Modular Reduction in Abstract Polytopes

Mo dular reduction in abstract p olytop es B. Monson ∗ and Egon Sc h ulte † No vem b er 9, 2018 Abstract The pap er studies mo dular reduction tec hniques f or abstract regular and chiral p oly- top es, w ith t w o purp oses in mind: firs t, to survey the literat ur e ab out mo dular re- duction in p olytopes; a nd second, to ap p ly mo dular r ed uctio n, with mo duli giv en b y primes in Z [ τ ] (with τ the golden ratio), to construct new r eg ular 4-p olytop es of h yp er- b olic types { 3 , 5 , 3 } and { 5 , 3 , 5 } with automorphism grou p s giv en b y fi nite orthogonal groups. Key W ords: abstract p olytop es, regular and c hiral, Coxet er groups, mo dular reduction AMS Sub ject C la ssification (2000) : Primary: 51M20. S eco ndary: 20F55. 1 In tro ducti on P olytop es and their sy mmetry ha v e inspired mathematicians since an tiquit y . In t he past three decade s, a mo dern abstract theory of po lytopes has emerged featuring an attractive in terpla y of mat hematical ar eas, including geometry , com binatorics, group theory , graph theory and top ology (see [21]). Abstract p olytop es share man y prop erties with ordinary con v ex p olytop es but a priori are not em bedded in the geometry of an ambie n t space. The presen t pa p er studies mo dular reduction techn iques for regular and c hiral abstract p olytop es . Mo dular reduction has prov ed to b e a p o w erful to ol in the construction and analysis of new classes of p olytop es. Our paper serv es t w o purp oses: first, to surv ey the literature ab out mo dular reduction tec hniques in p olytop es; a nd second, to apply mo dular reduction, with mo duli g iven b y primes in the ring of in tegers Z [ τ ] of the quadratic num b er field Q ( √ 5), to construct new infinite classes of regular 4- polytop es of h yperb olic t ypes { 3 , 5 , 3 } and { 5 , 3 , 5 } with automorphism groups give n b y o r t ho gonal groups o v er finite fields. Last, but not least, as a birthday greeting, w e wish to ac kno wledge the many con tributions b y our friend and colleague T ed Bisztriczk y in con v exit y , p olytop e theory and combinatorial geometry (see also [2]). ∗ Suppo rted by NSERC of Canada Gr an t # 4818 † Suppo rted by NSA-grant H9823 0-07-1-0 005 1 2 Basic notion s and metho ds F or general back ground material on abstract p olytop es w e refer the reader to [21, Chs. 2,3]. Here we just review some basic terminology . An ( abstr act ) p olytop e o f r ank n , or an n -p olytop e , is a partially ordered set P with a strictly monot one rank function having range {− 1 , 0 , . . . , n } . An elemen t of rank j is called a j - f a c e of P , a nd a face of rank 0 , 1 o r n − 1 is a vertex , e dge or fac et , resp ectiv ely . The maximal c hains, or flags , of P all con tain exactly n + 2 faces, including a unique least fa ce F − 1 (of ra nk − 1) and a unique greatest face F n (of rank n ). Tw o flags are said to b e adjac ent ( i - adjac ent ) if they differ in a single face (just their i -face, respectiv ely). Then P is required to b e s tr ongly flag-c onn e cte d (see [2 1, Ch.2]). Finally , P has the following homogeneit y prop ert y: whenev er F ≤ G , with F a ( j − 1)-face and G a ( j + 1)- face for some j , then there are exactly tw o j - faces H with F ≤ H ≤ G . Whenev er F ≤ G are faces of ranks j ≤ k in P , the se ction G/ F := { H ∈ P | F ≤ H ≤ G } is thus a ( k − j − 1)-p olytop e in its ow n right. In particular, w e can iden tify F with F /F − 1 . Moreo v er, w e call F n /F the c o-fac e at F , or the v e rtex- fi gur e at F if F is a v ertex. Our in terest is primarily in regular or chiral p olytop es. A p olytop e P is r e gular if its automorphism gr oup Γ( P ) is transitiv e on the flags of P , and P is chir al if Γ( P ) has tw o flag o rbits suc h that adjacen t flags are alw a ys in distinct orbits. F or a regular p olytope P , the group Γ( P ) is generated by n in v olutions ρ 0 , . . . , ρ n − 1 , where ρ i maps a fixed, or b ase , flag Φ to the flag Φ i , i -adja ce nt to Φ. The se generators satisfy (at least) the standard Coxe ter-type relations ( ρ i ρ j ) p ij = ǫ for i, j = 0 , . . . , n − 1 , (1) where p ii = 1, p j i = p ij =: p i +1 if j = i + 1, and p ij = 2 o therwis e; thus the underlying Co xeter diagram is a string diagr a m. Note t ha t p j := p j − 1 ,j ∈ { 2 , 3 , . . . , ∞} for j = 1 , . . . , n − 1. These n um b ers determine the ( Sc h l¨ afli ) typ e { p 1 , . . . , p n − 1 } of P . Moreo v er, the f ollo wing interse ction c ondition holds: h ρ i | i ∈ I i ∩ h ρ i | i ∈ J i = h ρ i | i ∈ I ∩ J i ( I , J ⊆ { 0 , 1 , . . . , n − 1 } ) . (2) The rotations σ i := ρ i ρ i − 1 ( i = 1 , . . . , n − 1) generate the r otat ion sub gr oup Γ + ( P ) of Γ( P ), whic h is of index at most 2. W e call P dir e ctly r e gular if this index is 2. A group Γ = h ρ 0 , . . . , ρ n − 1 i whose generators satisfy (1) and ( 2), is called a string C- gr oup ; here, the “C” stands for “Coxe ter”, though not ev ery C-group is a Cox eter gr oup. These string C-g roups are precisely the automorphism gro ups of r egula r polytop es, since, in a natural w ay , suc h a p olytop e can be uniquely reconstructed from Γ (see [21, § 2E]). Therefore, w e often iden tify a regular p olytope with its a uto morphism (string C-) gr oup. The gro up Γ( P ) of a c hiral p olytop e P is g ene rated b y n − 1 elemen ts σ 1 , . . . , σ n − 1 , whic h again are asso ciated with a base flag Φ = { F 1 , F 0 , . . . , F n } , suc h t ha t σ i fixes all the faces in Φ \ { F i − 1 , F i } and cyc lically p erm utes (“ro t ates”) consecutiv e i -faces of P in the (p olygonal) section F i +1 /F i − 2 of r a nk 2. By replacing a generator b y its inv ers e if need b e, w e can further require t hat, if F ′ i denotes the i -face of P with F i − 1 < F ′ i < F i +1 and F ′ i 6 = F i , then σ i ( F ′ i ) = F i . The resulting distinguished generators σ 1 , . . . , σ n − 1 of Γ( P ) then satisfy 2 relations σ p i i = ( σ j σ j +1 . . . σ k ) 2 = ǫ for i, j, k = 1 , . . . , n − 1 , with j < k , (3) where again the n um bers p i determine the t yp e { p 1 , . . . , p n − 1 } of P . Moreo v er, Γ( P ) and its generators satisfy a certain in tersection condition resem bling that for C-gr o ups. Conv ersely , if Γ is a group g enerated b y σ 1 , . . . , σ n − 1 suc h tha t the relatio ns (3) and the new in tersection condition hold, then Γ is the group of a ch iral p olytop e, or the rotation subgroup for a directly regular p olytop e; here the p olytop e is regular if and only if Γ admits an in v olutory automorphism ρ such that ρ ( σ 1 ) = σ − 1 1 , ρ ( σ 2 ) = σ 2 1 σ 2 , and ρ ( σ j ) = σ j for j ≥ 2. F or a chiral p olytop e P , the tw o flag o rbits yield tw o sets of g ene rators σ i whic h are not conjuga t e in Γ( P ); th us a c hiral p olytop e o ccurs in tw o enantiomorphic (mirror image) f orms. W e now describ e t he basic idea of mo dular reduction. W e b egin with a linear group G o v er a ring D , c hoo se an ideal J of D , and try to construct a p olytop e fro m the quotient group o f G o btained b y viewing G a s a linear g r oup o v er D /J . F or the latter step the main obstruction is ty pically the in tersection condition for the resulting quotien t g r o up of G . In most applications, G itself is alr eady a string C-group (often a Coxete r gr oup), D is the ring of integers in an algebraic n um b er field, and J is an ideal of D . More precisely , let D b e a comm utativ e ring with identit y 1, let V b e a fr ee mo dule ov e r D of rank n with basis b 0 , . . . , b n − 1 , and let G b e a subgroup of the general linear group GL n ( D ) o v er D , whose elemen ts w e ma y view as in v ertible linear transformations (mo dule isomorphisms) of V . No w let J b e an ideal of D . Then the natural ring epimorphism D → D /J defined b y a → a + J induces a group epimorphism G → G J , where G J is a subgroup of GL n ( D /J ), the m o dular r e duction of G ; in other w ords, w e obtain G J b y simply viewing the matrix en tries of the elemen ts in G as entries in D /J . Then G J naturally acts on the free mo dule V J o v er D /J of rank n with basis b J 0 , . . . , b J n − 1 . W e often abuse notatio n b y referring to the mo dular images of ob j ects b y t he same name (suc h as V or b 0 , . . . , b n − 1 , etc.); that is, w e drop the sup erscript J . Mo dular reduction is, of course, a natural idea and has been used in v arious wa ys t o construct maps a nd p olytop es [28, 29, 3 5 ]. Here we b egin b y applying the idea to crystallo- graphic Co xeter groups. 3 P olytop es from cryst allographic Co xeter groups Let Γ b e an abstract string Co xeter group with generator s ρ 0 , . . . , ρ n − 1 and presen tation as in (1), where aga in p ii = 1 and p ij = 2 for | i − j | ≥ 2. Let V b e real n -space, with basis α = { a 0 , . . . , a n − 1 } and symmetric bilinear form x · y defined by a i · a j := − 2 cos π p ij , 0 ≤ i, j ≤ n − 1 . (4) Let R : Γ → G b e the (faithful) standard represen tation of Γ in V , where G = h r 0 , . . . , r n − 1 i is the isometric refle ction group on V generated b y the reflections with r o ots a i (see [14, § 5.3–5.4]); th us, r i ( x ) = x − ( x · a i ) a i ( i = 0 , . . . , n − 1) . No w let m = 2 m ′ , where m ′ is the low es t common multiple of all p ij whic h are finite. Let ξ b e a primitiv e m - th ro ot of unity , and let D := Z [ ξ ]. Then, with resp ect to the basis α of V , the 3 reflections r j are represen ted b y matrices in GL n ( D ) so that w e may view G a s a subgroup of GL n ( D ). (By [9, Th. 21.1 3], D is the ring of integers in the a lg ebraic num ber field Q ( ξ ); and D has (finite) rank φ ( m ) as a Z -mo dule.) No w w e can r e duc e G mo d p , for any prime p , here a llo wing p = 2 [9, ch . XI I]. More precisely , supp ose tha t p is a ratio nal prime and tha t J is a maximal ideal in D with p D ⊆ J ⊂ D . Then K := D /J is a finite field of characteris tic p , and reduction mo d p of G is ac hiev ed b y a pply ing the natura l epimorphism D → K to the matrix entries of the elemen ts in G . This then defines a represen tation κ : G → GL n ( K ) with imag e g roup G p := G J = κ ( G ). (This construction is essen tially indep enden t of the c hoice of J .) Note that κ is faithful when G is finite and p ∤ | G | (in fact, κ often is f a ithful ev en when p divides | G | ). In any case, k er κ is a p -subgroup of G . The mo dular reduction tec hnique is considerably mor e straigh tforw ard fo r crystal lo- gr aphic Coxe ter groups, meaning that G (or Γ) lea v es in v ariant some lattice in V . A (string) Co xeter group G is kno wn to b e crystallographic if and only if p ij = 2, 3, 4, 6 or ∞ for all i 6 = j . In fact, ev en allow ing non-string diag rams, it is also true that G is crystallographic if and only if there is a b asic system β = { b 0 , . . . , b n − 1 } , with b i := t i a i for certain t i > 0, suc h that m ij := − t − 1 i ( a i · a j ) t j ∈ Z for 0 ≤ i, j ≤ n − 1. Then, fo r the rescaled ro ots b i , we ha v e r i ( b j ) = b j + m ij b i , (5) so that t he generators r i are represen ted by in tegral mat rice s with resp ect to the basis β , and the corresp onding r o ot lattic e ⊕ j Z b j actually is G -inv ariant. Th us we may take D = Z , and G can be reduced mo dulo any integer d ≥ 2. W e fo cus primarily on the case when p is an o dd prime, but address some questions regarding comp osite mo duli as w ell. F or a comp osite mo dulus d ≥ 2, the reduced g roup G d generally do es not “split” according to the prime f actorization of d , and its structure dep ends more heavily on the diagram ∆( G ) used in the reduction pro cess (t w o diagrams whic h mo dulo a prime are equiv alen t, may not b e equiv alen t mo dulo a comp osite mo dulus). W e now describ e these diagrams. The v arious p ossible basic systems { t i a i } for a crystallographic Cox eter