Ideal Uniform Polyhedra in H and Covolumes of Higher Dimensional Modular Groups

Home , Ideal point, Ideal polyhedron, Rectification (geometry)

Canad. J. Math. ¨¨, pp. Ë–Ç http://dx.doi.org/Ë¨. Ëþq/S¨¨¨Ç Ë X¨¨¨¨¨qB ©Canadian Mathematical Society ¨¨

n Ideal Uniform Polyhedra in H and Covolumes of Higher Dimensional Modular Groups

Ruth Kellerhals

In memoriam Norman Johnson

Abstract. Higher dimensional analogues of the modular group PSL(, Z) are closely related to hyperbolic reection groups and Coxeter polyhedra with big symmetry groups. In this context, we develop a theory and dissection properties of ideal hyperbolic k-rectiûed regular polyhedra, which is of inde- pendent interest. As an application, we can identify the covolumes of the quaternionic modular groups with certain explicit rational multiples of the Riemann zeta value ζ(q).

1 Introduction PSL First and eminent prototypes of arithmetic groups are the modular gr√oup (, Z), PSL( [ω]) ω = Ë (− + i ) the Eisenstein modular group , Z , where Ë q is a primi- tive third root of unity, and the Hamilton modular group PSL(, Z[i, j]). hey act by orientation preserving isometries on real hyperbolic n-space for n = , q, and , respectively, and they are isomorphic to ûnite index subgroups of discrete hyperbolic reections groups (see [ËË, Ëq, Ë]). In this way, their arithmetic, combinatorial, and geometric structure can be characterised by means of ûnite volume hyperbolic Coxeter polyhedra of particularly nice shape. Best known is the geometry of PSL(, Z), which is related to the Coxeter group with Coxeter symbol [q, ∞] and an π π PSL( [ω]) ⊂ PSL( ) ideal hyperbolic triangle of angle q and area . he group , Z , C is a Bianchi group and isomorphic to a subgroup of index in the hyperbolic Coxeter simplex group with Coxeter symbol [q, q, B]. In this way, the volume of a fundamental domain, or the covolume of PSL(, Z[ω]), can be expressed by means of Humbert’s volume formula for imaginary quadratic number ûelds [B] as well as by means of Lobachevsky’s volume formula (see [8, part I,Chapter 8] and [], for example) according to q~ π PSL ω q ζ √ Ë covolq( (, Z[ ])) = () = JI . π Q( −q) q Here, ζk (s) denotes the Dedekind zeta function of the algebraic number ûeld k, x and JI( ) is Lobachevsky’s function (see Section . ).

Received by the editors June Ë , ¨Ë; revised December q¨, ¨Ë. Published online on Cambridge Core. AMS subject classiûcation: 51M10, 20F55, 51M25, ËËF¨B, ¨G¨, ËËM¨B. Keywords: Ideal hyperbolic polyhedron, k-rectiûcation, Napier cycle, Coxeter group, quaternionic modular group, hyperbolic volume, zeta value. R.Kellerhals

he aim of this work is twofold. First, we study the geometry of the higher dimensional modular and pseudo-modular groups of × matrices whose coeõcients form a basic system of algebraic integers in an associative normed real division algebra. In this way, they act by fractional linear transformations on the boundary n− R Ë ∪ {∞} and—by Poincaré extension—as hyperbolic isometries in the upper half n space U . Since the octonion multiplication is no longer associative, such groups can be realised by means of quaternionic matrices with unit Dieudonné determinant ∆ and by certain Cliòord matrices for n ≤ þ only.Of particular interest is the PS L hybrid in- quaternionic modular group ∆ (, Hyb) with coeõcients in the ring of tegers Hyb = Z[ω, j]. his group is closely related to the Coxeter pyramid group with Coxeter symbol [B, q, q, q, q, B] whose explicit covolume computation, however, is very diõcult. In this context, we discovered a beautiful, highly symmetric, hyperbolic polyhe- ideal birectiûed -cell r S dron, the B re g, which can be interpreted as a þ-dimensional analogue of the ideal regular tetrahedron Sre g (see Example .q). Based on this, and as our second and main achievement, we introduce and develop the theory of ideal hyperbolic k-rectiûed regular polyhedra rk P in hyperbolic n-space viewed in the projective model of Klein-Beltrami. Such a polyhedron is uniform; that is, its symmetry group acts transitively on the set of its vertices. For the important families of regular simplices P = Sre g and regular orthoplexes (or cross-polytopes) P = Ore g, we derive explicit dissection relations by means of certain truncated characteristic sim- ̂ n ̂ n n plices Ui (αk ) and Vi (αk ) characterised by a dihedral angle αk depending on the dimension n and on the rectiûcation degree k (for notation and the construction, see Section q.). Denote by δik ∈ {¨, Ë} the Kronecker-Delta function deûned for elements i, k in an index set I. hen one of our main results can be stated as follows (see heorem q.B). heorem Ë.Ë Let n ≥ q and Ë ≤ k ≤ n − be integers. For a regular polyhedron P n with Schläi symbol p p , the ideal k-rectiûed regular n-polyhedron ⊂ E { Ë, ... , n−Ë} ̂ n P = rk P ⊂ H admits the following dissections. If P is a simplex S with p p , then r S admits for i n a (i) re g Ë = ⋅ ⋅ ⋅ = n−Ë = q k re g ¨ ≤ ≤ dissection into i n i of δ -truncated orthoschemes isometric to Û αn , !( + Ë − )! ( − ¨i ) i ( k ) and each of these splits into n+Ë simply-truncated orthoschemes isometric to Û (αn), ¼ i ¨ k where αn = n−k . k arccos (n−k−Ë) If P is an orthoplex O with p p and p , then r O (ii) re g Ë = ⋅ ⋅ ⋅ = n− = q n−Ë = k re g admits for i n a dissection into n i n i of δ -truncated orthoschemes ¨ ≤ ≤ −Ë !( − )! (− ¨i ) isometric to V̂ αn , and each of these splits into n simply-truncated orthoschemes i ( k ) i n n isometric to V̂ α , where α √ Ë . ¨( k ) k = arccos n−k−Ë

Our proof is based on Debrunner’s heorem [ ] and the theory of Napier cycles as introduced by Im Hof []. r S As a consequence, the ideal birectiûed B-cell re g admits a dissection into B! isometric copies of the Coxeter prism [B, q, q, q, q, B], which in turn is part of a crystallographic Napier cycle (see Section .q). his relationship allows us to determine the r S PS L volume of re g and the covolume of ∆ (, Hyb) as follows (see heorem .Ë). Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups q heorem Ë. For the hybrid modular group PS L PS L ω j , ∆ (, H yb) = ∆ (, Z[ , ])

PS L Ëq ζ covolþ( ∆ (, H yb)) = (q). ËÇ¨

As a curious by-product, we obtain the following and seemingly new expression for ζ(q) by combining the two diòerent volume representations for the Coxeter polyhedron [B, q, q, q, q, B] (see Section , (.ËË)):

