ). (Note, how ever, that x · y is p ositiv e definite if and only if G is finite [4, Th. 6.4 ].) W e shall let e denote the iden tity in the group GL ( V ). Recalling our ear lier description of the regular n -p olytop e P , we now hav e an epimorphism G → Γ( P ) r j 7→ ρ j . In tuitiv ely then, w e may think of regular p olytop es as having maximal reflection symmetry . 3 Crystallog raphic Co xeter g roups and their mo du lar reductio ns No w let us sp ecialize. W e say that the linear Co xeter g roup G is crystal lo gr aphic (with resp ec t to the standard represen tation) if it lea v es in v ariant some la t tic e P n − 1 j =0 Z b j generated by a basis β = { b j } for V . As describ e d in [5 ] or [8, Prop. 4.1], there is no loss of generalit y in assuming that β is a b a s i c system fo r G , meaning that eac h b j is a r o ot for the corresp onding reflection r j . Th us, r i ( b j ) = b j + m i j b i (3) 3 for certain Cartan inte gers m i j , 0 ≤ i, j ≤ n − 1 , with a ll m i i = − 2 a nd m i j = 0 for | i − j | ≥ 2. No w recall t hat the string Coxete r group G = [ p 1 , . . . , p n − 1 ] is crystallographic if and only if a ll p j ∈ { 2 , 3 , 4 , 6 , ∞} [8, Prop. 4.1(c)]. If the corresp onding C oxeter diagr am ∆ c ( G ) is connected, then G admits only a finite num b er of essen tially distinct basic systems β . As w e o bs erv ed in [8, § 4], eac h basic system and corresp onding latt ice can b e enco ded in a new diagram ∆( G ), a v aria n t of ∆ c ( G ). Briefly , the branc hes of ∆( G ) are no longer lab elled; instead, eac h no de j of ∆( G ) is lab elled by the real n um b er b 2 j = b j · b j . Eac h sub diagram on t w o no des i and j must then b e o ne of those app earing in T able 1 b elo w. P erio d of r i r j Sub diagram o n no des Cartan integers i (left), j (rig h t) m ij , m j i 2 a • c • 0 , 0 3 a • a • 1 , 1 4 a • 2 a • 2 , 1 6 a • 3 a • 3 , 1 ∞ a • 4 a • 4 , 1 ∞ a • = = = a • 2 , 2 T a ble 1. P ossible diagra ms for dihedral subgroups h r i , r j i o f G F or each i 6 = j , we hav e m ij m j i = 4 cos 2 ( π /p ij ), where p ij is t he p erio d of t he r otation r i r j . (In particular, p j − 1 ,j = p j .) Note that no des i and j m ust be clearly distinguished, say as left and right in the T a ble 1, whenev er m ij 6 = m j i . By suitably r escaling the no de lab els on each connected comp onen t of ∆( G ), we can assume that these lab els a re a set o f relativ ely prime p ositiv e in tegers. As a f amiliar example, consider the usual tessellation P of the Euclidean plane b y congruen t squares. Then P is an infinite regular 3-p olytop e, and G = [4 , 4] ≃ Γ( P ) admits the diagrams 1 • 2 • 1 • , 1 • 2 • 4 • and 2 • 1 • 2 • . (4) Ha ving fixed suc h a basic system for a crystallographic Co xeter group G = [ p 1 , . . . , p n − 1 ], w e can reduce G mo dulo an y in teger s ≥ 2: the natural epimorphism Z → Z s induces a homomorphism of G on to a subgroup G s of GL n ( Z s ), the group of n × n in v ertible matrices o v er Z s . Our hop e, of course, is that the finite group G s will b e the automor phism group of a regular n -p olytop e. (In [8, 9, 10] we examined such groups in the case that s is an o dd prime, so as to exploit the structure of orthogonal groups o v er finite fields.) W e shall often abuse notation b y referring to the mo dular images of ob jects by the same name (suc h as r i , e , b i , V , etc.). In particular, { b i } will denote the standard basis for V = Z n s , whic h in general w e m ust now view as a f re e mo dule ov er the ring Z s . W e shall see in Lemma 3.1 that r i usually con tinu es to act as a reflection after reduction; in any case, w e can compute it using (3). How ev er, the situation for metrical quan tities suc h a s b i · b j , a ra t ional n um b er whic h o ccasionally has denominator 2 , is more intricate [8, Eq. 4 10]. Nev ertheless, at least when gcd (6 , s ) = 1 , w e can in terpret G s as a subgroup of the orthogonal group O ( Z n s ) for t he symmetric bilinear fo r m defined on Z n s b y means of t he Gram matrix [ b i · b j ]. Moreov er, we can then write r i ( x ) = x − 2 x · b i b i · b i b i since b 2 i will b e inv ertible (mo d s ). In our earlier work with prime mo duli, these issues w ere a concern only fo r ‘non-generic’ groups, where s = 3 a nd G has some p eriod p j = 6. Here, with more general mo duli, the analysis is more complicated. It of t e n happ ens , for instance, that the g roup G s dep en ds essen tia lly on the c hoice of ba sic system and the corresp onding diagram ∆( G ). F or example, for the mo dulus s = 4, the group G 4 corresp onding to the three diagrams in (4) has order 32, 128 and 64, resp ectiv ely . These ar e, in fact, the automorphism groups of the regular toroidal maps { 4 , 4 } (2 , 0) , { 4 , 4 } (4 , 0) and { 4 , 4 } (2 , 2) (see T able 4 b elo w). Clearly , w e m ust no w confront a crucial question: when is G s = h r 0 , . . . , r n − 1 i s a string C -group (i.e. t he automorphism gro up of a finite, abstra ct regular n -p olytop e P = P ( G s ))? Unfortunately , w e cannot provide an ything lik e a comprehensiv e a ns w er here. Instead, f o r classes of groups G of par t ic ular interest, we shall ha v e to rely more on ad ho c tec hniques than w e did for prime mo duli, without trying to exploit in a n y deep w a y the structure of orthogonal groups o v er general rings. Occasionally , we emplo y GAP [2] to settle ‘small’ cases. Certainly , the generators r j of G s satisfy the Coxete r-type relations inherited from G (see (1), with ρ j replaced b y r j ). H ow ev er, b efore confron ting the interse ction condition (2) for G s , w e m ust take a closer lo ok. F or example, it might happ en that r j = e (mod s ). Notation . W e say that no de i of ∆( G ) is e -e if b oth Cartan in t e gers m i,i − 1 and m i,i +1 are ev en; o-e if just one of the in tegers is ev en; a nd o-o if b oth are o dd. F or the terminal no des 0 and n − 1 on the string we shall agree tha t m 0 , − 1 = m n − 1 ,n = 0. Note tha t end no des can nev er b e o - o . Lik ewise, a no de is e-e if it is lab elled a , while an y adja cent no des are lab elled 4 a , 2 a or a (after a double branc h), as in . . . 2 a • a • 2 a • . . . , a • = = = a • . . . , . . . 2 a • a • , etc. T ypical o- e no des are the middle no des in the sub diagrams . . . 