group G can b e represen ted b y a dia g ram ∆( G ) ( se e [5, p. 415] or [22]): for 0 ≤ i, j ≤ n − 1, no de i is lab elled 2 t 2 i (= b 2 i ); and distinct no des i 6 = j a r e joined b y λ ij := min { m ij , m j i } unlab el le d branche s. Note that λ ij = λ j i = 0, 1 or 2, so the underlying gra ph is essen tially that o f the underlying Co xeter diagra m ∆ c ( G ) of G , except that a mar k p ij = ∞ is indicated b y a double d branch in the case tha t m ij = m j i = 2. In T able 1 we displa y the p ossible sub diagrams corresp onding to the dihedral subgroups h r i , r j i . F or simplicit y w e hav e replaced the no de lab els 2 t 2 i , 2 t 2 j b y s, t o r s, k s ( k = 1 , 2 , 3 , 4), as appropriate. Note here that m ij m j i = 4 cos 2 π p ij , so that , if m ij ≥ m j i ≥ 1, w e hav e t 2 j /t 2 i = m ij /m j i = 1 , 2 , 3 , 4 (or 1) for p ij = 3 , 4 , 6 , ∞ , resp ectiv ely . No des i, j P arameters λ ij s • t • p ij = 2 0 s • − − − k s • p ij = 3 , 4 , 6 , ∞ 1 ( k = 1 , 2 , 3 , 4) s • = = = s • p ij = ∞ 2 ( k = 1 ) T able 1. Ba sic Systems for the Crystallographic Dihedral G r oups h r i , r j i . 4 The Gram matrix B = [ b ij ] := [ b i · b j ] is easily computed from ∆( G ), since b ii = 2 t 2 i is simply the lab el attac hed to no de i , and b ij = − λ ij max { b ii , b j j } / 2 for i 6 = j . Moreov er, m ij = λ ij max { 1 , b j j /b ii } fo r i 6 = j . F or a connected Co xeter diagram ∆ c ( G ), the corresp onding group G has to scale o nly finitely many basic systems β . An y suc h syste m is represen ted b y an essen tially unique diagram ∆( G ), in whic h no de lab els form a set of relative ly prime in tegers. F or tw o suc h basic systems β , β ′ , w e can con v ert from ∆( G ) to ∆ ′ ( G ) by consecutiv ely adjusting the lab els and branc hes of v ario us pairs of adjacen t no des b y op erations of the following kind: in v erting the ratio of the no de lab els; doubling a single branc h and balancing its lab els, or conv erting a double branch to a single branch, with ratio 4, if the corresp onding br a nc h in ∆ c ( G ) is mark ed ∞ . F ollowing these adjustmen ts on pairs of no des, w e may finally ha v e to r escale the en tire set of lab els to obtain a set of relatively prime integers. Then w e ma y reduce G mo dulo an o dd prime p t o obtain a subgroup G p of GL n ( Z p ) generated b y the mo dular images of the r i ’s. W e call the prime p ge n eric for G if p ≥ 5, or p = 3 but no branch of ∆ c ( G ) is mark ed 6. In the g ene ric case, no no de la bel of ∆( G ) v anishes mo d p a nd a c hange in the underlying basic system for G has the effect of merely conjugating G p in GL n ( Z p ). On the other hand, in the non- generic case, the group G p ma y dep end essen tially on the actual diagram ∆( G ) used in the reduction mo d p . F or an y o dd prime p , we alw a ys find that G p = h r 0 , . . . , r n − 1 i is a subgroup of t he o rthogonal group O ( Z n p ) of isometries for the (p ossibly singular) symmetric bilinear form x · y , the latter b eing defined on Z n p b y means of (the mo dular image of ) the Gram matrix B ; in particular, r i is the orthogonal reflection with r o ot b i if b 2 i 6 = 0. No w recall f r o m [23, Thm. 3.1] that an irreducible group G p of the ab ov e sort, generated b y n ≥ 3 reflections, must necessarily b e one of the following: • an o r thogonal gr oup O ( n, p, ǫ ) = O ( V ) or O j ( n, p, ǫ ) = O j ( V ), excluding t he cases O 1 (3 , 3 , 0), O 2 (3 , 5 , 0), O 2 (5 , 3 , 0) (supp osing for thes e three that disc( V ) ∼ 1), and also excluding the case O j (4 , 3 , − 1); or • the reduction mo d p of one of the finite linear Cox eter groups of type A n ( p ∤ n + 1) , B n , D n , E 6 ( p 6 = 3), E 7 , E 8 , F 4 , H 3 or H 4 . W e shall say in these t w o cases that G p is of ortho gona l or sp h eric al typ e , respectiv ely , although there is some ov erlap for small primes. (The notation a ∼ b means t ha t a = t 2 b for some t in the field.) Our description in [2 3 , Thm. 3.1] rests on the classification of the finite irreducible reflection g r o ups ov er any field, obtained in Z aless ki ˘ ı & Sere ˇ zkin [37] (se e also [17, 32, 33, 36]). F or a non-singular space V , recall tha t ǫ = 0 if n is o dd, and ǫ = 1 o r − 1 if n is ev en and the Witt index of V is n/ 2 or ( n/ 2) − 1. When n is ev en, disc( V ) ∼ ( − 1) n/ 2 if ǫ = 1, and disc( V ) ∼ ( − 1) n/ 2 γ , with γ a non-square, if ǫ = − 1. Moreo v er, O 1 ( V ) and O 2 ( V ) are the subgroups o f O ( V ) generated b y the t w o distinct conjugacy classes of reflections, eac h c haracterized b y the quadratic character of the spinor norm of its reflections, namely 1 or γ , resp ectiv ely . The a b ov e analysis sets up the stage for the construction of finite regular p olytop es from crystallographic string Cox eter g r o ups G . Clearly , the generators r i of the reduced g r oup G p satisfy (at least) the Cox eter-ty p e r elations inherited from G . In most cases it also is more or less straig h tforw ard to determine the o v erall structure of G p b y app ealing to the 5 ab o v e description of p ossible reflection groups. Ho w ev er, the main c hallenge is to determine when G p has the in tersection pro p erty (2) for its standard subgroups. Here the outcome is often quite unpredictable, with the result dep end ing on the underlying gro up G , the diagram ∆( G ) used in the reduction, a nd the prime p . The reduced gro up G p ma y turn out to b e a C-g r o up for all primes, or for all primes in certain congr uence classes, or for only finitely man y primes (for example, only fo r p = 3), or for no prime at all. Th us there is no theorem that cov ers all cases sim ultaneously . Here we describ e a quite g eneral theorem (see [25, Thm. 2.3]) whic h, in the three basic cases considered, t ypically enables us to determine (often after additio nal considerations) whether or not the reduced group G p is a C-group. F or an y k , l ∈ { 0 , . . . n − 1 } w e let G p k := h r j | j 6 = k i and G p k ,l := h r j | j 6 = k , l i . Then G p is a string C-group if and only if G p 0 and G p n − 1 are string C-groups and G p 0 ∩ G p n − 1 = G p 0 ,n − 1 . (This enables an inductiv e attack on the problem.) Similarly we let V k and V k ,l denote the subspace of V = Z n p spanned b y the v ectors b j with j 6 = k or j 6 = k , l , resp ective ly . F o r a singular subspace W of a non-singular space V , w e also let b O ( W ) denote the subgroup of O ( W ) consisting of tho se isometries which act trivially on the radical of W . Theorem 3.1 L et G = h r 0 , . . . , r n − 1 i b e a crystal lo gr a p hic line ar Co xeter g r oup with string diagr am. Supp ose that n ≥ 3 , that the prime p is generic for G and that ther e is a squar e among the lab els of the no des 1 , . . . , n − 2 of the diagr am ∆( G ) (this c an b e achieve d by r e adjusting the no de lab els). F or v arious subsp ac es W o f V we identify O ( W ) , b O ( W ) , etc. with suitable sub gr oups of the p o intwise stabilizer of W ⊥ in O ( V ) . (a) L et the subsp ac e s V 0 , V n − 1 and V 0 ,n − 1 b e non-sin gular, and let G p 0 , G p n − 1 b e of ortho gonal typ e. (i) Then G p 0 ∩ G p n − 1 acts trivial ly on V ⊥ 0 ,n − 1 and O 1 ( V 0 ,n − 1 ) ≤ G p 0 ∩ G p n − 1 ≤ O ( V 0 ,n − 1 ) . (ii) If G p 0 = O ( V 0 ) and G p n − 1 = O ( V n − 1 ) , then G p 0 ∩ G p n − 1 = O ( V 0 ,n − 1 ) . (iii) If either G p 0 = O 1 ( V 0 ) or G p n − 1 = O 1 ( V n − 1 ) , then G p 0 ∩ G p n − 1 = O 1 ( V 0 ,n − 1 ) . (b) L et V , V 0 , V n − 1 b e non-sin g ular, let V 0 ,n − 1 b e s ingular (so that n ≥ 4 ), and let G p 0 , G p n − 1 b e of ortho gonal typ e. (i) Then G p 0 ∩ G p n − 1 acts trivial ly on V ⊥ 0 ,n − 1 , and b O 1 ( V 0 ,n − 1 ) ≤ G p 0 ∩ G p n − 1 ≤ b O ( V 0 ,n − 1 ) . (ii) If G p 0 = O ( V 0 ) and G p n − 1 = O ( V n − 1 ) , then b O ( V 0 ,n − 1 ) = G p 0 ∩ G p n − 1 . (iii) If either G p 0 = O 1 ( V 0 ) or G p n − 1 = O 1 ( V n − 1 ) , then b O 1 ( V 0 ,n − 1 ) = G p 0 ∩ G p n − 1 . (c) Supp ose V , V 0 ,n − 1 ar e non-singular while at le ast one of the subsp ac es V 0 , V n − 1 is s i n gular. A lso supp ose that G p 0 ,n − 1 is of ortho gonal typ e, with G p = O 1 ( V ) wh en G p 0 ,n − 1 = O 1 ( V 0 ,n − 1 ) . Then G p 0 ∩ G p n − 1 = G p 0 ,n − 1 . W e briefly discuss the mo dular p olytop es asso ciated with some in teresting classes of crys- tallographic string Co xeter groups. Recall that [ p 1 , p 2 , . . . , p n − 1 ] denotes the Coxete r g roup with a string Cox eter diagram on n no des and branc hes lab elled p 1 , p 2 , . . . , p n − 1 . They a r e the automorphism groups of the univ ersal regular p olytop es { p 1 , p 2 , . . . , p d − 1 } ( se e [21 , § 3D]). 3.1. Gr oups in which G has sph e ric al or Euclide an T yp e 6 The mo dular reduction G p of a n y spherical or euclidean (crystallographic) group G is a string C-group for any prime p ≥ 3, and G p ∼ = G if G is spherical. There a re fo ur kinds of (connected) spherical string diag r a ms (up t o duality ), namely A n , B n , F 4 and I 2 (6) (dihedral of order 12). The corresp onding mo dular p olytop es are isomorphic to the n -simplex, n -cube, 24-cell and hexagon, resp ectiv ely , and admit “mo dular realizations” in the finite space Z n p . F or the euclidean groups [4 , 3 n − 3 , 4] ( n ≥ 3) , [3 , 4 , 3 , 3] and [ ∞ ], w e obtain the regular toroids { 4 , 3 n − 3 , 4 } ( p, 0 n − 2 ) of r ank n and { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) of r ank 5, and the regular p - gon { p } . F or a spherical group G , we hav e G d ≃ G for an y mo dulus d ≥ 3, and sometimes eve n for d = 2 (here depending on choice of diagra m). F or a euclide an gr o up G and d ≥ 3 (again, sometimes for d = 2 as well), the reduced group G d is the group of a regular toro id o f rank n , but now its type v ector (suffix) dep ends on ∆( G ) (in particular, on the parity of n ) and, as w ell, on the parity of d ; the details are quite inv olv ed (see [26]). 3.2. Gr oups of r anks 3 or 4 The g roups of ra nks 1 or 2 are subsumed by our discussion in the previous paragraph. F or rank 3, any g roup G p is a string C-g r o up. F or example, the h yp erbolic group G = [3 , ∞ ] with diagram 1 • 1 • 4 • (disc( V ) = − 1) , yields a regular map of t yp e { 3 , p } . F or p ≥ 5 w e find that G p = O 1 (3 , p, 0) ≃ P S L 2 ( Z p ) ⋊ C 2 of order p ( p 2 − 1). Whe n p = 3, 5 or 7, resp ectiv ely , w e obtain the regular tetrahedron { 3 , 3 } , icosahedron { 3 , 5 } and the Klein p olyhedron { 3 , 7 } 8 . This construction redescrib es the family of r egular maps discussed (and g ene ralized) in [1 9 ] or [20]. The situation c hanges drastically for groups G of higher ranks, with obstructions already o ccurring in rank 4. W e shall not attempt to fully sum marize the findings of [24], whic h settle all groups G = [ k , l, m ] of rank 4 (emplo ying results such as Theorem 3.1). Suffice it here to men tion some p ossible scenarios illustrating tha t the outcome is often unpredictable. One p ossible scenario is that G p is a string C-group for all primes p ≥ 3; this happ ens, for instance, if the subgroup [ k , l ] or [ l , m ] is spherical, if l = ∞ , or if [ k , l ] or [ l , m ] is euclidean and l = 4 or 6. F or example, from [3 , 3 , ∞ ] w e obtain a r egular 4-p olytop e of type { 3 , 3 , p } whose ve rtex-figures are isomorphic to the maps of t yp e { 3 , p } deriv ed from [3 , ∞ ]; its group G p is isomorphic to S 5 if p = 3, O 1 (4 , 3 , 1) if p ≡ 1 mo d 4, a nd O 1 (4 , 3 , − 1) if p ≡ 3 mo d 4, p 6 = 3. When p = 3, 5 or 7, resp ectiv ely , this giv es the 4-simplex { 3 , 3 , 3 } , t he 6 00-cell { 3 , 3 , 5 } , and the unive rsal 4-p olytop e {{ 3 , 3 } , { 3 , 7 } 8 } first describ ed in [27]. On the other extreme, there are g roups G for which G p is a C-group for only finitely man y primes p . F or example, fo r G = [6 , 3 , 6] only G 3 is a C-group. More import an t examples arise when the four subspaces V , V 0 , V 3 , V 0 , 3 are all no n-singular. F or example, when G = [ ∞ , 3 , ∞ ] the reduced group G p is a C-g roup only for p = 3, 5 or 7 ; from G 3 and G 5 w e obtain the 4- simple x { 3 , 3 , 3 } a nd the regular star- polytop e { 5 , 3 , 5 2 } , and from G 7 a (non-univ ersal) 4-p olytop e with facets and v ertex-figures giv en b y a dual pair of Klein maps { 7 , 3 } 8 and { 3 , 7 } 8 . Another p ossible scenario is that G p is a C-group only for certain congruence classes of primes. If G = [6 , 3 , ∞ ], then G p is a C-group if and only if p = 3 or p ≡ ± 5 mo d 1 2. 