π qB¨ π π π π ζ(q) = JI + S JI + ω(t) + JI − ω(t)dt , where π Ëq q q B B sin t cos ω(t) = √ . sin t − Ë his work is organised as follows. In Section , we provide the basic concepts about hyperbolic polyhedra and Coxeter orthoschemes. We discuss Napier cycles and present some distinguished examples. At the end of the section, we supply the volume identities that will play a crucial role. In Section q, we present Debrunner’s classical dissection result for regular simplices and orthoplexes in a standard geometric space. hen we develop the theory of ideal hyperbolic k-rectiûed regular polyhedra and provide an interpretation by (polarly) truncated polyhedra.Our key dissection result as given by heorem Ë.Ë can then be established. In the last part, in Section , we exploit the relation between certain quaternionic (pseudo)-modular groups and arithmetic hyperbolic Coxeter groups as described by Johnson [ËË]. By combining various of our results and applying them to the ideal birectiûed B-cell, we will be able to establish our second main result as stated in heoremË. . he work ends with the Remark . about þ PS L the incommensurability of the modular þ-orbifolds given by H ~ ∆ (, H am) and þ PS L H ~ ∆ (, H yb). In fact, there is no (orientable) hyperbolic þ-manifold that is a ûnite cover of both orbifolds. 2 Napier Cycles and Volumes in Hyperbolic 5-space

In this section, we present the necessary background about hyperbolic polyhedra and their description as fundamental polyhedra for discrete hyperbolic reection groups. Of particular interest are regular polyhedra and their characteristic simplices. he truncation by (polar) hyperplanes will lead us to the notion of Napier cycles. Finally, some related volume formulas will be presented that will form a key ingredient in the proofs of our main results.

2.1 Hyperbolic Polyhedra and Coxeter Orthoschemes

n n n Denote by X either the Euclidean space E , the sphere S , or the hyperbolic space n n H , together with its isometry group Isom(X ). In the sequel, we will focus on the hyperbolic case assuming that the corresponding classical concepts in the euclidean- aõne and spherical cases are well known. n he hyperbolic space H can be viewed in diòerent ways, for example, by means of n n n the Poincaré models in the upper half space U and in the unit ball B of E , as well R.Kellerhals

n as the projective unit ball model K of Klein-Beltrami, closely related to the vector n n space model H in the space E ,Ë of Lorentz–Minkowski. n For the description of polyhedral objects, the vector space model H (and its n projective counterpart K ) of hyperbolic space is most convenient (see [B,Chap- n n ter I], for example). More concretely, let H be deûned in the quadratic space E ,Ë = ( n+Ë ⟨x y⟩ = n x y − x y ) (n ) R , , n,Ë ∑i=Ë i i n+Ë n+Ë of signature , Ë , that is, n x n,Ë x x x H = { ∈ E S⟨ , ⟩n,Ë = −Ë, n+Ë > ¨}, d x y x y x n,Ë with distance function H( , ) = arcosh(−⟨ , ⟩n,Ë). A non-zero vector ∈ E is time-like, light-like, or space-like, if its square Lorentzian norm is negative, zero, or n positive, respectively. he boundary ∂H can be identiûed with the set of light-like n,Ë n vectors x ∈ ∩ on the unit sphere such that xn+ > ¨. Furthermore, for an E S n Ë integer Ë ≤ k ≤ n − Ë, a hyperbolic k-plane in H is the non-empty intersection of a n+ n (k+Ë)-dimensional subspace of R Ë with H .Of importance will be the fact that the Lorentz-orthogonal complement of a hyperbolic hyperplane is generated by a space- n like vector.Let us add that the group Isom(H ) coincides with the group PO(n, Ë) of positive Lorentzian matrices. For more details, see [q,Chapter q] and [B]. n Now, any discrete subgroup of Isom(H ) has a convex fundamental domain in n H whose closure can be assumed to be polyhedral. Of particular interest will be n discrete groups generated by ûnitely many reections in hyperplanes of H . For their description, we rely on the combinatorics and geometry of their fundamental polyhedra. n To this end, represent a hyperplane H ⊂ H by a unit normal vector, that is, by a n space-like vector e ∈ E ,Ë of Lorentzian norm Ë such that H x n x e = { ∈ H S⟨ , ⟩n,Ë = ¨}, and which bounds two closed half-spaces, for example, H− x n x e = { ∈ H S⟨ , i ⟩n,Ë ≤ ¨}. n A (convex, closed, and indecomposable) polyhedron P ⊂ H is of the form − P = Hi , i∈I where the index set I is always supposed to be ûnite. A polyhedron P is compact n (or of ûnite volume) if P is the convex hull of ûnitely many points of H , called the n n vertices of P (or of H ∪ ∂ H ). he polyhedron P is ideal if all vertices belong to n ∂ H . In the sequel, we will consider acute-angled polyhedra, that is, polyhedra with (interior) dihedral angles ∡(Hi , H j) ≤ π~ for i, j ∈ I. Consider the Gram matrix G P P e e i j I ( ) of formed by the products ⟨ i , j⟩n,Ë, , ∈ .It is known that an indecomposable real symmetric N × N matrix A with diagonal elements [A]ii = Ë and non- diagonal elements [A]i j ≤ ¨, i =~ j, is the Gram matrix G(P) of an acute-angled poly- n hedron P ⊂ H (uniquely determined up an isometry) if and only if the signature of A n n equals (n, Ë). Similar results for acute-angled polyhedra in E and S are well known (see [8, Part I,Chapter B]). Furthermore, many of the combinatorial, metrical, and arithmetic properties of P can be read oò from G(P). In particular, for i =~ j, Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups þ e e H H the coeõcients ⟨ i , j⟩n,Ë characterise the mutual position of the hyperplanes i , j as follows: ⎧ n ⎪cos αi j if Hi , H j intersect at the angle αi j in H , ⎪ n −⟨ei e j⟩n = ⎨ Hi H j ∂ H (.Ë) , ,Ë ⎪Ë if , meet at , ⎪ n ⎩⎪cosh li j if Hi , H j are at distance li j in H . For more details, see Vinberg’s seminal work [B], [8, Part I,Chapter B]. In case of many orthogonal bounding hyperplanes and small SIS, it is convenient n to represent a given polyhedron P ⊂ H in terms of a (weighted) graph Σ = Σ(P) n of order SIS. With each bounding hyperplane H with normal vector e ∈ E ,Ë directed outwards with respect to P, we associate a node ν in Σ. Two diòerent nodes νi , ν j are connected by an edge with a weight ci j if the hyperplanes Hi , H j are not orthogonal in n c e e c c α H . he weight i j is given by ⟨ i , j⟩n,Ë. However, if −Ë < i j < ¨, then i j = −cos i j, and we replace ci j by αi j. An edge with weight −Ë will be decorated by the symbol ∞. Edges with weights Sci jS > Ë are replaced by a dashed edge, and the weights are omitted in most cases. More speciûcally, if P is a Coxeter polyhedron having by deûnition dihedral angles of the form αi j = π~mi j for integers mi j ≥ , only, the corresponding weights traditionally carry only the label mi j > q.Hence, simple edges indicate an intersection angle equal to π~q, and nodes not connected by an edge symbolise orthogonal hyperplanes. In order to depict a Coxeter graph in an abbreviated way, we oen use Coxeter symbol p p q q the . In particular, [ Ë, ... , k ] or [ Ë, ... , l , ∞] with integer labels pi , q j ≥ q are associated with linear Coxeter graphs with k +Ë or l + edges marked by r the respective weights. he Coxeter symbol [(p, q)[ ]] describes a group with cyclic Coxeter graph consisting of r ≥ consecutive Coxeter graphs [p, q] (see [Ë,Appen- dix], for example). In the sequel, we oen represent a Coxeter polyhedron by quoting its Coxeter symbol. Recall that the reections with respect to the bounding hyperplanes of a Coxeter n n polyhedron P ⊂ H generate a discrete group ΓP ⊂ Isom(H ) that is called a hyperbolic Coxeter group. he group ΓP is denoted by the Coxeter graph and Coxeter symbol of P, and we do not distinguish between Coxeter polyhedron and reection group. Notice that in contrast to the hyperbolic ones, the irreducible spherical (û- nite) and euclidean (or aõne) Coxeter groups are completely classiûed. For a list, see [Ç,Chapter ] or [8,Chapter þ]. R n α α n I A polyhedron ⊂ X with linear graph Σn = Σn( Ë, ... , n) of order +Ë = S S ≥ given by Figure Ë:

n Figure Ë: he graph of an n-orthoscheme R ⊂ X . is a geometric n-simplex whose bounding hyperplanes are indexed by Hi , ¨ ≤ i ≤ n, in such a way that Hi ⊥ H j for Si − jS > Ë. he polyhedron R is parametrised by the α H H i n orthoschemes dihedral angles i = ∡( i−Ë, i ), Ë ≤ ≤ . hese simplices are called BR.Kellerhals

and were introduced by L. Schläi [þ] in the spherical case. hey play an important role when studying polyhedra with high degree of symmetry and provide the n characteristic simplices in the barycentric decomposition of regular polyhedra in X (see Section q.Ë,[q, Section 8.], and [8, Part II,Chapter þ]). Consider the special case of a graph Σn(α, β) as given in Figure , where the pa- rameters α, β ∈ [¨, π~) are such that the graph Σn(α, β) relates to a polyhedron n R(α, β) ⊂ X (possibly of inûnite volume).

Figure : Graphs related to certain characteristic n-orthoschemes.

π ● he graph Σn(α, ) describes the characteristic (or barycentric) simplex of a û- q n nite volume regular n-simplex Sre g(α) ⊂ X with dihedral angle α, which is spheri- − < ( α) < Ë ( α) = Ë Ë < ( α) ≤ Ë cal, euclidean, or hyperbolic if Ë cos n , cos n , or n cos n−Ë , respectively (see [Ë8], for example). Indeed, by barycentric decomposition from its π in-center, Sre g(α) can be decomposed into (n +Ë)! isometric copies of R(α, ).Ob- n q serve that ûnite volume regular simplices tesselating hyperbolic space H exist only for n = , q, and (see [8, Part II,Chapter þ, Section q]). A particular role is played by R( π π ) [ ] the non-compact, ûnite volume orthoscheme B , q with Coxeter symbol B, q, q , which is the characteristic simplex of an ideal regular hyperbolic tetrahedron or -cell π q Sre g( ) ⊂ H . q π ● he graph Σn(α, ) arises with respect to the barycentric decomposition into n n! isometric copies of the characteristic simplex of an n-dimensional regular cross- polytope or n-orthoplex Ore g(α) (in the terminology of J. Conway) with dihedral angle α, which is hyperbolic of ûnite volume if √Ë < cos α ≤ √ Ë . In particular, the n n−Ë π q Coxeter orthoscheme [ , , q] is related to an ideal regular octahedron Ore g( ) ⊂ H . π he graph Σn(α, ) (read from right to le in Figure ) describes the dual polyhedron n of Ore g(α), that is, a regular n-cube Tre g(α) ⊂ X with dihedral angle α.Of n course, the polyhedron Tre g(α) exists with ûnite volume in H under the identical angle constraint.

2.2 Napier Cycles

n An n-orthoscheme R ⊂ H as given by the graph in Figure Ëi s characterised by n + Ë e e n,Ë n,Ë outer (unit) normal vectors ¨, ... , n ∈ E , forming a basis of E and satisfying e e e e i j −Ë < ⟨ i−Ë, i ⟩n,Ë < ¨ and ⟨ i , j⟩n,Ë = ¨ for =~ . In this respect, an orthoscheme is a part of and generates a so-called Napier cycle as introduced by Im Hof [].

n n Deûnition .Ë A Napier cycle in E ,Ë is a set N = {ei ∈ E ,Ë S i ∈ Z~(n + q)} of n + q vectors subject to the conditions c e e i (.) i ∶= ⟨ i−Ë, i ⟩n,Ë < ¨ for all , e e j i i i ⟨ i , j⟩n,Ë = ¨ for =~ − Ë, , + Ë. Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups 8

n Any n+Ë consecutive vectors in a Napier cycle N form a basis of E ,Ë. Furthermore, the deletion of two non-adjacent vectors from N deûnes two Lorentz-orthogonal sub- n spaces of E ,Ë. Either these subspaces are both light-like, or one of them is space-like, while the other one is time-like (or negative). A Napier cycle N can be of three diòer- ent types. Either all vectors of N are space-like, or exactly one vector is not space-like, or exactly two vectors are not space-like. For n ≥ , the vectors in N admit a num- i n e e bering and can be normalised in such a way that, for ¨ ≤ ≤ , ⟨ i , i ⟩n,Ë = Ë with c e e −Ë < i = ⟨ i−Ë, i ⟩n,Ë < ¨. Consider such a normalised Napier cycle N with given c c type d n d negative numbers Ë, ... , n. he set N is of if it contains precisely + vectors of positive Lorentzian norm, equal to Ë, say, whose non-vanishing Lorentzian products ci admit the interpretation according to (.Ë). Moreover, the respective additional c e e c e e c e e products n+Ë = ⟨ n , n+Ë⟩n,Ë, n+ = ⟨ n+Ë, n+⟩n,Ë and ¨ = ⟨ n+, ¨⟩n,Ë can be easily computed by means of a suitable Gram determinant calculation in terms of the values c c δ c c Ë, ... , n. In fact, let ( Ë, ... , n) be the determinant of the Gram matrix c ⎛ Ë Ë ⎞ ⎜c ⋱ ⎟ ⎜ Ë Ë ⎟ G(e e ) = ⎜ ⋱ ⋱ ⋱ ⎟ ¨, ... , n ⎜ ⎟ ⎜ ⋱ Ë cn⎟ ⎝ ⎠ cn Ë e e of the vectors ¨, ... , n, which satisûes nice recursion formulas. In particular, one easily veriûes that n + δ c c Ë δ Ë c c Ë c Ë (.q) ( Ë, ... , n) = n , −√ , , ... , n = n if all i = − . −Ë In the same spirit, by [, Proposition Ë.B] (or [þ, Section 8]), one can show that