3 a • a • 2 a • . . . or . . . a • = = = a • c • . . . (where the integer lab el c divides a ). Let us now summarize basic prop erties of the generator s r i for G s . Using (3 ) , the calculations are straigh tforw ard, if a bit in v olv ed. Lemma 3.1 L et G = h r 0 , . . . , r n − 1 i ≃ [ p 1 , . . . , p n − 1 ] b e any crystal lo gr aphic line ar Coxeter gr oup with string diagr am. Supp ose s ≥ 2 , and r e duc e G mo dulo s . The n (a) Ea c h r i ∈ G s has p erio d 2 , exc ept that r i = e when s = 2 and no de i of ∆( G ) i s e-e . (b) r i and r j c ommute in G s when i < j − 1 . (c) Supp ose p i = 2 , 3 , 4 o r 6 . If s > 2 , then r i − 1 r i has p erio d p i in G s (unchange d fr om char acteristic 0 ). 5 Now let s = 2 . If p i = 3 or 6 , the p eri o d of r i − 1 r i is always 3 . If p i = 4 , the p erio d c ol lap s es to 2 if a nd only if one of no des i − 1 or i is e-e . F or p i = 2 , the p erio d c ol lapses to 1 if and only if b oth no des ar e e-e (so that r i − 1 = r i = e ). (d) Supp os e p i = ∞ . Then r i − 1 r i has p erio d s in G s , exc e p t in the fol lowi ng c ases, e ach when s is ev en: for the sub dia gr am a • = = = a • , the p erio d b e c omes s 2 when b oth no des ar e e-e ; for the sub diagr am a • 4 a • the p erio d b e c omes 2 s when the no de lab el l e d a is o-e . Remarks . In the typic al case, when all r i ha v e p erio d 2, w e sa y that G s is a string gr oup gener ate d by involutions . Ev en fo r mo dulus s = 2, it is quite p ossible that all r i b e inv olutions (though not geometrical r eflections), so long as ∆( G ) has sp ecial features, as explained later. Assuming now that all r i are inv olutions, we conclude that G s is a string C- g roup if and only if it satisfies the intersec tion condition (2), with r i = ρ i . Our main problem is therefore to determine when G s satisfies (2). W e hin ted earlier at the definite adv an tages of w orking with prime mo duli. F o r a com- p osite mo dulus s , w e w ould at least hop e t ha t G s someho w splits according to the prime decomp os ition o f s . Ho w ev er, our hop es for a simple approa c h a r e dashed b y examples suc h as the follo wing. Let G ≃ [4 , 6 , 4] b e the gro up with diagram 2 • 1 • 3 • 6 • . First of all, we find f or p = 2 that G 2 is a string C -g r oup o f order 9 6 . The middle ro tation order collapses and w e actually obta in the group for the unive rsal lo cally pro jectiv e p olytop e { { 4 , 3 } 3 , { 3 , 4 } 3 } . F or p = 3 we get a gro up G 3 of order 51 8 4 for a self-dual p olytop e of t yp e { 4 , 6 , 4 } (see [9, Eq. (33)]). No w for mo dulus s = 6 w e find t ha t G 6 has order 2488 32 = 1 2 (96 × 5184) . But the in tersection condition fa ils, since h r 1 , r 2 i 6 has index 3 in h r 0 , r 1 , r 2 i 6 ∩ h r 1 , r 2 , r 3 i 6 . In other w o rds , the p olytopalit y o f G s is not determined through the prime factorizatio n of s . Since, in the end, w e are more concerned with lo cally toroidal groups G , whic h do fall to a more direct atta c k, we shall mainly ignore the prime factorization of s . (W e note, how ev er, that precisely that appro ac h w o rk ed in [11, 12]. But fo r the 4-p olytop es considered there, the rotation groups w ere cov ered b y sp ecial linear groups ov er certain rings of algebraic integers; and the resulting mo dular g r oups do split according to the prime factorizatio n.) Before pro ceeding, let us set do wn some useful nota tion. F or an y J ⊆ { 0 , . . . , n − 1 } , w e let G s J := h r j | j 6∈ J i ; in particular, fo r k , l ∈ { 0 , . . . n − 1 } w e let G s k := h r j | j 6 = k i and G s k ,l := h r j | j 6 = k , l i . W e also let V J b e the submo dule of V = Z n s spanned by { b j | j 6∈ J } , and similarly for V k , V k ,l . Note that V J is G s J -in v ariant. In particular, G s j acts on V j , fo r j = 0 or n − 1; how ev er, this a ction need not b e faithful (see [9, Lemma 3.1]). 4 Mo dular p o lytop es of spheric a l t yp e When G = [ p 1 , . . . , p n − 1 ] is finite, the in v arian t for m x · y on real n -space V is p ositiv e definite, so t hat G a c ts in a natur a l wa y on any sphere S n − 1 with centre o ∈ V . Accordingly , w e also sa y that G is of spheric al typ e . If the spherical group G has a connected diagram, then 6 up to isomorphism P ( G ) is one of the familiar con v ex regular n -p olytop es [8, § 5-6]. After cen tra l pro jection, suc h p olytopes can usefully b e view ed a s regular spherical tessellations of the circumsphere S n − 1 . In [8, § 5-6] w e show ed that G ≃ G p , f o r any o dd prime mo dulus p and crystallogr a phic string Co xeter group G of spherical ty p e and in any rank n ≥ 1. When s is divisible by an o dd prime p , the natural epimorphisms G → G s → G p immediately giv e G ≃ G s . Here w e tak e a differen t approa ch, w orking explicitly with the underlying represen ta t ion o f the spherical group G in GL n ( Z ). W e confirm that G s ≃ G for any mo dulus s ≥ 3 , and sometimes eve n for s = 2. (F or n = 1 , 2 suc h isomorphisms follow at once from Lemma 3.1.) Ho wev er, since the a c tual calculations for general rank n are quite tiresome, we shall simply summarize the results, with brief commen ts, for eac h of the relev ant families of spherical groups. In fa ct, to serv e later a pplic ations, w e mus t generalize a little and consider how a spherical gro up can embed as a string subgroup of some group G of higher rank. In o ther words, w e consider certain spheric al sub diagr ams o f ∆( G ). Of course, when G s ≃ G w e also kno w the structure of the mo dular p olytop e P ( G s ), whic h is merely a cop y of P ( G ). (a) The group of the m -simplex: A m ≃ S m +1 , for m ≥ 1. Here, for some lab el a ≥ 1, ∆( G ) has the sub diagram . . . a • a • . . . a • a • . . . , (5) on m consecutiv e no des j, . . . , j + m − 1. F or all m ≥ 2 and all s ≥ 2, w e then hav e h r j , . . . , r j + m − 1 i s ≃ A m . P a rt (c) of Lemma 3.1 prov ides t he base step of a n induction on m ≥ 2. As in [8, § 6.1], w e then exploit the contragredien t represen tation o f A m . ( Alternat ively , we could use the fact that the eve n subgroup o f h r j , . . . , r j + m − 1 i is the alternating group of degree m + 1, whic h is simple if m ≥ 4 ; the cases m = 2 , 3 are straigh tforward.) F or m = 1, we note that A 2 1 = { e } ; otherwise, for s ≥ 3, A s 1 ≃ A 1 ≃ C 2 . (b) The group of the m -cub e: B m , for m ≥ 2. W e m ust accommo date t w o distinct basic systems for B m . Consider first the sub diagram . . . a • 2 a • 2 a • . . . 