3.3. Gr oups of higher r an k s 7 The lar g e n um b er of crystallographic Co xeter gro ups G o f ranks n ≥ 5 mak es it difficult to f ully en umerate the regula r p olytop es o btained b y o ur metho d. F or the group [4 , 3 , 4 , 3] of rank 5 we obtain regular 5-p olytop es with toroids { 4 , 3 , 4 } ( p, 0 , 0) as facets and 24-cells { 3 , 4 , 3 } as v ertex-figures; their gro up G p is giv en b y O 1 (5 , p, 0) if p ≡ ± 1 mo d 8 and O (5 , p, 0) if p ≡ ± 3 mo d 8. These are examples of lo c al ly tor oidal regular p olytop es , meaning t ha t their facets and v ertex-figures are spherical or toroida l (but not all sphe rical). Suc h p olytop es hav e no t b een fully classified (see [21, Ch.12]). The three closely related gr o ups [3 , 4 , 3 , 3 , 3], [3 , 3 , 4 , 3 , 3] and [4 , 3 , 3 , 4 , 3] of rank 6 similarly give rise to lo cally toroida l 6-p olytop es with groups O 1 (6 , p, +1) if p ≡ ± 1 mo d 8 and O (6 , p, +1) if p ≡ ± 3 mo d 8. Eac h p olytop e is cov ered by the resp ectiv e unive rsal p olytope { { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) , { 4 , 3 , 3 , 3 } } , { { 4 , 3 , 3 , 4 } ( p, 0 , 0 , 0) , { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) } , { { 3 , 3 , 4 , 3 } ( p, 0 , 0 , 0) , { 3 , 4 , 3 , 3 } ( p, 0 , 0 , 0) } , whic h has b een conjectured to exist fo r all primes p ≥ 3 and to b e infinite for p > 3 (see [21, Ch.12]). F or p = 3 (a nd p = 2, but this is outside our discussion), these three univ ersal p olytop es are known to b e finite. The first t w o ha v e group Z 3 ⋊ O (6 , 3 , +1), and the last has group ( Z 3 ⊕ Z 3 ) ⋊ O (6 , 3 , + 1) (see [25]). More generally w e can reduce the fo ur groups G mo dulo any in teger d ≥ 2 to obtain other kinds o f lo cally toroida l regular p olytop es of ranks 5 or 6, no w with type v ectors for facets and v ertex-figures dep ending on ∆( G ) and its sub diagrams for facets and vertex -figures, as w ell a s on the parity of d (see [26]). In particular, this establishes [21, Conjecture 12C2] concerning the existence of lo cally toroidal regular 6-p olytop es of t yp e { 3 , 4 , 3 , 3 , 3 } , sayin g that the uni- v ersal regular 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 for all d ≥ 2; this then settles the existence of the first kind of 6-p olytop es men tioned in the prev ious paragraph. The corresponding conjec tures for the lo cally t o roidal regular 6-p olytop es of ty p es { 3 , 3 , 4 , 3 , 3 } and { 4 , 3 , 3 , 4 , 3 } are still op en (see [21, 1 2 D3,12E3]). Another in teresting sp ecial class consists of the 3 –infin i ty gr oups G = [ p 1 , . . . , p n − 1 ], for whic h all p erio ds p j ∈ { 3 , ∞} . T ypically then we hav e an alternating string of 3’s and ∞ ’s, and the outcome dep ends on the nature of the string. F or the prime p = 3 w e alw ay s hav e a C-group, namely S n +1 . If G = [3 k , ∞ l ], with k + l = n − 1 , then for all primes p ≥ 3, the group G p is a string C -gr oup. W e hav e seen examples with n = 3 or 4. F o r G = [3 , ∞ l , 3], with n = l + 3 , l ≥ 1, we ha v e a string C -group, except p ossibly when p = 7 and l ≥ 4, with l ≡ 1 mo d 3. On the other hand, if G = [3 k , ∞ l , 3 m ], with k > 1 or m > 1, and l ≥ 1, then G p is a string C - group for a ll but finitely man y primes p . Ho w ev er, G p v ery often is not a C-group. F or example, supp ose t ha t G has a string subgroup of the fo rm [ . . . , ∞ , 3 k , ∞ , . . . ], with k ≥ 1, and that p ≥ 5; then G p is not a string C - group, except p ossibly when p = 5 or 7 a nd k ≤ 1 for all suc h string subgroups [ . . . , ∞ , 3 k , ∞ , . . . ]. 4 P olytop es from h yp erb olic Co xeter group s F ollowing [31] we b egin with the symmetry group Γ = [ r , s, t ] of a regular ho ney com b { r , s, t } in h yp erb olic 3-space H 3 and consider a f a ithful represen tation of Γ as a group of complex M¨ obius transformations. Recall here that the absolute of H 3 can b e iden tified with the com- plex inv ers iv e plane (compare the P oincare halfspace mo del of H 3 ), and that ev ery gro up o f 8 h yp erbolic isometries is isomorphic to a group o f M¨ obius transformations o v er C . In partic- ular, a plane reflection in H 3 corresp onds to an in v ersion in a circle (o r line) determined as the “interse ction” of the mirr o r plane in H 3 with the absolute. Under this corresp ondence the generating plane reflections ρ 0 , . . . , ρ 3 of the hyperb olic reflection group Γ b ecome in v er- sions in circles, a g ain denoted b y ρ 0 , . . . , ρ 3 , cutting one another a t the same angles as the corresp onding reflection planes in H 3 . The g roup Γ is o ne of ten p ossible groups, namely [4 , 4 , 3] , [4 , 4 , 4] , [6 , 3 , 3] , [6 , 3 , 4] , [6 , 3 , 5] , [6 , 3 , 6] , [3 , 6 , 3] , [3 , 5 , 3] , [5 , 3 , 4] , [5 , 3 , 5]; see [34] for a complete list of the generating inv ers ions for these groups. Next recall that M¨ obius transformations may con v enien tly b e represen ted (uniquely up to scalar m ultiplication) by 2 × 2 matrices, namely az + b cz + d ← →  a b c d  and a z + b cz + d ← → #  a b c d  , with appropria te in terpretations for multiplication of suc h matrices. W e then obta in matrices r 0 , r 1 , r 2 , r 3 and s 1 , s 2 , s 3 , resp ectiv ely , for the generators ρ 0 , ρ 1 , ρ 2 , ρ 3 of Γ and σ 1 , σ 2 , σ 3 of Γ + . Let G denote t he group of complex 2 × 2 matrices generated b y