δ(c cn) δ(c cn) c = Ë, ... , c = Ë, ... , (. ) ¨ δ c c , n+Ë δ c c , ( , ... , n) ( Ë, ... , n−Ë) δ(c cn)δ(c cn ) − c = Ë, ... , , ... , −Ë Ë n+ δ c c δ c c . ( Ë, ... , n−Ë) ( , ... , n) Now, since the Lorentzian orthogonal complement of each unit space-like vector e in n N deûnes a hyperplane H bounding two half-spaces in H , it is not diõcult to under- stand the polyhedral conûguration provided by a Napier cycle of type d. In fact, type n e ËNapier cycles correspond to orthoschemes in H . Type Napier cycles (with n+ H H not space-like, say) are inûnite volume orthoschemes bounded by ¨, ... , n and H simply-truncated n truncated by n+Ë to yield ûnite volume, orthoschemes in H . In a similar way, Napier cycles of type q are ûnite volume, doubly-truncated orthoschemes H H n bounded by hyperplanes ¨, ... , n+ in H . hey arise from inûnite volume orthoschemes bounded by n + Ë hyperbolic hyperplanes by truncation by means of the two remaining ones. For details, see [].

2.3 Crystallographic Napier Cycles

n Consider a Napier cycle of type d in ,Ë and assume that it yields a (d − Ë)-truncated n E Coxeter orthoscheme in H . his means that for each weight ci with −Ë < ci < ¨, Ç R.Kellerhals c π m one has that i = −cos m for some integer i ≥ q. In [], Im Hof completely classi- i ûed these particular Coxeter polyhedra, referring to them as crystallographic Napier n cycles, and showed that they exist in H for n ≤ , only. In the sequel, the following examples will be of particular interest.

Example . he Coxeter polyhedron [B, q, q, q, ∞] in H with graph given by e Figure q gives rise to a crystallographic Napier cycle of type (here with B not space- like) and describes a simply-truncated Coxeter orthoscheme. Underlying is the inû- R H H nite volume orthoscheme ¨ = [B, q, q, q] bounded by the hyperplanes ¨, ... , p R according to the graph in Figure q. Denote by k the vertices of ¨ opposite to the bounding hyperplane Hk for ¨ ≤ k ≤ þ. By construction, they form an or- p p p p ,Ë R thogonal edge path ¨ Ë, ... , þ in E . he polyhedron ¨ can be interpreted as the characteristic simplex (see Section q.Ë) of an inûnite volume regular hyperbolic S π p v v -simplex re g( ) with in-center equal to ¨ and all of whose vertices ¨, ... , q π are given by space-like vectors. In particular, one vertex of Sre g( ) corresponds to p R q whose neighborhood in ¨ is a cone over the hyperbolic Coxeter tetrahedron H H p [B, q, q]. he truncating hyperplane þ intersects at the point q on the boundary ∂ H , indicated by a circle in Figure q. In this way, the polyhedron [B, q, q, q, ∞] is a p Coxeter pyramid with apex at inûnity q over a product of two (euclidean) Coxeter simplices with symbols [B, q] and [∞] (see [8], for example). In particular, all edges π viv j Sre g( ) qi j ∂ H of q are bisected at an ideal point, denoted by , on . he convex hull of qi j, ¨ ≤ i < j ≤ , gives rise to an ideal polyhedron of ûnite volume in ∂ H , π π ideal rectiûed þ-cell r Sre g with dihedral angles q and , called the , denoted by Ë (see Section q.).

Figure q: he -dimensional ûnite volume analogue of [B, q, q].

Example .q Consider the Coxeter polyhedron [B, q, q, q, q, B] in Hþ given by Figure .It is a simply-truncated Coxeter orthoscheme which belongs to a crystallo- R graphic Napier cycle of type . he Coxeter orthoscheme ¨ = [B, q, q, q, q] bounded H H p p by the hyperplanes ¨, ... , þ and with vertices ¨, ... , þ opposite to them is of in- p p R ûnite volume. he vertices , þ of ¨ are given by space-like vectors described by G R p principal submatrices of ( ¨) of signature ( , Ë), while the vertex q is given by a p light-like vector. he neighborhood of q is a cone over the product of two (euclidean) Coxeter triangles with symbol [B, q] as indicated by a circle in Figure . he polyhe- R dron ¨ is the characteristic simplex arising in the barycentric decomposition of an S π p inûnite volume hyperbolic regular þ-simplex re g( ) with in-center ¨. All vertices π q Sre g( ) v v of q are given by space-like vectors ¨, ... , þ such that they span space-like planes span(vi , v j) in Eþ,Ë. By applying Vinberg’s theory about the face complex of acute-angled hyperbolic polyhedra, the graph in Figure indicates that the barycen- π p Sre g( ) ter of each -face (the vertex q, for example) of q is a point on the boundary ∂ Hþ. Taking the convex hull of all these ideal barycenters yields a ûnite volume Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups

Figure : he þ-dimensional ûnite volume analogue of [B, q, q].

π π ideal hyperbolic polyhedron, of dihedral angles equal to q and , which is called the birectiûed B-cell r S þ and denoted by re g ⊂ H (see Section q.).

Example . Similarly to Example ., the Coxeter pyramid [ , , q, q, ∞] in H π Ore g( ) associated with an inûnite volume regular -orthoplex describes a simply- ideal rectiûed -orthoplex r O truncated orthoscheme that decomposes the Ë re g ⊂ H π with all dihedral angles equal to into qÇ isometric copies (see Section q.). Similarly to Example .q, the Coxeter polyhedron [ , , q, q, q, B] in Hþ associated π Ore g( ) with an inûnite volume regular þ-orthoplex describes a simply-truncated ideal birectiûed þ-orthoplex r O þ orthoscheme which decomposes the re g ⊂ H with π π dihedral angles and q into q,Ç ¨ isometric copies (see Section . , Remark .8, and Section q.).

Example .þ he Coxeter polyhedron [q, , qþ, B] in H8 is a simply-truncated or- ideal birectiûed 8-orthoplex r O thoscheme that decomposes barycentrically the re g ⊂ H8 π π π with dihedral angles q , q and into B þ, Ë¨ isometric copies (see Section q. with Corollary q.8).