2 a • 2 a • . . . , (6) on no des j, . . . , j + m − 1 of ∆( G ). Then h r j , . . . , r j + m − 1 i s ≃ B m , for all s ≥ 3, and f or s = 2 so long a s no de j (lab elled a ) is o-e . If, how ev er, s = 2 and no de j is e-e , then r j = e and the giv en generators do not give a string C - g roup. Instead, fro m (a) w e see that the subgroup collapses in rank to a cop y of A 2 m − 1 . 7 Here, and in similar situations b elo w, w e obtain dual v ersions of these results by flipping the diagr a m end-for-end. Consequen tly , w e may supp ose that m ≥ 3 for t he alternative basic system . . . 2 a • a • a • . . . a • a • . . . , (7) on no des j, . . . , j + m − 1. Again w e ha v e h r j , . . . , r j + m − 1 i s ≃ B m , whenev er the mo dulus s ≥ 3. F or s = 2, the subgroup h r j , . . . , r j + m − 1 i s is isomorphic either to B m (the group of t he cub e), or to B m / {± e } (the g r oup of the h emi-cub e ), as detailed in T a ble 2. no de m ≥ 3 j j + m − 1 ev en o dd o-o o-o B m B m o-o o-e B m / {± e } B m o-e o-o B m B m o-e o-e B m / {± e } B m / {± e } T a ble 2. The group B 2 m for the diagram (7) (Since m ≥ 3, no de j + m − 1 cannot b e e-e .) Note that the b ottom row co v ers t he case that G actually equals B m , for whic h there is inevitably a collapse when s = 2. A crucial step in the v erification emplo ys a small observ ation concerning B m ≃ [4 , 3 , . . . , 3] = h r 0 , r 1 , . . . , r m − 1 i : if ϕ : B m → H is a homomorphism whic h is 1 − 1 on the subgroups h r 0 , r 1 i and h r 1 , . . . , r m − 1 i , then k er ϕ ⊆ {± e } . The pro of follo ws from explicit calculation in B m , tak en as the semidirect pro duct C m 2 ⋊ S m . Not e here that H is isomorphic t o B m if a nd only if ( r 0 r 1 . . . r m − 1 ) m 6 = e . (T o argue f rom a top ological p erspectiv e, the regular m -p olytop e asso ciated with the gro up h r j , . . . , r j + m − 1 i s m ust b e a regula r tessellation on an ( m − 1) - dime nsional spherical space- form and hence necessarily b e isomorphic t o a r egula r tessellation on the ( m − 1)-sphere or real pro jectiv e ( m − 1)-space (see [6, 6C2]). This observ ation also applies to the next group.) (c) The group of t h e 24 -cell: F 4 . W e m ust consider a subdia g ram suc h as . . . a • a • 2 a • 2 a • . . . (8) on no des j, . . . , j + 3 in ∆( G ). By part (b), the natural mapping ϕ : F 4 → h r j , r j +1 , r j +2 , r j +3 i is 1 − 1 on subgroups h r j , r j +1 , r j +2 i and h r j +1 , r j +2 , r j +3 i . A similar small observ ation no w giv es ker ϕ ⊆ {± e } . No matter how the sub diagram is em b edde d in ∆( G ) w e find that h r j , r j +1 , r j +2 , r j +3 i s ≃  F 4 , if s ≥ 3 ; F 4 / {± e } , if s = 2 . (9) 8 5 Mo dular p o lytop es of Euclidean t yp e Supp ose no w that G = [ p 1 , . . . , p n − 1 ] is a string Co xeter group of Euclide an ( or affine ) t yp e, with connected diagram (no p j = 2). Then G a c ts as the full symmetry group of a certain regular t e ssellation T ≃ P ( G ) of Euclidean space A n − 1 . Indeed, G mus t b e one of the Co xeter g r o ups displa y ed in the left column of T a ble 3, though p erhaps with generators sp ec ified in dual order. Note tha t eac h of these g roups is crystallographic. A r e gular n - tor oid P is the quotien t of suc h a t essellation T by a non-trivial normal subgroup L of translations in G . Th us e v ery toroid can b e view ed as a finite, regular tessellation of the ( n − 1)-torus. W e refer to [6, 1 D and 6D-E] for a complete classification; briefly , for each group G the distinct toro ids are indexed b y a typ e ve ctor q := ( q k , 0 n − 1 − k ) = ( q , . . . , q , 0 , . . . , 0), where q ≥ 2 a nd k = 1 , 2 or n − 1. (F or G = [3 , 3 , 4 , 3], the case k = 4 is subsumed by the case k = 1.) Anyw a y , L is generated (as a normal subgroup of G ) b y t he translation t := t q 1 · · · t q k , where { t 1 , . . . , t n − 1 } is a standa r d set of generators fo r the full group T of translations in G . The mo dular to r o ids P ( G p ) described in [8 , § 6B] are sp ecial instances; with one exception, w e had t he re q = ( p, 0 , . . . , 0). F or completeness w e also list in T able 3 the infinite dihedral group [ ∞ ], whic h of course has rank 2 and a cts on the Euclidean line A 1 . The corresp onding 2-toroids are then r egula r p olygons inscrib e d in a ‘1-torus’, namely , in a n o r din ary circle. Before pro ceeding t o a classification of the groups G s , we tak e a closer lo ok at the geo- metric action of groups of a ffin e Euclidean isometries. Supp ose then that G = h r 0 , . . . , r n − 1 i is of Euclidean type (here alwa ys with connected diagram). F rom [4, § 6.5] w e recall that the in v arian t quadratic form x · y on real n - s pace V m ust b e p ositiv e semidefinite, so t hat the r adic al subsp ac e rad( V ) = h c i is 1-dimensional. Since r j ( c ) = c , f o r 0 ≤ j ≤ n − 1, G is in fact a subgroup of b O ( V ), the p oin t wise stabilizer of rad( V ) in O ( V ). T o actually exploit the structure of G as a group of (affine) isometries on Euclidean ( n − 1)- s pace, w e pass to the contragredien t represen tation of G in the dual space ˇ V (as in [4, 5.1 3]). Since c is fixed by G , w e see that G leav es in v ariant an y translate of the ( n − 1)-space U = { µ ∈ ˇ V : µ ( c ) = 0 } . Next, for each w ∈ V define µ w ∈ ˇ V b y µ w ( x ) := w · x . T he mapping w 7→ µ w factors to a linear isomorphism b et w een V / r ad ( V ) and U , and so w e transfer to U the p ositive d e finite form induced by V on V / rad( V ). No w c ho ose an y α ∈ ˇ V suc h that α ( c ) = 1, and let A n − 1 := U + α . Putting all this t o gethe r w e ma y now think of A n − 1 as Euclide an ( n − 1) - sp ac e , with U as its sp ac e of tr ans l a tion s . Inde ed, eac h fixed τ ∈ U defines an isometric translation on A n − 1 : µ 7→ µ + τ , ∀ µ ∈ A n − 1 . It is easy to c hec k that this mapping on A n − 1 is induced by a unique isometry t ∈ b O ( V ), namely the tr ansve ction t ( x ) = x − τ ( x ) c, = x − ( x · a ) c, 9 where τ = µ a for suitable a ∈ V . (Remem b er here tha t we employ the contragredien t represen tation of b O ( V ) on ˇ V , not just that of G .) In summary , we can t herefore safely think of translations as transve ctions. In the f ollo wing table w e list those Euclidean Co xeter groups whic h are relev ant to our analysis (see [8, § 6B]). Concerning the group G = [4 , 3 n − 3 , 4 ] (for the familar cubical tessel- lation o f A n − 1 ), w e recall our conv en tion that 3 n − 3 indicates a string of n − 3 ≥ 0 consec utiv e 3’s. The gro up G dim( A n − 1 ) One p ossible diagra m The corresp onding v ector ∆( G ) c ∈ rad( V ) [4 , 3 n − 3 , 4 ] n − 1 ≥ 2 2 • 1 • 1 • · · · 1 • 1 • 2 • c = b 0 + 2( b 1 + . . . + b n − 2 ) + b n − 1 [3 , 3 , 4 , 3] 4 1 • 1 • 1 • 2 • 2 • c = b 0 + 2 b 1 + 3 b 2 + 2 b 3 + b 4 [3 , 6] 2 1 • 1 • 3 • c = b 0 + 2 b 1 + b 2 [ ∞ ] 1 1 • = = = 1 • c = b 0 + b 1 T a ble 3. Euclidean Coxe ter Groups An inv estigation of t he action of these discrete reflection g roups on the Euclidean space A n − 1 sho ws, in eac h case, that G ≃ T ⋊ H splits as the semidirect pro duct of the (normal) subgroup T of translations with a certain (finite) p oint g r oup H ( see [4, Prop. 4.2]). W e can and do displa y eac h group in the ta ble so that H = G 0 = h r 1 , . . . , r n − 1 i . No w w e are in a p osition to surv ey t he mo dular reduction of the Euclidean groups in T a ble 3. Again w e more generally consider Euclidean subgroups E = h r j , . . . , r j + m i ≃ T ⋊ h r j +1 , . . . , r j + m i (10) of our usual group G ; and once more w e allow v ar io us p oss ible basic systems. Notice that w e sp ec ifically assume tha t E is embedded in G so that the p oin t subgroup (of spherical t yp e) is h r j +1 , . . . , r j + m i . Because of this, w e can use the splitting in (10 ) to actually p erform explicit calculations, although the details are quite in v o lv ed. W e b egin with Lemma 5.1 L et G b e a c rystal lo g r aphic line ar C o x et er gr oup with string diagr am. Supp ose that E = h r j , . . . , r j + m i is the (Euclid e an) sub gr oup of G c orr esp onding to one of the sub di - agr ams displaye d in T able 4 or T able 5, so that E = T ⋊ H , with tr a n slation gr oup T and (spheric al) p oint gr oup H = h r j +1 , . . . , r j + m i . Also supp ose that s, m , and the no des j, j + m ar e r estricte d in one of the v arious ways indic a te d in the T ables, s o in p articular H ≃ H s . L et ϕ : E → E s ⊆ G s b e the natur al epimorphis m for mo dulus s ≥ 2 . (a) Then k er( ϕ ) ⊂ T . 10 (b) E s is a string C -gr oup, nam ely the automorphi s m gr oup of a r e gular m -tor oid. (c) If T s acts faithful ly on the Z s -submo dule sp an n e d by b j , . . . , b j + m , then T s ∩ h r j +1 , . . . , r j + m , . . . , r j + l i s = { e } , for any l ≥ m . Pro of . As alw a ys, our calculations ma y w ell dep end on the underlying c hoice of basic system { b i } fo r G , as enco ded in the diagram ∆( G ). By insp ection of the v arious diagrams in T ables 4 and 5, w e confirm in each case that E = T ⋊ H , with H = h r j +1 , . . . , r j + m i . F urthermore, w e also observ e that the radical of P j + m k = j R b k is spanned by an in te gr al v ector c = P j + m k = j x k b k , in which the co effic ien t of b j is x j = 1. No w for pa r t (a) let g = th ∈ k er( ϕ ), with t ∈ T , h ∈ H , so that t ≡ h − 1 (mo d s ). F o r j ≤ i ≤ j + m , we hav e t ( b i ) = b i + z i c , with z i ∈ Z (the co efficien t of b j in c is 1), since t is a translation and the lattice P j + m k = j Z b k is inv ariant under E ; likew ise h − 1 ( b i ) = b i + v i , with v i ∈ P j + m k = j +1 Z b k , since h ∈ h r j +1 , . . . , r j + m i . Th us z i ≡ 0 (mo d s ), so that h − 1 ≡ e (mo d s ). Since reduction mo dulo s is fait hf ul on H , we ha v e h = e (in c hara cteristic 0), and g = t ∈ T . F or par t (b) w e first of all note that the subgroups H = h r j +1 , . . . , r j + m i and A := h r j , . . . , r j + m − 1 i are spherical, since the v arious constrain ts on s, m j,j − 1 , m j + m,j + m +1 in T a- bles 4 and 5 g uaran tee t hat b oth subgroups are fa it hf ully represen ted mo d s ; see Section 4. No w (b) f o llo ws at once fro m (a), since k er( ϕ ) is a normal subgroup of translations; see [6, 6D-E]. Here we also need to mak e a forw ard app eal to the computatio n of the type v ector q of T ables 4 and 5, eliminating the p ossibilit y that the index of ker( ϕ ) in T is to o small for E s to b e p olytopal. (W e can also give a direct pro of of the inters ection prop ert y of E s using [6, Prop. 2E16(a)]. Since the subgroups A, H are b oth (spherical) string C -groups, w e need only sho w that A s ∩ H s ⊆ h r j +1 , . . . , r j + m − 1 i s . So supp ose g ∈ A and h ∈ H (b oth in c ha racteris tic 0) suc h that g ≡ h mo d s . Then h − 1 g =: t ∈ k er( ϕ ) ⊆ T . Now let T b e the regular tessellation in Euclidean m -space asso ciated with E , let o b e t he base v ertex of T , and let z b e the cen ter o f the base facet (tile) F of T . Then t − 1 ( h − 1 ( z )) = g − 1 ( z ) = z , so t m ust b e the translation b y the v ector h − 1 ( z ) − z . Since h − 1 ( z ) is the cen ter of the facet h − 1 ( F ) of T and o is a v ertex of h − 1 ( F ) , t he t w o vertice s h − 1 ( z ) and z of the dual of the v ertex-figure of T at o are equiv a lent under t and thus under ke r ( ϕ ). Hence, if t is non-trivial, then reduction mo dulo s collapses the v ertex-figure of T at o , contrary to t he fact that H s is isomorphic to H . Therefore, t m ust b e trivial and g = h ∈ A ∩ H = h r j +1 , . . . , r j + m − 1 i . It follo ws that the mo dular imag es of g and h a re in h r j +1 , . . . , r j + m − 1 i s , as required. Alterna- tiv ely w e can argue here as follows. The tra ns lation v ectors of the conjugat es of t under H generate a sublattice of ker( ϕ ) with ve ry small index in T ; how ev er, our computatio n of the t yp e v ectors q has show n that this cannot o ccur.) F or part (c) w e let ϕ ( t ) = ϕ ( h ) ∈ T s ∩ h r j +1 , . . . , r j + l i s . Again t ( b i ) ≡ b i (mo d s ) for j ≤ i ≤ j + m , so that by h yp othesis w e ha v e t ≡ e (mo d s ).  