s 1 , s 2 , s 3 . When considered mo dulo scalars, G (or rather the corresp onding pro jectiv e group P G ) is isomorphic to Γ + , the ro tation group of { r , s, t } . It turns out that the matrices in G all ha v e en tries in a certain subring D of C dep ending on [ r, s, t ]. Th us mo dular reduction a pplies . W e choose appropriate ideals J in D and consider the matrices s 1 , s 2 , s 3 o v er the quotient ring D /J , again mo dulo scalars (determined by a suitable subgroup o f the cen ter). Then under certain conditions, the resulting pro jectiv e group G J (sa y) is either the rotation subgroup for a directly regular p olytop e, or the full automor phis m group of a ch iral p olytop e. W e illustrate the metho d for the h yp erb o lic group [4 , 4 , 3]. Whe n Γ = [4 , 4 , 3] is view ed as a group of complex M¨ obius transforma t io ns, the generators may b e ta k e n as ρ 0 ( z ) = z , ρ 1 ( z ) = iz , ρ 2 ( z ) = 1 − z , ρ 3 ( z ) = 1 /z . Then the rotation subgroup Γ + of Γ is generated b y σ 1 = ρ 0 ρ 1 = − iz , σ 2 = ρ 1 ρ 2 = − iz + i, σ 3 = ρ 2 ρ 3 = 1 − 1 /z , and cons ists only of prop er M¨ obius transformations (not in v olving complex conjuga t io n). Here the ma t rice s s 1 , s 2 , s 3 for σ 1 , σ 2 , σ 3 are given by s 1 =  − i 0 0 1  , s 2 =  − i i 0 1  , s 3 =  1 − 1 1 0  , and G is the gr o up of a ll inv ertible 2 × 2 matrices o v er the G a uss ian in tegers D = Z [ i ] ( with determinan ts ± 1 , ± i ). In particular, [4 , 4 , 3] + ∼ = P S L 2 ( Z [ i ]) ⋊ C 2 , where the first factor is also kno wn as the Gaussian mo dular group or Picard group (see also [15]). Since we wish to obtain toroidal facets of t yp e { 4 , 4 } + ( b,c ) , w e m ust consider the imp osition of the extra relation ( σ − 1 1 σ 2 ) b ( σ 1 σ − 1 2 ) c = 1 on the generators o f Γ + (see [8]). Note here that the corresp onding matrix pro duct in G is ( s − 1 1 s 2 ) b ( s 1 s − 1 2 ) c =  1 − 1 0 1  b  1 − i 0 1  c =  1 − ( b + ci ) 0 1  . 9 Th us the ideals J of Z [ i ] must b e c hosen in suc h a w a y that b + ci = 0 in Z [ i ] /J . One natural ch oice of ideal is J = m Z [ i ], where m ≥ 3 is an integer in Z . This choice of ideal typically pro duces directly regular p olytop es of type {{ 4 , 4 } ( m, 0) , { 4 , 3 }} whose rotation subgroup is P S L 2 ( Z m [ i ]) or a closely related group, with the exact structure determined b y the prime factorizatio n of m in Z (see [31, p.238]) . F or example, if m ≡ 3 mo d 4 is a prime, w e obtain P S L 2 ( Z m [ i ]) ∼ = P S L 2 ( m 2 ). A more interesting choice o f ideal arises from a solution of the equation x 2 = − 1 mo d m , with m as ab ov e. Let m = 2 e p e 1 1 . . . p e k k b e the prime factorization in Z . The n the equation is solv able if and only if e = 0 , 1 and p j ≡ 1 mo d 4 for eac h j . If ˆ i ∈ Z m is such that ˆ i 2 = − 1 mo d m , then there exists a unique pair of p ositiv e in tegers b, c suc h that m = b 2 + c 2 , ( b, c ) = 1 and b = − ˆ ic mo d m . W e now take J = ( b + ci ) Z [ i ], whic h is the kernel of the ring epimorphism Z [ i ] → Z m that maps the complex n um b er x + y i to the elemen t x m + y m ˆ i in Z m , wh ere x m ≡ x and y m ≡ y mo d m . This choice of ideal “destro ys” the reflections in the o v erlying reflection group and typically yields c hiral p olytopes o f type {{ 4 , 4 } ( b,c ) , { 4 , 3 }} whose ro tation subgroup is P S L 2 ( Z m ) o r a closely related gro up, with the exact structure again determined by the prime fa cto r izat io n of m . F or example, if m ≡ 1 mo d 8 is a prime, then the g roup is P S L 2 ( m ). Note that t he structure of the p olytop e also dep ends on the solution ˆ i . F or example, when m = 65, the solution ˆ i = 8 giv es facets { 4 , 4 } (1 , 8) , while ˆ i = 18 leads t o facets { 4 , 4 } (4 , 7) . With similar tec hniques w e can construct a host of regula r or c hiral 4-p olytop es of t yp es { 4 , 4 , 4 } , { 4 , 4 , 4 } , { 6 , 3 , 3 } , { 6 , 3 , 4 } , { 6 , 3 , 5 } , { 6 , 3 , 6 } o r { 3 , 6 , 3 } (see [30, 31]). Ho w ev er, the underlying ring D will dep end on the Sc hl¨ afli sym b ol. The Gaussian in tegers Z [ i ] and Eisenstein in tegers Z [ ω ] (with ω a cub e ro ot of unit y) suffi ce in all case s except { 6 , 3 , 4 } and { 6 , 3 , 5 } (this is based on subgroup relatio ns hips b et w een the rotatio n subgroups for the v ar ious t yp es). F or { 6 , 3 , 4 } and { 6 , 3 , 5 } , resp ectiv ely , we can work ov er the ring Z [ ω , √ 2] and Z [ ω , τ ] (with τ the golden ratio; see the next section). Chiral p olytop es also exist in ranks larger than 4, but explicit constructions of finite examples (o r rank 5) w e re only disco v ered quite recently in [4]. Mo dular reduction is an effectiv e method to pro duce examples in rank 4. In addition to the symmetry groups of 3-dimensional hyperb olic ho ney com bs, sev eral other discrete hy p erb olic groups a dmit rep- resen tations as groups of line ar fractional transformations o v er other rings of complex or quaternionic in tegers (describ ed in detail in the fo rthcoming b o ok b y Johnson [15]). Al- though the a r it hme tic in v olv ed is likely to be considerably more complicated than in the rank 4 case, there is a go o d c hance that the reduction metho d will carry o v er to pro duce examples of finite c hiral p olytop es o f rank larg er than 4. 