2.4 Some Volume Identities

n Consider a normalised Napier cycle N = {ei ∈ E ,Ë S i ∈ Z~(n+q)} of type d, Ë ≤ d ≤ q, n,Ë in such that −Ë < ci = ⟨ei− , ei ⟩n < ¨ for Ë ≤ i ≤ n. As described in Section ., E Ë ,Ë n N yields a k-truncated orthoscheme Rk in H for k = d − Ë. We are interested in ûnding explicit volume expressions for crystallographic Napier cycles containing the n building blocks of tesselating k-rectiûed regular polyhedra in H (see Section q.). By [Ë , Ëþ], we dispose of closed volume formulae available for dimension n = q and for even dimensions. ● For n = q, the volume of any k-truncated orthoscheme can be expressed in x terms of its dihedral angles and by means of the Lobachevsky function JI( ), which is deûned as follows (see [Ë ]): Ë ∞ sin(rx) x (x) = Q = − S S tSdt x ∈ JI r log sin , R. r=Ë ¨ For example, by the classical Lobachevsky formula [Ë , ()], the volume of a non- R ⊂ q ( π − α α β) compact (¨-truncated) orthoscheme H with linear graph Σq , , (see Figure Ë) is given by the expression π π R Ë α Ë α β α β volq( ) = JI ( ) + JI − + − JI + + . Ë π In particular, the volume of the Coxeter orthoscheme [B, q, q] equals JI( ) so that π π Ç q (Sre g( )) = ( ) volq q q JI q . Ë¨ R.Kellerhals

● For n = m even, the volume of a k-truncated hyperbolic n-orthoscheme Rk with graph Σk = Σ(Rk ) can be expressed in terms of certain real-valued modiûed n l volume functions Fn and fl deûned on (the set of measurable subsets of) H and S , f F respectively, in a more elegant way. More concretely, let ¨ = ¨ = Ë, and deûne for l = Ë, ... , n − Ë, n ≥ , the volume functions

l+Ë f ν F in ν i ν l ∶= l voll , n ∶= n voln with = −Ë, l = l . voll (S ) hen the so-called Reduction Formula as presented in [Ëþ, Section q] allows one to express the volume of a k-truncated n-orthoscheme Rk with graph Σk in terms of its dihedral angles as follows: m r (−Ë) r F m( k ) = Q Q f m r (σ) Q f ∶= (.þ) Σ r r −( +Ë) , −Ë Ë. r=¨ + Ë σ σ Here, σ runs through all spherical subgraphs of order (m − r) of Σk all of whose connected components are of even order.Observe that, for m > r, each such compo- nent describes a spherical orthoscheme of odd dimension < (m − r). By means of formula (.þ) for n = m and by Schläi’s results about the order of a ûnite Coxeter group providing the values fl for l ≤ m − Ë(see [þ, No. q, p. BÇ ò]), the volumes of all k-truncated Coxeter n-orthoschemes were determined in [Ëþ,Appendix] (up to some minor calculation errors). In particular, for n = , the simply-truncated Coxeter orthoscheme [B, q, q, q, ∞] is of volume π~þ ¨, while for n = B, the simply-truncated Coxeter orthoscheme [q, , q, q, q, q, ∞] is of volume πq~þ, ¨¨. In the sequel, we are particularly interested in the volume computation for those polyhedra in Hþ that are related to orbit spaces by certain quaternionic modular groups. In general, it is very diõcult to ûnd a closed volume formula for a family of polyhedra of ûxed combinatorial-metrical type in Hþ. However, there are some partial but very useful results for k-truncated þ-orthoschemes. For k = ¨, consider a þ-orthoscheme − (.B) R(α, β, γ) = Hi such that cos α + cos β + cos γ = Ë, ¨≤i≤þ which is deûned by the graph in Figureþ .

Figure þ: he orthoscheme R(α, β, γ) ⊂ Hþ with cos α + cos β + cos γ = Ë.

p p R α β γ he angle condition in (.B) ensures that the vertices ¨ and þ of ( , , ) opposite H H to the bounding hyperplanes ¨ and þ are ideal points. In [ËB, heorem, p. Bþ], we obtained the following volume formula: π R α β γ Ë α β Ë γ (.8) volþ ( , , ) = JI ( ) + JI ( ) − JI − q q q Ë π π q − JI + α + β + JI − α + β + ζ(q). ËB q q B Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups ËË ω ω Here, the Lobachevsky function of order three JIq( ), ∈ R, is related to the classi- (z) = ∞ zr SzS ≤ ( ) = ζ( ) cal trilogarithm function Liq ∑r=Ë rq , Ë, satisfying Liq Ë q , as follows:

∞ Ë ix Ë cos(rx) (x) = R (e ) = Q x ∈ (.Ç) JIq Liq rq , R. r=Ë x π he function JIq( ) is even, -periodic, and fulûls the distribution law

m Ë −Ë rπ (mx) = Q x + x ∈ m ∈ > (.) m JIq JIq m , where R and Z ¨, r=¨ x which allows one to identify some special values of JIq( ) with certain rational multiples of ζ(q) (see [ËB, Section .q]). As a consequence of the identities (.8), (.Ç), and (.), the Coxeter orthoscheme R( π π π ) Hþ q , , q in has volume

8 ζ (.Ë¨) volþ([q, , q, q, ]) = (q). , B¨Ç However, the volume computation for the simply-truncated Coxeter orthoscheme [B, q, q, q, q, B] in Hþ discussed in Example .q cannot be performed in an exact man- ner by exploiting results in the spirit of (.8). Nevertheless, based on Schläi’s volume diòerential formula (see [ËÇ, Section .Ë], for example), its volume can be represented by a single integral as follows. Denote by R(ω) ⊂ Hq a non-compact orthoscheme ω π π ω with dihedral angles , q , B (see Figure Ë). As a function of , its volume is given by the classical Lobachevsky formula [Ë , ()] according to π π π R ω Ë Ë ω ω volq( ( )) = JI + JI + + JI − . q B B With these preparations, the volume of [B, q, q, q, q, B] can be written as follows:

(.ËË) volþ([B, q, q, q, q, B]) π Ë Ë π = S vol (R(ω(t)))dt − JI π q q Ç q π Ë π π π π = JI + S JI + ω(t) + JI − ω(t)¡dt π ËB q q B B sin t ≃ ¨.¨¨8Ë8þ8, where cos ω(t) = √ . sin t − Ë Remark .B By a structural result of Prasad (see [þ, Proposition .Ë (Ë)], for example), the volume of the non-compact arithmetic orbifold deûned over Q as given by the orbit space of [B, q, q, q, q, B] is a rational multiple of ζ(q). Now, numerical evidence suggests that the value (.ËË) for the volume of [B, q, q, q, q, B] is equal to Ëq ζ( ) þ,8B¨ q . By means of a combinatorial-metrical argument, we will prove rigorously ([ ]) =? Ëq ζ( ) the conjectural identity volþ B, q, q, q, q, B þ,8B¨ q (see heorem .Ë). Ë R.Kellerhals

Remark .8 In a similar way, the volume of the simply-truncated Coxeter orthoscheme [B, q, q, q, , ] in Hþ can be identiûed as follows:

π Ë qπ π π π vol [B, q, q, q, , ] = JI + S JI + ω(t) + JI − ω(t)¡dt . þ π ËB Ç q B B In [ ], Ratcliòe and Tschantz determined—among other things—the covolume of f n(x) = x +⋅ ⋅ ⋅+x − x the group of units of the quadratic form q Ë n q n+Ë, which is known to contain a (maximal) reection subgroup Γ of ûnite index for n ≤ Ëq. In the case of n = þ, the Coxeter group Γ ⊂ Isom(Hþ) is given by [B, q, q, q, , ]. heir volume computation yields the precise expression √ q L volþ([B, q, q, q, , ]) = (q, Ë) ≃ ¨.¨¨þqþÇ8 ÇÇ, q¨ L(s D) = ∞ D Ë L where , ∑r=Ë r rs denotes the Dirichlet -series with Kronecker symbol D ( r ) (see [ , Table Ë] and compare with [þ, Proposition .Ë ()]). Notice that the two arithmetic hyperbolic Coxeter groups [B, q, q, q, , ] and [B, q, q, q, q, B] have non- compact fundamental polyhedra of identical combinatorial type being pyramids over a product of two simplices.If the two groups were commensurable, the volumes would be necessarily Q-proportional. However, in [8], we proved that the groups are incommensurable.