Remarks . W e ha v e seen that H ≃ H s alw a ys holds when s ≥ 3 and o ccasionally when s = 2; under the constrain ts on m indicated in T ables 4 and 5 , it also holds for s = 2. A consequence of our calculations is that, for all the cases detailed in T ables 4 and 5, the semidirect splitting (10) of E = h r j , . . . , r j + m i (in c haracteristic 0) surviv es reduction mo dulo s . Thus , E s ≃ T s ⋊ H s , although it is not necessarily the case that T s ≃ Z m s . 11 Of course, taking the r i ’s in rev erse order, we o bta in a dual v ersion of L emma 5.1 . In applications, w e m ust then tak e care that the sub diagrams in T ables 4 and 5, along with the attached constraints, really ha v e b een flipp ed end-fo r - end. Next w e m ust deal with the sp ecific features o f eac h group G . Guided b y [6 , 6D-E], w e can, with some effort, write out explicit matrices for standard generators t 1 , . . . , t m of the translation subgroup T ⊂ h r j , . . . , r j + m i . Suc h matrices incorp orate the unsp ecified, but crucial, Cartan in tegers m j,j − 1 and m j + m,j + m +1 and furthermore v ary a little with the choice of the underlying basic system. But fro m Lemma 5 .1(b) we kno w t ha t E s is a string C - group. T o finish off its description, w e identify the t yp e v ector q by calculating the p erio ds of the k ey translations t 1 , t 1 t 2 and t 1 t 2 . . . t m . It is conv enien t now to separate our results into tw o lots: (a) The groups [4 , 3 m − 2 , 4 ] ( m ≥ 2 ) . When h r j , . . . , r j + m i ≃ [4 , 3 m − 2 , 4 ], w e must con tend with the three distinct basic sys tems sho wn in T a ble 4. F or an y s ≥ 3, w e observ e that h r j , . . . , r j + m i s is the group of a suitable cubic toroid { 4 , 3 m − 2 , 4 } q of ra nk m + 1 (on the m -torus), w hose t yp e vector q is a lso displa yed in the T a ble . The same holds for s = 2, so long as terminal no des j and j + m are constrained as indicated. This restriction guaran tees tha t t he facet a nd v ertex-figure subgroups ar e spherical, with the correct rank m (see Section 4 a bov e). F or an y other terminal no de types when s = 2, one finds that h r j , . . . , r j + m i 2 either fails to ha ve inv olutory generators (so is not a string C -group) or is lo c al ly p r oje ctive rather than toroidal (see [6, 14A] and [3]). Sub diagram of ∆( G ) Mo dulus Affine Constrain ts o n T yp e v ector on no des j, . . . , j + m s dim. m ≥ 2 no des j, j + m q 2 a • a • · · · a • 2 a • o dd s ≥ 3 an y — ( s, 0 , . . . , 0) ev en s ≥ 4 m o dd at least one o- o ( s, 0 , . . . , 0 ) ev en s ≥ 4 m o dd b oth o-e ( s 2 , s 2 , . . . , s 2 ) ev en s ≥ 4 m even — ( s 2 , s 2 , . . . , s 2 ) s = 2 m o dd b oth o - o (2 , 0 , . . . , 0) a • 2 a • · · · 2 a • a • o dd s ≥ 3 an y — ( s, 0 , . . . , 0) ev en s ≥ 4 an y a t least one o-e ( s, 0 , . . . , 0) ev en s ≥ 4 an y b oth e-e ( s 2 , 0 , . . . , 0) s = 2 an y b oth o -e (2 , 0 , . . . , 0) 4 a • 2 a • · · · 2 a • a • o dd s ≥ 3 an y — ( s, 0 , . . . , 0) ev en s ≥ 4 an y j + m is e-e ( s, 0 , . . . , 0) ev en s ≥ 2 an y j + m is o-e ( s, s, 0 , . . . , 0) T a ble 4. Groups fo r the cubic toroids (b) The sp ecial groups [3 , 3 , 4 , 3] ( m = 4 ), [3 , 6] ( m = 2 ) and [ ∞ ] ( m = 1 ) . 12 Similar remarks apply to the remaining Euclidean groups h r j , r j +1 , r j +2 , r j +3 , r j +4 i ≃ [3 , 3 , 4 , 3], h r j , r j +1 , r j +2 i ≃ [3 , 6] or h r j , r j +1 i ≃ [ ∞ ] (and their duals). F or the first tw o groups w e may exclude the mo dulus s = 2, for whic h t he re is a colla ps e in either the facet or v ertex-figure. Our calculations are summarized in T able 5. The resulting p olytop es are regular toroids { 3 , 3 , 4 , 3 } q of rank 5 (on the 4- torus), { 3 , 6 } q of rank 3 (on the 2- torus), and regular p olygons { q } (on the 1-t o rus ), when q = ( q ) in the latter case. Note for the group [3 , 6] that the residue of the Carta n in teger m j,j − 1 (mo d 3) is a consideration (see [8, 5.6]). Remark . W e ha v e surv ey ed here the Euclidean subgroups E of G . W e emphasize that an y reduced subgroup E s not explicitly co vere d (up to dualit y) b y an en tr y in T able 4 or T able 5 will fail in some w ay to b e the group of a regular toroid. Sub diagram o f ∆( G ) Mo dulus Affine Constrain ts on T yp e v ector on no des j, . . . , j + m s dim. m n o des j, j + m q a • a • a • 2 a • 2 a • o dd s ≥ 3 4 — ( s, 0 , 0 , 0) ev en s ≥ 4 4 no de j is o-o ( s, 0 , 0 , 0) ev en s ≥ 4 4 no de j is o-e ( s 2 , s 2 , 0 , 0) 2 a • 2 a • 2 a • a • a • an y s ≥ 3 4 — ( s, 0 , 0 , 0) a • a • 3 a • s ≡ ± 1 (mo d 3) 2 — ( s, 0) ( s > 2) s ≡ 0 (mo d 3) 2 m j,j − 1 ≡ ± 1 (mo d 3 ) ( s, 0) s ≡ 0 (mo d 3) 2 m j,j − 1 ≡ 0 ( m o d 3) ( s 3 , s 3 ) 3 a • 3 a • a • an y s ≥ 3 2 — ( s, 0) a • = = = a • o dd s ≥ 3 1 — ( s ) ev en s ≥ 4 1 some no de o- e ( s ) ev en s ≥ 4 1 b oth no des e-e ( s 2 ) s = 2 1 b oth no des o-e (2) 4 a • a • o dd s ≥ 3 1 — ( s ) ev en s ≥ 4 1 no de j + 1 is e-e ( s ) ev en s ≥ 2 1 no de j + 1 is o-e (2 s ) T a ble 5. Groups for the sp ecial toroids 13 6 The Quotient C riterion The following result is a mo dular v ariant of the quotient criterion in [6, 2E17]. As usual there is a dual vers ion with subgroups G n − 1 and G 0 in terc ha nged . Theorem 6.1 L et G = h r 0 , . . . , r n − 1 i b e a crystal lo gr aphic line ar Coxeter gr o up with string diagr am, and supp ose G s is a s t ring C -gr oup for mo dulus s ≥ 2 . Supp ose also that s | d and that either (a) G n − 1 is of spheric al typ e and that G n − 1 ≃ G s n − 1 (so that the underlying b as i c system of G is r estricte d as e x plaine d in § 4 when s = 2 ); or (b) G n − 1 = T ⋊ G 0 ,n − 1 is of Euclide an typ e, with tr a nslation gr oup T an d (f a it hful ly r ep r e- sente d) sp h eric al p oin t gr oup G 0 ,n − 1 ≃ G s 0 ,n − 1 (so that n ≥ 3 an d the underlying b asic system of G is r estricte d a s explaine d in § 5 ). Also assume in this c ase that T d ∩ h r 1 , . . . , r n − 1 i d = { e } . (11) Then G d is a string C -gr oup. Pro of . W e adapt the pro of of [6, 2 E17 ]. Since s | d w e ha v e natura l epimorphisms η : G → G d and ϕ : G d → G s . F or clarit y w e av oid our customary a bu se o f notatio n and tak e care to distinguish the standard generators q j := η ( r j ) o f G d and s j := ϕ ( q j ) o f G s . Since G s is a string C -group, eac h s j and hence eac h q j is an in volution. By [6, 2E16(b)], w e need only sho w that G d n − 1 is a string C- g roup and, for 1 ≤ k ≤ n − 1, that G d n − 1 ∩ h q k , . . . , q n − 1 i ⊆ h q k , . . . , q n − 2 i . So, b eginning with the latter, let g ∈ G d n − 1 ∩ h q k , . . . , q n − 1 i ; then ϕ ( g ) ∈ h s k , . . . , s n − 2 i ⊆ G s n − 1 , since G s is a string C -g roup. In t he spherical case (a), ϕ is 1–1 on G d n − 1 , since G n − 1 ≃ G s n − 1 ( ≃ G d n − 1 ). Th us g ∈ h q k , . . . , q n − 2 i . Consider the Euclidean case (b). There exists (a unique) h ∈ h q k , . . . , q n − 2 i with ϕ ( h ) = ϕ ( g ). Applying Lemma 5.1 to ϕ ◦ η (restricted to G n − 1 ), w e hav e g = th for some translation t ∈ T d . By (11) w e get t = e , so that g ∈ h q k , . . . , q n − 2 i . Finally , G d n − 1 is a string C-group in eac h case. This follo ws from applying our consider- ations in Sections 4 and 5 to G n − 1 , since switc hing from s to a multiple d merely eases a ny constrain ts whic h could prev en t G d n − 1 from b eing a string C -g roup.  