5 The groups [3 , 5 , 3] and [5 , 3 , 5] In Section 3 w e observ ed t ha t any crystallographic string Cox eter group G has rotat ional p erio ds p j ∈ { 2 , 3 , 4 , 6 , ∞} and can b e represen ted fa it hf ully a s a mat r ix g roup o v er the domain Z . Here w e widen the discussion a little b y allo wing the p erio d p j = 5. Keeping (4) in mind, we note that 2 cos π 5 = τ , where the golden ratio τ = (1 + √ 5) / 2 is the p ositiv e 10 ro ot of τ 2 = τ + 1 . W e therefore mov e to the larger co efficien t domain D := Z [ τ ] = { a + bτ : a, b ∈ Z } , and so on find that w e need only a dd the sub diagram s • − − − τ 2 s • to those already listed in T able 1 in order f or the Cartan in tegers m ij of (5) to be in D for all i, j . This sub diagram, say on no des i, j , do es indeed define the no n- crys tallogra phic dihedral group h r i , r j i with order 1 0 a nd p erio d p ij = 5. (In other not ation, this is the group H 2 ≃ I 2 (5).) Naturally , w e m ust no w allo w rescaling of no des by an y ‘in teger’ s ∈ D o r its inv erse . F urthermore, referring bac k to ( 5 ), w e find that m ij = τ 2 ∈ D , so that G is represen ted as a matrix gro up o v er D through its action on the D -mo dule ⊕ j D b j . Let us no w summarize the k ey ar it hmetic prop erties of the domain D . (W e refer to [10] for a detailed accoun t of this, and to [3] for a deep er discussion of ‘ D -lattices’ and t he related finite Cox eter groups H k , k = 2 , 3 , 4.) First of all, w e recall that D is the ring of algebraic in tegers in the field Q ( √ 5). The non-trivial field automorphism mapping √ 5 7→ − √ 5 induces a ring automorphism ′ : D → D , whic h in this section we shall call c onjugation . Th us ( a + bτ ) ′ = ( a + b ) − bτ . In par t icular, τ ′ = 1 − τ = − τ − 1 . Recall that z = a + bτ ha s norm N ( z ) := z z ′ = a 2 + ab − b 2 . W e note tha t D is a Euclidean domain, through a division algor it hm based on | N ( z ) | . The set of units in D is {± τ n : n ∈ Z } = { u ∈ D : N ( u ) = ± 1 } . Recall that in tegers z , w ∈ D are asso ciates if z = u w for some unit u . Up to asso ciates , the primes π ∈ D can b e classified as follows: • the prime π = √ 5 = 2 τ − 1, whic h is self-conjugate (up to asso ciates: π ′ = − π ); • rational primes π = p ≡ ± 2 mo d 5, also self-conjugate; • primes π = a + bτ , for whic h | N ( π ) | equals a ratio nal prime q ≡ ± 1 mo d 5. In this case, the conjug a te prime π ′ = ( a + b ) − bτ is not an asso ciate of π . Let us now turn to the group G = [3 , 5 , 3], here acting as an o r t hogonal group o n real 4-space V (in con trast to the conformal represen tation on H 3 men tioned in the previous section). Since τ 2 is a unit, there is essen tially only one choice of diagra m, namely ∆( G ) = 1 • 1 • τ 2 • τ 2 • . The discriminan t is disc( V ) = − 1 16 (2 + 5 τ ) ∼ − (2 + 5 τ ) , where the prime δ := − (2 + 5 τ ) has norm − 11. No w consider any prime π ∈ D . Our goals are to show that G π = h r 0 , r 1 , r 2 , r 3 i π is a string C -g r oup and to determin e its structure, then sa y a little ab out the corresponding p olytop e P π := P ( G π ). 11 In fact, we can a lmost immediately apply a suitable generalization of Theorem 4 .2 in [23]. First note that the subgroup G π 3 = h r 0 , r 1 , r 2 i π is obv iously some quotien t of the spherical group [3 , 5] ≃ H 3 . Now it is easy t o che c k tha t after reduction mo dulo an y prime π , ev en for asso ciates o f 2, the reflections r j still hav e p erio d 2. Next, we consider the isometry z := ( r 0 r 1 r 2 ) 5 =     − 1 0 0 τ 4 0 − 1 0 2 τ 4 0 0 − 1 3 τ 2 0 0 0 1     . (6) in G . Since τ 4 is a unit, this means that r 0 r 1 r 2 still has p erio d 10 in G π . Th us h r 0 , r 1 , r 2 i π ≃ [3 , 5] and dually h r 1 , r 2 , r 3 i π ≃ [5 , 3]. Consulting the pro of of [2 3, Th. 4.2], w e see that we need only sho w that the orbit of µ 0 := [1 , 0 , 0 , 0] under the rig ht action of the matrix group h r 0 , r 1 , r 2 i has the same size mo dulo π as in characteris tic 0, namely 12. This is routinely v erified, so w e hav e prov ed mo st of Prop osition 5.1 L et G = [3 , 5 , 3] . F or any prime π ∈ D , the gr oup G π = h r 0 , r 1 , r 2 , r 3 i π is a finite string C -gr oup. The c orr esp ondin g finite r e gular p olytop e P π is self-dual an d has ic osahe dr al fac ets { 3 , 5 } and do de c ahe dr al vertex figur es { 5 , 3 } . Pro of . T o v erify self-duality we define g ∈ GL ( V ) by g : [ b 0 , b 1 , b 2 , b 3 ] 7→ [ τ − 1 b 3 , τ − 1 b 2 , τ b 1 , τ b 0 ]. Then g 2 = 1, g r 0 g = r 3 and g r 1 g = r 2 . (See [21, 2E12].)  A more detailed description of G π m ust dep end on the nature of the prime π . (Of course, our results are t ypically unaff ected b y replacing π by any asso ciate ± τ m π .) In all cases the underlying finite field K := D / ( π ) has order | N ( π ) | , so that G π acts as an orthog onal group on the 4-dimensional v ector space V ov e r K preserving the mo dular image of the bilinear form f or G . Case 1 : π = 2. Here an easy calculation using GAP confirms that G π is the orthog onal group O (4 , 2 2 , − 1) with Witt index 1 o v er K = GF (2 2 ). Since | G 2 | = 816 0 , the p olytop e P 2 has 6 8 vertic es and 68 icosahedral facets. Henceforth we supp ose that π is not a n asso ciate of 2. T o w ork with suc h primes, we need a generalization of the rational Legendre sym b ol ( p | q ). Th us for any α ∈ D a nd prime π w e set ( α | π ) D :=  +1 , if α is a quadrat ic residue (mo d π ); − 1 , otherwise. (Compare [10, Ch. VI I I].) W e a re mainly in terested in computing ǫ := ( δ | π ) D where δ = − (2 + 5 τ ) is the discriminan t. Since ev ery lab el in ∆( G ) is square, w e conclude that G π is a subgroup of O 1 (4 , | N ( π ) | , ǫ ), so long as δ a nd π are relative ly prime. Indeed, G π will almost alw a ys equal suc h an orthogonal group. 