We ûnish the volume considerations by quoting and applying the following result for doubly-truncated þ-orthoschemes as proved in [ËÇ, ( )].

Figure B: A doubly-truncated þ-orthoscheme with cyclic graph Ω(α).

Proposition .Ç Consider a doubly-truncated orthoscheme in Hþ with cyclic graph Ω(α) of order Ç as given in Figure B. hen its volume equals π α Ë ζ Ë α α volþ(Ω( )) = (q) − JI ( ) + JI − . q q q

Example . By the above proposition together with the identity (.), we obtain that

[ ] Ëq ζ Ëq (.Ë) volþ [(q, B) ] = (q) = ¨ ⋅ . ÇÇ þ, 8B¨ Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups Ëq

By comparing (.Ë) with the result of Remark .B, we deduce the following conjec- ( π ) = [( )[ ]] [ ] tural volume identity between Ω q q, B and B, q, q, q, q, B :

[ ] ? B (.Ëq) volþ([(q, B) ]) = ⋅ volþ [B, q, q, q, q, B]. q his volume identity will be a direct consequence of a dissection result for an ideal birectiûed regular simplex in Hþ (see Section q.þ, heorem q.B(i) for n = þ and k = ). 3 Rectifying Hyperbolic Regular Polyhedra

In this section, we present Debrunner’s dissection result for regular simplices and orthoplexes by means of certain orthoschemes. hen we develop the theory of ideal hyperbolic k-rectiûed regular polyhedra in the projective model of hyperbolic n-space and provide an interpretation by polarly truncated polyhedra. In this way, we can describe the two families of ideal k-rectiûed regular simplices and orthoplexes by means of Napier cycles and prove one of our main results as given by heorem q.B.

n 3.1 Regular Polyhedra in X and Debrunner’s Result P n ags F F F Consider a polyhedron ⊂ X and its of the form F = { ¨, ... , n−Ë}, −Ë ∶= ∅, k F P F F k n consisting of -dimensional faces k of such that k−Ë ⊂ k for = ¨, ... , − Ë. he polyhedron P is regular (and denoted by Pre g at times) if its symmetry group Sym(P) acts (simply) transitively on its ags (see [8, Part II,Chapter þ, Section q], for example). It follows that each face of P is itself a regular polyhedron and that the symmetry group of P has a unique ûxed point, the (bary- or in-)center of P, denoted by bn. he point bn is the center of the diverse in- and circumspheres attached to P. Fix a ag F of P and consider the centers bk of its k-faces (¨ ≤ k ≤ n − Ë). In particular, b v P b b b b the point ¨ coincides with a vertex ∈ . Each sequence ¨, ... , i−Ë, i+Ë, ... , n − deûnes an (aõne) hyperplane Hi , ¨ ≤ i ≤ n − Ë, which bounds the half-space Hi in n n− − bi C = ∩ ËH bn X containing the point . Consider the polyhedral cone i=¨ i with apex n b b in X whose edges pass through ¨, ... , n−Ë.It provides a fundamental domain for Sym P n H H ( ), which is generated by the reections in the hyperplanes ¨, ... , n−Ë.It is not diõcult to see that R ∶= C ∩ P is an n-orthoscheme, called the characteristic simplex of P. he regular polyhedron P is of dihedral angle α if the hyperplane Hn b R P H opposite to n in the boundary of (and of ) and the hyperplane n−Ë form the angle H H α k n p k ∡( n−Ë, n) = . For = Ë, ... , − Ë, let k denote the number of -dimensional P F F F P faces of containing the face k− and being contained in the face k+Ë where n ∶= . Schläi symbol P p p hen the of is the ordered set { Ë, ... , n−Ë}. Reading a given Schläi symbol in reversed order yields the Schläi symbol of the regular polyhedron dual to P.Observe that the cone C and the reection group generating Sym(P) can be represented by the Coxeter graph ΣC as given in Figure 8. Furthermore, the graph ΣC A B D E E E F H H I m relates to a ûnite Coxeter group (of type n, n, n, B, 8, Ç, , q, , or ( ); see [Ç, part I, Section ]). he vertex ûgure Pv at v of an arbitrary polyhedron P is the intersection of P n with a sphere centered at v of suõciently small radius in X (intersecting only the edges passing through v). Up to a normalisation, each vertex ûgure of an (ordinary) n vertex v in X is a spherical (n − Ë)-polyhedron. In the hyperbolic case, a vertex q of Ë R.Kellerhals

Figure 8 C n : he Coxeter graph ΣC associated with the cone ⊂ X .

n a ûnite volume polyhedron P may be on the boundary at inûnity ∂ H , in which case its vertex ûgure—as intersection of P with a suitable horosphere centered at q—turns into a Euclidean (n − Ë)-polyhedron. It is easy to see that each vertex ûgure Pv of a regular polyhedron P is itself a regular polyhedron and admits a barycentric decomposition into (n−Ë)-orthoschemes isometric to R∩Pv . In this way, the Schläi symbol p p p p { Ë, ... , n−Ë} can be interpreted as the union of the Schläi symbol { Ë, ... , n−} p p of a facet with the Schläi symbol { , ... , n−Ë} of a vertex ûgure of a regular polyhedron P. he graph ΣC is the Coxeter graph associated with the spherical vertex n R ∩Pb P ⊂ ûgure n . Finally and most importantly, a regular polyhedron X of dihedral angle α can be entirely described by means of its characteristic simplex R with graph π π α Σn( p , ... , p , ) according to Figure Ë( see [q, Sections 8.þ–8.]). Ë n−Ë Examples are the Schläi symbols {q, ... , q} and { , q, ... , q}, which describe n (self-dual) regular simplices and regular hypercubes in X , while the Schläi symbol {q, ... , q, } describes a regular orthoplex (or cross-polytope). Notice that these are the only regular polyhedra existing in every dimension. For these types of reg- n ular polyhedra, realised on the sphere S , Schläi [þ, Sections and qË] obtained volume identities for certain related orthoscheme families. More precisely, these are the orthoscheme family Ui (α), ¨ ≤ i ≤ n + Ë, associated with a regular n-simplex n Sre g(α) ⊂ X and the orthoscheme family Vi (α), ¨ ≤ i ≤ n, associated with a reg- n ular orthoplex Ore g(α) ⊂ X as given by the graphs in Figures Ça nd . For their realisation conditions, see Section .Ë and [ , Remark (8.þ), Remark (8.)]. Further- U α R α π V α R α π more, the orthoschemes ¨( ) = ( , ) and ¨( ) = ( , ) are isometric to the U α V α q orthoschemes n+Ë( ) and n( ), respectively, the latter being the corresponding characteristic simplices (see Section .Ë and Figure ).