Example and R em arks . In general, some condition lik e (1 1) is necessary . Consider, for instance, the diagrams a • 1 • = = = 1 • and 1 • a • = = = a • . (12) F or a ∈ { 1 , 2 , 3 } , the corresp onding groups of rank 3 reduce to string C -gro ups for any mo dulus d > 2. In the left diagram w e can ev en ta k e a = 4 and so o bta in a p olyhedron of t yp e { d, d } , for o dd d ≥ 3, or t yp e { d, d 2 } , for ev en d ≥ 4 . How ev er, taking a = 4 in the right diagram, w e find that the interse ction condition fails precisely when the mo dulus d = 2 s , with s o dd: for then t = ( r 0 r 1 ) s = ( r 1 r 2 ) s 6 = e (mo d d ); and t ∈ T d ∩ h r 1 , r 2 i d directly con tradicts (11). W e shall see that the fault lies in the embedding of the sub diagrams for facet and vertex -figure. 14 T o explain what is g oing on w e use Lemma 5.1(c) (with s = d, j = 0 , m = n − 2 , l = m + 1). Th us (11) is fulfilled whenev er T d acts faithfully on the Z d -submo dule V n − 1 . This holds, for example, when dropping no de n − 1 has no effect on the em b edding constrain ts for G n − 1 , as describ ed in the T ables. T o see this, note tha t r i induces a mapping ˜ r i on V n − 1 , for 0 ≤ i ≤ n − 2. Clearly , K d := h ˜ r 0 , . . . , ˜ r n − 2 i is just the (toroidal) gro up corresp onding to the the sub diagram of ∆( G ) o btained b y deleting no de n − 1. If, as w e supp os e, this deletion has no effect on the constrain t s on no de n − 2, it mus t b e that G d n − 1 and K d ha v e the same t yp e vector q , as given in the T ables. Since the corresp onding spherical p oin t gro up s a r e isomorphic, it follows that G d n − 1 ≃ K d and that T d acts faithfully on V n − 1 . Th us G d is a string C -group. In particular, we now see that (1 1) is redundan t whenev er d is o dd and in sev eral other instances. This leads to a n imp ortan t simplification: f o r d o dd w e need only c hec k that G s is a string C -group for some o dd prime divisor s = p . Occasionally , the mo dulus s = 4 is anot her k eystone. 7 Lo cally to roidal p olytop es In this Section, w e consider lo c al ly tor oidal regular p olytop es, that is p o ly top es of rank n ≥ 4 whose facets a nd v ertex-figures are globally spherical or toroidal, as describ ed ab o ve, with at least one kind toroidal. The n -p olytop es o f this kind ha v e no t y et b een fully classified, although quite a lot is know n (see [6, Chs. 10-12]). As usual, we b egin with a crystallogra ph ic linear Cox eter group G = h r 0 , . . . , r n − 1 i , but immediately discard degenerate cases in whic h the underlying diagr a m ∆( G ) is disconnected. (In suc h cases G is reducible; a nd P ( G ) and its quotien ts hav e the sort of ‘flat ness’ describ e d in [6, 4E].) In [9] we discuss ed a ll lo c ally tor oidal 4- polytop es P ( G p ) whic h arise from our construction with prime mo dulus p . Since our metho ds for general mo duli s add little to t he discussion of suc h p olytop es in [6, Chs. 1 0–11] and [9], w e examine here just one group of rank 4, namely G = [3 , 6 , 3], with diagram 3 • 3 • 1 • 1 • . When s = 4 we find that G 4 has o rde r 7680 and is the automorphism g roup o f a lo cally toroidal 4-p olytop e in the class h { 3 , 6 } (4 , 0) , { 6 , 3 } (4 , 0) i . Next w e note in T able 5 that there are no em b edding con traints on no de 2. W e conclude from Theorem 6.1(b) (and the subseque n t remarks) and from [9, p. 345] that G d is a string C -group whenev er the mo dulus d is divisible b y either 4 or an o dd prime, that is, whenev er d ≥ 3. The p olytop e P ( G d ) is in the class h { 3 , 6 } q , { 6 , 3 } r i , where alw ays q = ( d, 0), but r = ( d , 0) when 3 ∤ d a nd r = ( d 3 , d 3 ) when 3 | d . This construction complemen ts the approac h in [6, 11E]. T urning to higher rank n > 4, w e observ e that an y spherical facet, o r ve rtex-figure, m ust b e of type { 3 n − 2 } , { 4 , 3 n − 3 } , { 3 n − 3 , 4 } or { 3 , 4 , 3 } ( n = 5 only). Lik ewise, the required Euclidean section m ust hav e type { 4 , 3 n − 4 , 4 } or when n = 6, { 3 , 3 , 4 , 3 } or { 3 , 4 , 3 , 3 } . As described in [6, Lemma 1 0 A1], these constrain ts sev erely limit the p ossibilities: in rank 5, w e hav e just G = [4 , 3 , 4 , 3] acting on hy p erb olic space H 4 ; and in rank 6 w e ha v e G = [4 , 3 , 3 , 4 , 3] , [3 , 4 , 3 , 3 , 3] or [3 , 3 , 4 , 3 , 3], all acting on H 5 . Thus w e ma y complete our discussion by examining the mo dular p olytop es whic h r e sult from these g roups in ranks 5 15 and 6. 