12 Case 2 : π = √ 5 = 2 τ − 1. Here | π π ′ | = 5 ≡ 0 mo d π , so that the discriminan t δ = − (2 + 5 τ ) ≡ 3 mo d π , whic h is non-square in K = GF (5). Thus ǫ = − 1 and G π = O 1 (4 , 5 , − 1) has order 1560 0. In fact, the p olytop e P √ 5 is isomorphic t o that obtained in [24, p. 347] thro ug h reduction mo d 5 of the crystallographic gro up [3 , ∞ , 3]. Case 3 : π is an asso ciate of an o dd ra tional prime p ≡ ± 2 mo d 5. Since K = GF ( p 2 ), G π = G p is, for suitable ǫ , a subgro up of O 1 (4 , p 2 , ǫ ), whose order w e recall is p 4 ( p 4 − ǫ )( p 4 − 1). Consulting [23, Th. 3.1 ], w e see that G p = O 1 (4 , p 2 , ǫ ) so long as w e can rule out tw o remote alternativ es. First o f all, it is conceiv able that G p ≃ H 4 = [3 , 3 , 5]. But here it is easy to c hec k directly that H 4 cannot b e generated by reflections r j satisfying the Co xeter-t ype relations inherited from [3 , 5 , 3], let alone the indep enden t relations induced b y reduction mo dulo p . Secondly , we m ust sho w that G p is not isomorphic to some o rthogonal group O 1 (4 , p, η ), η = ± 1, ov e r the subfield GF ( p ). If this w ere so, then Theorem 3.1 in [23] w ould actually imply that G p is similar to O 1 (4 , p, η ) under extension of scalars. More precisely , if L is a n algebraic closure o f K = GF ( p 2 ), t hen there w ould exist some g ∈ GL ( V L ) with g G p g − 1 = O 1 (4 , p, η ). Using (5), w e compute with resp ect to the new basis { c i } = { g ( b i ) } f or V L . Th us the reflection ˜ r i := g r i g − 1 satisfies ˜ r i ( c j ) = g ( b j + m ij b i ) = c j + m ij c i . W e conclude that the field of definition for G p m ust alw a ys con tain the subfield generated b y the Cartan in tegers m ij . In our case, m 12 = τ 2 6∈ GF ( p ), so that g G p g − 1 cannot p ossibly b e a group O 1 (4 , p, η ). Ha ving sho wn that G p = O 1 (4 , p 2 , ǫ ), w e next determine ǫ . F rom [10, Th. 8 .5(a)] w e hav e ( δ | π ) D = ( N ( δ ) | p ) = ( − 11 | p ) = ( − 1 | p )( 11 | p ) = ( p | 11 ) , b y (rational) quadratic recipro cit y . Since the non-zero squares (mo d 11) are 1 , 3 , 4 , 5 , 9, w e conclude that ǫ :=  +1 , if p ≡ 3 , 12 , 23 , 27 , 37 , 38 , 42 , 47 , 48 , 53 mo d 55; − 1 , if p ≡ 2 , 7 , 8 , 13 , 17 , 18 , 28 , 32 , 43 , 52 mo d 55 . Case 4 : π = a + bτ , where N ( π ) = a 2 + ab − b 2 = q , where the ra tional prime q ≡ ± 1 mo d 5; ho w ev er, π is not an asso ciate of δ = − (2 + 5 τ ). W e no w hav e K = GF ( q ). An eve n easier app eal to [23, Th. 3.1] gives G π = O 1 (4 , q , ǫ ). W e need only determine ǫ = ( δ | π ) D . Since a + bτ ≡ 0 mo d π , we ma y supp ose τ = − a/b ∈ K . Th us δ = − (2 + 5 τ ) ∼ − b 2 (2 + 5 τ ) ≡ 5 ab − 2 b 2 mo d π . By [10, Th. 8.5(a)], w e obtain ǫ = ( δ | π ) D = ( (5 ab − 2 b 2 ) | q ) = ( b | q )( (5 a − 2 b ) | q ) . Using the rational Legendre sym b ol, we can th us compute ǫ for any prime π = a + bτ . 13 It is p ossible to say when π and its conjugate π ′ giv e opp osite ǫ ’s, so that t he corresp onding orthogonal spaces ha v e, in some order, Witt indices 1 and 2. This happ ens if and only if q is a square mo d 11 , since ( δ | π ) D ( δ | π ′ ) D = ( ( b (5 a − 2 b )( − b )(5 a + 7 b )) | q ) = ( − 11 | q ) = ( q | 11 ) . One notable instance here is π = δ ′ = − 7 + 5 τ , whic h is relatively prime to the discrimi- nan t δ . W e hav e G δ ′ = O 1 (4 , 11 , − 1), of order 1771 440. Case 5 : π = δ = − (2 + 5 τ ). This is the only cas e in whic h the or thogonal space V is singular. No w K = GF (11) and τ = − 2 / 5 = 4. W e find tha t r a d( V ) is spanned by c = 7 b 0 + 3 b 1 + 2 b 2 + b 3 , and that V = ra d( V ) ⊥ V 3 , where V 3 is the non-singular subspace spanned b y b 0 , b 1 , b 2 . It is then not hard to see that O ( V ) ≃ ˇ V 3 ⋊ ( K ∗ × O ( V 3 )) , where K ∗ ≃ GL (rad( V )) and ˇ V 3 is dua l to V 3 . W e observ e that the ab elian gr oup ˇ V 3 ≃ K 3 consists of all transv ections r ( x ) = x + ϕ ( x ) c , where ϕ ∈ ˇ V 3 (with ˇ V 3 view ed as a subspace of ˇ V fix ing c ). Now since ev ery r j fixes c , G δ m ust b e a subgroup of the p oin t wise stabilizer of ra d( V ). In fact, anot her calculation with GAP confirms that G δ = b O 1 ( V ) ≃ ˇ V 3 ⋊ O 1 ( V 3 ) , whic h has order 1 1 3 · 11 · (11 2 − 1) = 17569 20. No w consider the isometry z ∈ G defined in (6). It is easy to c hec k that z ( c ) ≡ c mo d δ , so t hat z = 1 rad(V) ⊥ − 1 V 3 ∈ b O 1 ( V ) a cts as the c entr a l invers i o n in the gro up O 1 ( V 3 ) f o r the icosahedral facet. Th us G δ has a normal subgroup A isomorphic to ˇ V 3 ⋊ h z i and so of order 2 · 11 3 . Using O 1 (3 , 11 , 0) ≃ P S L 2 (11) ⋊ C 2 (see [1, Th. 5.20]), w e conclude tha t G := G δ / A ≃ P S L 2 (11) , of o rder 6 6 0. Remark ably , G is also a string C -group. The resulting p olytop e is the 11 - cell indep ende n tly disco v ered b y Coxe ter in [7] and Gr ¨ unbaum in [11]. Indeed, b oth r 0 r 1 r 2 and r 1 r 2 r 3 ha v e p erio d 5 in the quotien t (see (6)), and P ( G ) = { { 3 , 5 } 5 , { 5 , 3 } 5 } is the univ ersal 4- polytop e with hemi-icosahedral facets and hemi-do decahedral v ertex- figures. This finishes our in v estigation of the group [3 , 5 , 3]. Eviden tly a somewhat similar analysis is p ossible fo r the gr oup H = [5 , 3 , 5] with diagram ∆( H ) = 1 • τ 2 • τ 2 • 1 • and corresp onding discriminan t − 1 16 (3 + 7 τ ) ∼ − (3 + 7 τ ) =: λ . Since N ( λ ) = − 19, w e see that λ is also prime. W e note o nly that the group H λ for the singular space V again has an in teresting quotien t. In fact, H ≃ P S L 2 (19) 14 is the automorphism gro up for the univ ersal regular p olytop e P ( H ) = { { 5 , 3 } 5 , { 3 , 5 } 5 } , with hemi-do decahedral f acets and hemi-icosahedral ve rtex-figures. This is the 5 7 -cell de- scrib ed by Co xeter in [6]. With the exception of the 11 -cell and 57 -cell, the p olytop es describ ed here can b e view ed as regular tessellations on hy p erb olic 3-manifo lds (see [21 , 6J]). Moreov er, the tw o exceptions are the o nly regular p olytop es or rank 4 (or higher) with automorphism group isomorphic to P S L 2 ( r ) for some prime p ow e r r (see [18]). F or related work see also [12, 13, 16]. References [1] E. Artin, Geometric Algeb ra , In terscience , New Y ork (1957 ). [2] T. Bisztriczk y , P . McMullen, R. Sc hneider and A. Ivi´ c W eiss (Eds.), Polytop es – A bstr act, Convex and Computational , NA TO ASI Series C440 (Klu w er, 1994) . 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(English t ranslation in Math. USSR Izv estija 17 (1981) , 477 –503.) B. Monson, University of New Brunswick, F r e dericton, New Brunsw i c k, Canada E3B 5A3, bmonson@unb.c a Egon Schulte, Northe astern University, Boston, Mass a chussetts, USA, 02115, schulte@neu.e du 16