Figure Ç U α i n S α n : Orthoschemes i ( ), ¨ ≤ ≤ + Ë, tiling re g ( ) ⊂ X .

Figure V α i n O α n : Orthoschemes i ( ), ¨ ≤ ≤ , tiling re g ( ) ⊂ X .

In [ , heorems (8. ) and (8.Ç)], Debrunner provided an alternative proof for Schläi’s n volume identities, in the cases of Sre g(α) and Ore g(α) in X , by using only a dissection argument. More precisely, he deduced the following result (see also the proof of heorem q.B below). Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups Ëþ

heorem q.Ë (Debrunner) n (i) A regular simplex Sre g(α) ⊂ X of dihedral angle α admits for ¨ ≤ i ≤ n a dissection into i!(n + Ë − i)! orthoschemes isometric to Ui (α), and each of these splits into n+Ë orthoschemes isometric to U α . i ¨( ) n (ii) A regular orthoplex Ore g(α) ⊂ X of dihedral angle α admits for ¨ ≤ i ≤ n −Ë n a dissection into i!(n − i)! orthoschemes isometric to Vi (α), and each of these splits into n orthoschemes isometric to V α . i ¨( )

In the next section, we will extend these results to their ideal truncated or k-rectiûed hyperbolic counterparts.

3.2 Rectification of Regular Polyhedra

n Let P ⊂ E be a regular Euclidean polyhedron with fk faces Fk of dimension k for k n F F F F b ¨ ≤ ≤ . Consider a ag ¨, ... , n−Ë, n of faces k with centers k as above. hen the orbit Mk ∶= Sym(P)bk consists of the centers (or midpoints) of all k-dimensional faces of P and has cardinality equal to fk . n For ¨ ≤ k ≤ n−Ë, the k-rectiûed regular polyhedron rk P ⊂ E of P is the (Euclidean) convex hull of the fk points in Mk . he polyhedron rk P arises from P by shrinking k P r P P r P all the -dimensional faces of to their centers.Hence, ¨ = , while Ë is the P v P v M result of the truncation from of each vertex cone, denoted by cone( , v ), ∈ ¨, by the aõne hyperplane Ev determined by the centers of the edges of P ending at v. r P E P v M he facets of Ë consist of the polyhedra v ∩ associated with the vertices ∈ ¨, and the truncated facets of P. As a reference, see [q,Chapter VIII]. r P Observe that the polyhedron n−Ë coincides with the dual of the regular poly- n hedron P. Consequently, we will consider k-rectiûed regular polyhedra in E for Ë ≤ k ≤ n − , only. n A k-rectiûed regular polyhedron rk P ⊂ E gives rise to an ideal hyperbolic n-polyhedron (of ûnite volume) in the following way. Interpret hyperbolic n-space n n H in the projective unit ball model K of Klein–Beltrami. Next, consider the in- n S rk P ⊂ bn m m f Mk sphere of E centered at , which touches all points Ë, ... , k of . By normalising appropriately, the sphere S can be identiûed with the unit boundary n− n sphere S Ë of K . Since k ≥ Ë, the vertices of P are ultra-ideal points, lying outside n n of S with respect to K , and P ∩ H is a convex region bounded by the hyperbolic Sym P F hyperplanes ( ) n−Ë, which is of inûnite volume.

Deûnition q. Let Ë ≤ k ≤ n − be an integer. he ideal hyperbolic k-rectiûed ̂ n regular polyhedron P ⊂ H associated with P is the (hyperbolic) convex hull of the n− n m m f ∈ Ë P̂ rk P P̂ = rk P ideal points Ë, ... , k S of H . We identify with and write accordingly. ̂ n An ideal hyperbolic k-rectiûed regular polyhedron P ⊂ H is of ûnite volume and determined uniquely up to isometry by the Schläi symbol of the underlying regular polyhedron P and the degree of rectiûcation k.Of particular interest will be the ideal hyperbolic k-rectiûed regular simplex rk Sre g and the ideal hyperbolic k-rectiûed n regular orthoplex rk Ore g in H (for k = and n = þ, 8, see Examples .q and .þ). ËBR.Kellerhals

We will provide a complete description of their facial structure and dihedral angles (see Section q. , Remark q.þ).

3.3 Rectification and Polar Truncation

Our aim is to extend Debrunner’s heorem q.Ë to the families of ideal k-rectiûed reg- n ular simplices rk Sre g and orthoplexes rk Ore g in H , forming the single categories of such polyhedra (up to duality) existing in all dimensions. To this end, we exploit the n n properties of the Klein–Beltrami model K in real projective space P in order to adjust Debrunner’s proof appropriately. In fact, this approach will allow us to interpret an ideal rectiûed regular polyhedron as a regular polyhedron, which is suitably (polarly) truncated. n First, there is the well known relationship between points X = [x] ∈ P , repre- x n,Ë π y Pn x y sented by non-zero vectors ∈ E , and hyperplanes X = {[ ] ∈ S⟨ , ⟩n,Ë = ¨} Q x Pn x x relative to the quadric n,Ë = {[ ] ∈ S⟨ , ⟩n,Ë = ¨}, which yields a bijection between the set of all points or poles X =∶ pol(πX ) and the set of all hyperplanes or polar hyperplanes π X Pn Pn Q X =∶ pol( ) of . his duality principle for relative to n,Ë is characterised by the following important properties (see [Ë , Section Ë], for example).