7.1 Rank 5: the group G = [ 4 , 3 , 4 , 3] Here w e m ust con tend with the four distinct basic system s enco ded in the diag rams 1 • 2 • 2 • 4 • 4 • 1 • 2 • 2 • 1 • 1 • ( a ) ( b ) 2 • 1 • 1 • 2 • 2 • 4 • 2 • 2 • 1 • 1 • ( c ) ( d ) (13) When the mo dulus is an o dd prime p , the four corresp onding finite groups G p are isomorphic string C -gr o ups ; and w e r e call from [10, § 4.1] that G p =  O 1 (5 , p, 0 ) , if p ≡ ± 1 (mo d 8) O (5 , p, 0) , if p ≡ ± 3 (mo d 8) (14) Note that O 1 (5 , p, 0 ) has order p 4 ( p 4 − 1)( p 2 − 1) and index tw o in O (5 , p, 0) (see [8, pp. 3 0 0- 301]). The facets of the corresp onding regular 4-p olytop e P ( G p ) a r e toroids { 4 , 3 , 4 } ( p, 0 , 0) , whic h one could construct by iden tifying opp osite square faces of a p × p × p cub e [8, 6.4]. Of course, the v ertex-figures are copies of the 24-cell { 3 , 4 , 3 } . Next, f or mo dulus s = 4, we ma y c hec k directly on GAP that G 4 is a string C - group for eac h of the basic systems in (13). Diagrams (a), (b), (c) giv e p olytop e s of t yp e { { 4 , 3 , 4 } (4 , 0 , 0) , { 3 , 4 , 3 } } , whose r esp ectiv e a ut o morphis m groups hav e orders g = 2 16 · 3 2 , g , and 4 g . On t he other hand, dia gram (d) gives a p olytop e of t yp e { { 4 , 3 , 4 } (4 , 4 , 0) , { 3 , 4 , 3 } } whose group has order 16 g . By [6, 1 2B1], none of these p olytop es can b e univ ersal for their t yp e. Ho w ev er, with differen t generato r s , the third group, of order 4 g = 2 359 296 , is the automorphism gr oup for the univ ersal p olytop e of t yp e { { 4 , 3 , 4 } (2 , 2 , 2) , { 3 , 4 , 3 } } and hence is kno wn to b e isomorphic to ( Z 6 2 ⋊ Z 5 2 ) ⋊ F 4 (see [6, Thm. 8F19 and T able 12B1]). No w consider any mo dulus d > 2, whic h a gain is divisible either b y an o dd prime s or b y s = 4. W e immediately conclude fro m Theorem 6.1(a), in its dual form, that G d is a string C -group for eac h diagram in (13) a nd fo r eac h mo dulus d > 2. If d is o dd , it is easy to c hec k t ha t the four diag rams deliv er isomorphic groups. Indeed, a c ha nge from any one of the four basic systems to a no the r is accomplished b y r escaling v ario us b j ’s b y p o we rs of 2 (see [8, p. 30 5]). Since 2 is inv ertible mo dulo d , the corresp onding linear groups are conjugate in GL 5 ( Z d ); and, crucially , suc h isomorphisms pair off the sp ecifie d generating reflections. Consulting T able 4 (with s replaced by d ), w e conclude that the resulting no n-univ ersal p olytope ha s t yp e { { 4 , 3 , 4 } ( d, 0 , 0) , { 3 , 4 , 3 } } . (15) F or d even , we hav e a lready observ ed that a c hange in basic system may w ell alter the corresp onding group and p olytop e . Referring again to T a ble 4, w e do find that diagrams (a), (b), (c) in (13) pro vide p olytop es of the type display ed in (1 5 ), now with d ev en. How ev er, diagram (13)( d) giv es a p olytop e of t yp e { { 4 , 3 , 4 } ( d,d, 0) , { 3 , 4 , 3 } } . 16 Of course, in a ll the ab o v e cases, w e just as easily obtain the dual p olytop e of ty p e { 3 , 4 , 3 , 4 } by flipping a diagram end-for-end. The univ ersal lo cally toro idal p olytop es of r a nk 5 are describ e d in [6, 12B]. There ar e just three finite instances, whose facets are toroids with t yp e v ector (2 , 0 , 0), ( 2 , 2 , 0) or (2 , 2 , 2). Unfortunately , we cannot get an y of these b y o ur construction, since for s = 2 w e alwa ys ha v e b y (9 ) that the 24- cell collapses to its central quotien t, the ‘hemi-24-cell’ { 3 , 4 , 3 } 6 . On the other hand, for d > 2 our construction give s finite p olytop es of the type indicated; in con trast, the metho ds in [6 , p. 452] are non-constructiv e and app eal t o the residual finiteness of certain groups to establish the existenc e of suc h p olytop e s. Finally , in t his subsection, it is of some in terest to further inv estigate the case s = 2. W e ma y discard diagra ms (a) a nd (b), in whic h r 0 = e (mo d 2). Ho w ev er, diagra m (c) do es giv e a string C - g roup G 2 of o rde r 23 0 4, for the univ ersal p olytop e { K , { 3 , 4 , 3 } 6 } , where K := { { 4 , 3 } 3 , { 3 , 4 } } , so that 3- faces and v ertex figures are of pro jectiv e t yp e. Diagram (d) lik ewise giv es a group G 2 of order 9216; and the cor r e sp onding p olytop e is doubly cov ered b y the univ ersal p olytope of type { { 4 , 3 , 4 } (2 , 2 , 0) , { 3 , 4 , 3 } 6 } , whose gro up is Z 5 2 ⋊ ( F 4 / {± e } ) (see [6, Thm. 8F2 1]). 7.2 Rank 6: the groups [3 , 4 , 3 , 3 , 3] , [ 3 , 3 , 4 , 3 , 3] and [4 , 3 , 3 , 4 , 3] In rank 6 w e m ust consider three closely related g roups, b eginning with G = h r 0 , r 1 , r 2 , r 3 , r 4 , r 5 i ≃ [3 , 4 , 3 , 3 , 3 ] . A basic system (of ro ots) for G is described b y one of the f ollo wing diagrams: 1 • 1 • 2 • 2 • 2 • 2 • 2 • 2 • 1 • 1 • 1 • 1 • . ( a ) ( b ) (16) Next w e turn to the subgroup H = h s 0 , . . . , s 5 i generated b y the reflections ( s 0 , s 1 , s 2 , s 3 , s 4 , s 5 ) := ( r 1 , r 0 , r 2 r 1 r 2 , r 3 , r 4 , r 5 ) , (17) whic h has index 5 in G and is isomorphic t o [3 , 3 , 4 , 3 , 3]. Starting with the diagram (16)(b), w e find that the basic system of ro ots attach ed to the s j ’s is now enco ded in the diagram 2 • 2 • 2 • 1 • 1 • 1 • . (18) (Diagram (16)(a) merely leads, in dual fashion, to (18) flipp ed end-for- end. This is the only other diagr am admitted b y H .) 17 The final subgroup K = h t 0 , . . . , t 5 i generated b y ( t 0 , t 1 , t 2 , t 3 , t 4 , t 5 ) := ( r 2 , r 1 , r 0 , r 3 r 2 r 1 r 2 r 3 , r 4 , r 5 ) (19) has index 10 in G a nd is isomorphic to [4 , 3 , 3 , 4 , 3 ]. Now diagrams (16)(a),(b) lead to diagrams (2 0)(a),( b) b elo w: 2 • 1 • 1 • 1 • 2 • 2 • 1 • 2 • 2 • 2 • 1 • 1 • ( a ) ( b ) 4 • 2 • 2 • 2 • 1 • 1 • 1 • 2 • 2 • 2 • 4 • 4 • ( c ) ( d ) (20) The group K a dmits the tw o other basic systems shown in (20)(c),(d). (See [6, 12A2]. Eac h group describ ed ab o v e acts on H 5 with a simplicial fundamental domain of finite v olume. In [7], these indices w ere computed b y dissecting a simplex fo r H (or K ) in to copies of the simplex f o r G .) In [10, § 4.2] w e sho w ed that G p , H p , K p are string C -gr oups for any o dd prime mo dulus p . In fact, all three are isomorphic to  O 1 (6 , p, +1 ) , if p ≡ ± 1 (mo d 8) O (6 , p, +1) , if p ≡ ± 3 (mo d 8) (21) Of course, w e require differen t generators in the three cases, as indicated in (17) and (19). Th us, the indices 5 and 10 in characteris tic 0 collapse to 1 under reduction mo d p . F or any prime p ≥ 3, O 1 (6 , p, +1 ) ha s order p 6 ( p 4 − 1)( p 3 − 1)( p 2 − 1) and index tw o in O (6 , p , +1) (see [8, pp. 300-301 ]). No w supp ose that the mo dulus is a ny o dd inte ger d ≥ 3. Just as in the previous sub- section, the t w o diagrams in (16) giv e isomorphic g roups, a s do the fo ur diagrams in (20). F urthermore, by the r emarks follo wing Theorem 6.1 w e see that G d , H d and K d are then string C -gr o ups . In each case, the t yp e v ector fo r a toroidal section is q = ( d , 0 , 0 , 0). The situation fo r even mo duli is more complicated. Once more, w e may discard the mo dulus d = 2, whic h in v ariably causes a collapse to the hemi-24- cell in an y section of type { 3 , 4 , 3 } . Let us consider the three groups in turn. The Polytopes P = P ( G d ). Using GAP , w e find that G 4 is a string C -group of order 2 26 · 3 2 · 5 for either dia gram in (16). It follows from Theorem 6.1(a) in its dual fo r m that G d is a string C -group for an y mo dulus d > 2. F rom either diagram in (16) we obtain a lo cally toroidal p olytop e in the class h{ 3 , 4 , 3 , 3 } ( d, 0 , 0 , 0) , { 4 , 3 , 3 , 3 }i . W e note that the toroidal fa c ets of P ( G d ) eac h ha v e 3 d 4 v ertices [6, T able 6E1]; and, of course, the v ertex-figures are 5-cub es { 4 , 3 , 3 , 3 } . Although the t w o admissible diagrams do yield string C -groups, w e hav e no general pro of that these groups are isomorphic when d is even , though t his is true for d = 4. The following theorem establishes [6, Conjecture 12C2] concerning t he existence of lo cally toroidal regular 6-p olytop es of t yp e { 3 , 4 , 3 , 3 , 3 } . 18 Theorem 7.1 The universal r e gular 6 -p olytop es {{ 3 , 4 , 3 , 3 } ( d, 0 , 0 , 0) , { 4 , 3 , 3 , 3 }} and {{ 3 , 4 , 3 , 3 } ( d,d, 0 , 0) , { 4 , 3 , 3 , 3 }} exist fo r al l d ≥ 2 . Pro of . First note that the case d = 2 w as settled in [6, pp.460- 461]. So let d > 2. W e no w app eal to our earlier remark that a non- e mpt y class o f regular p olytop es contains a (unique) univers al member (see [6 , 4A2]). Th us, the existence of a unive rsal p olytop e of the first kind (t yp e vec tor q = ( d, 0 , 0 , 0)) follow s directly from our construction of a mem b er of its class, namely P ( G d ). F or t he existenc e of the univ ersal p olytop es of the second kind (t yp e vec tor q = ( d, d, 0 , 0) ) w e refer to the discuss ion in [6, pp.460 -462], where it w as sho wn that the existenc e o f the univ ersal p olytop es of the second kind is implied by existence of univ ersal p olytop es of the first kind. (In fact, some of the a r g ume n ts pro vided there can no w b e simplified using prop erties of G d .)  The full classification of the finite univ ersal p olytop es o f eac h kind is still o p en, but three of these are know n to b e finite, including {{ 3 , 4 , 3 , 3 } (3 , 0 , 0 , 0) , { 4 , 3 , 3 , 3 }} , with automorphism group Z 3 ⋊ O (6 , 3 , +1) ( = Z 3 ⋊ G 3 ). See [10, § 4.2]. The Polytopes P = P ( H d ). W e ha v e already indicated that fo r d o dd the p olytope P ( H d ) lies in the class h{ 3 , 3 , 4 , 3 } ( d, 0 , 0 , 0) , { 3 , 4 , 3 , 3 } ( d, 0 , 0 , 0) i . In fact, P ( H d ) admits an order rev ersing bijection and so is self-dual . The mo dulus p = 3 is of particular in terest. In [10, § 4.2] we g a v e a new construction for the corresp onding (finite!) self-dual univers al p olytop e U H 3 := { { 3 , 3 , 4 , 3 } (3 , 0 , 0 , 0) , { 3 , 4 , 3 , 3 } (3 , 0 , 0 , 0) } . Indeed, Γ( U H 3 ) ≃ ( Z 3 ⊕ Z 3 ) ⋊ H 3 under a non-tr iv ial action o f H 3 on the ab elian f actor. Th us U H 3 is a 9- fold cov er of P ( H 3 ) ([6, T a ble 12D 1]); and trapp ed b et w een w e find a twin pair Q , Q ∗ of non-sel f - d u al p olytop es, with the same toroidal facets a nd v ertex-figures: Q 3:1 # # G G G G G G G G G U 3 3:1 > > } } } } } } } } 3:1 A A A A A A A A P ( H 3 ) Q ∗ 3:1 ; ; w w w w w w w w T urning to ev en mo duli, we again find t ha t H 4 is a string C -group (of index 5 in G 4 ); and w e note tha t t here are no em b edding constrain ts o n no de 4 (lo ok at the second diagram in T able 5). Th us, by the discussion follo wing Theorem 6.1, w e conclude that H d is a string C -group for all d > 2. When d is even , t he corresp onding p olytop e is in the class h{ 3 , 3 , 4 , 3 } ( d, 0 , 0 , 0) , { 3 , 4 , 3 , 3 } ( d 2 , d 2 , 0 , 0) i , 19 and hence is certainly not self-dual. Notice that the type v ectors for the facets and verte x-figures of the p olytop es P ( H d ) are related in that they in v o lv e the same parameter d . Thus w e cannot exp e ct our metho ds to completely settle Conjecture 1 2 D3 o f [6] concerning the existence of lo cally t o roidal regular 6-p olytop es o f t yp es { 3 , 3 , 4 , 3 , 3 } , for whic h t he parameters for the facets and v ertex-figures ma y v ar y indep ende ntly . The same remark applies to the p olytopes P ( K d ) studied next, and Conjecture 12E3 of [6] for the corresp onding type { 4 , 3 , 3 , 4 , 3 } . The Polytopes P = P ( K d ). F or o dd d ≥ 3 the four diagra ms in (2 0) giv e isomorphic p olytop es in the class h{ 4 , 3 , 3 , 4 } ( d, 0 , 0 , 0) , { 3 , 3 , 4 , 3 } ( d, 0 , 0 , 0) i . Here the f ace ts are cubical toroids; facets and v ertex-figures eac h ha v e d 4 v ertices. Supp ose then that d ≥ 4 is ev en. A calculation with GAP reve als the at first surprising result that the intersec tion condition (2) fa ils for dia grams (20)(b)(d), at least when d = 4 , 6. Noting that dropping the la st no de in each case alters the constrain ts o n node 4, w e therefore abandon these diagrams. F or diagr a m (20)(a) w e easily ve rify tha t K 4 is a string C -group (of index 10 in G 4 ). Note that there a re no embedding constrain ts on no de 4; see the first diagram in T able 4 , with m = 4 and s = d ev en. F rom Theorem 6.1, w e th us obtain a p olytop e in the class h { 4 , 3 , 3 , 4 } ( d 2 , d 2 , d 2 , d 2 ) { 3 , 3 , 4 , 3 } ( d 2 , d 2 , 0 , 0) i , ( even d ≥ 4 ) . Here the f ace ts ha v e d 4 / 2 v ertices; and eac h v ertex-figure has d 4 / 4 v ertices. The analysis for diagram (2 0 )(c) is similar, although the particular lo cation of the sub- group [3 , 4 , 3 ] preve n ts an automatic v erification of condition (11). Nev ertheless, b y brute- force calculation, w e find that (11) holds for an y mo dulus d ≥ 2. On t he other hand, f o r d = 4 with this basic system, we can independently chec k on GAP that K 4 is indeed a string C -group, with (unexp ected) order 2 29 · 3 2 . It f ollo ws from Theorem 6.1(b) that K d is a string C -group for an y mo dulus d ≥ 3. In particular, when d ≥ 4 is ev en w e obtain a p olytop e in the class h { 4 , 3 , 3 , 4 } ( d,d, 0 , 0) { 3 , 3 , 4 , 3 } ( d, 0 , 0 , 0) i . Here the f ace ts eac h hav e 2 d 4 v ertices. Ac kno wledgemen t . W e wish to thank an anonymous referee for the careful reading of the man uscript a nd sev eral helpful commen ts on it. References [1] E. Artin, G e ometric Algebra, Inte rscience, New Y ork, 1957. 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