n Properties q.q (i) he polar hyperplane πX of X = [x] ∈ P respectively in- Q x tersects, touches, or avoids the quadric n,Ë if and only if the vector is space-like, light-like, or time-like. (ii)I f two lines g, h in P intersect at I = g ∩ h, then pol(I) is the line determined by pol(g) and pol(h). π Pn π π (iii)I f a hyperplane Ë in contains the pole pol( ) of the hyperplane , then π π Ë ⊥ holds. P̂ r P Kn Consider an ideal hyperbolic Ë-rectiûed regular polyhedron = Ë ⊂ with underlying regular polyhedron P having the barycenter bn. For each vertex v ∈ P, n,Ë interpreted as a unit space-like vector in E , the polar hyperplane π[x] consists of all points [y] with y ∈ Ev where Ev is the hyperplane determined by the (ideal) centers mi of the edges vvi , Ë ≤ i ≤ N, of P ending at v (N ≥ n). Indeed, Ev is the hyperbolic vec- n tor subspace of E ,Ë generated by the (non-zero) light-like vectors v + vi representing the centers mi , up to normalisation (Ë ≤ i ≤ N). For a vector y = ∑i λi (v + vi ) ∈ Ev , ⟨y v⟩n = λi { +⟨vi v⟩n } λi { −Yvi Y YvY } = v vi one gets , ,Ë ∑i Ë , ,Ë = ∑i Ë n,Ë n,Ë ¨ since , lie on opposite sides of the light-like line generated by mi (see [q, heorem q.., heorem q..Ë¨], for example). As a consequence (see Property (iii) above), the hyperbolic hyperplane Ev intersects orthogonally all those hyperplanes, bounding P or bounding a characteristic R C P F F P simplex = ∩ for a ag F = { ¨, ... , n−Ë} of , which contain the vertex v = F ∈ P R = ∩n H− H ¨ (see Section q.Ë). Write i=¨ i , where ¨ is the hyperplane op- v b P̂ r P H posite to = ¨, as usually. Since = Ë is Ë-rectiûed, the hyperplanes ¨ and E b ∂Kn H E v are (hyperbolic) parallel (intersecting at Ë ∈ ) with ∡( ¨, v ) = ¨, while π ∡(Hi Ev ) = ≤ i ≤ n H− ∶= Ev , for Ë . For later purpose, it is convenient to write Ë . v =∶ v v P ≤ i ≤ f Next, consider all vertices Ë, ... , f¨ of . For Ë ¨, deûne (aõne) rays ρ b v t b ρ v ρ t i through n and i , parametrised by ≥ ¨ such that n = i (¨) and i = i ( Ë) Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups Ë8

Figure Ë¨ Û (αn) V̂ (αn) : he graphs of ¨ k and ¨ k .

⊥ t > Ei (t ) ∶= Ev = v for some Ë ¨. Moreover, intersecting hyperplanes Ë i i meet at the M Sym P b v v i j points of Ë = ( ) Ë, being the centers of the edges i j for =~ . Recall that the M Sym P b r P P̂ P̂ t (hyperbolic) convex hull of the points of Ë = ( ) Ë equals Ë = =∶ ( Ë). In ̂ this context, for k > Ë, the ideal hyperbolic k-rectiûed regular polyhedron rk P = P can P̂ t t t be interpreted as ( k ) for some (unique) k > k−Ë. Indeed, by Property (ii) (describ- ing a way to construct polar hyperplanes), for t↗tk , the points ρi (t) go uniformly away from the polyhedron P, while the polar hyperplanes represented by Ei (t) tend inwards of P until they meet points of Mk = Sym(P)bk at time t = tk .Observe that E t H R Ë( k ) intersects the hyperplane ¨ bounding under a certain non-zero dihedral E t E H angle, while Ë( Ë) = v and ¨ are (hyperbolic) parallel. he dihedral angle will be made explicit in the cases rk Sre g and rk Ore g (see (q.Ë)).

3.4 Napier Cycles Associated with rk Sre g and rk Ore g

Let n ≥ q and Ë ≤ k ≤ n − . he above considerations restricted to the special cases r S r O Û = k re g and k re g motivate¼us to look at the simply-truncated orthoschemes ¨ n n n k n n Û α α − V̂ V̂ α α √ Ë ¨( k ), where k = arccos n k , and ¨ = ¨( k ), where k = arccos ( − −Ë) n−k−Ë n c¨ (Û ) in H , whose graphs are given by Figure Ë¨ and carry the additional weights n+Ë ¨ c¨ (V̂ ) c¨ Û V̂ and n+Ë ¨ (denoted by n+Ë, for short). he polyhedra ¨ and ¨ are bounded by n + hyperbolic hyperplanes Hi whose intersection behavior is indicated in Figure Ë¨, by associating with Hi the node labeled by i, ¨ ≤ i ≤ n + Ë. he underlying U V H H orthoschemes ¨ and ¨, bounded by the hyperplanes ¨, ... , n, respectively, are such that the vertex opposite to the hyperplane Hn−k is an ideal point. Furthermore, H H the hyperplane n+Ë corresponds to the polar hyperplane (denoted earlier by −Ë), H U and it lies opposite to the ultra-ideal vertex opposite to n. he vertex ûgure of ¨ H A of the vertex opposite to ¨ is of type n−Ë = [q, ... , q], while the vertex ûgure of V H B ¨ of the vertex opposite to ¨ is of type n−Ë = [ , q, ... , q]. hese facts allow us to αn Û deduce the explicit formulas for k . Indeed, for ¨, for example, since the vertex opposite of Hn−k is an ideal point, having a Euclidean vertex neighborhood, the leading n k H H principal minor of order − of the Gram determinant associated with ¨, ... , n+Ë Û must vanish. For example, in the case of ¨, and based on (.q), the requirement δ(− αn − Ë − Ë ) = αn cos k , , ... , ¨ yields the expression for k as mentioned above (see also the proof of Lemma q. ). ¼ ¨ ̂ ¨ ̂ k+Ë Lemma q. Let n ≥ and ≤ k ≤ n − . hen, cn (U ) = cn (V ) = − q Ë +Ë ¨ +Ë ¨ k . ËÇ R.Kellerhals Û V̂ n H H Proof he polyhedra ¨ and ¨ are bounded by + hyperplanes ¨, ... , n+Ë n e e n,Ë in H , characterised by (space-like) normal vectors ¨, ... , n+Ë ∈ E , which are lin- n+Ë G G Û G G V̂ early dependent in R .Hence, their Gram matrices U = ( ¨) and V = ( ¨) m have vanishing determinant. For q ≤ m ≤ n, consider the principal submatrix GU G e e of U formed by the vectors ¨, ... , m. By some well known recursion formulas for the determinant of such matrices (see [q, Sections 8.8 –8.8B], for example), we easily deduce that m Gm A αk A Ë det U = det m − cos n det m−Ë = Ë − , m n − k − Ë l since det Al = (l + Ë)~ . herefore, we obtain Gn k + ¨ ̂ det U Ë cn (U ) = = ≤ +Ë ¨ n−Ë Ë, det GU k which implies the assertion. In a similar vein, one identiûes the determinant of the Gm G e e submatrix V of V formed by the vectors ¨, ... , m with m Gm B αk A Ë det V = det m − cos n det m−Ë = Ë − , m−Ë n − k − Ë l− since det Bl = Ë~ Ë.Hence, we deduce that Gn k + ¨ ̂ det V Ë cn (V ) = = ≤ +Ë ¨ n−Ë Ë, det GV k which ûnishes the proof. ∎

Remark q.þ Let P be a regular simplex Sre g or a regular orthoplex Ore g in Euclidean ̂ n n-space. he ideal hyperbolic k-rectiûed regular polyhedron P = rk P ⊂ H has facets (faces of codimension Ë) of two sorts: (truncated) facets belonging to P and polar facets, that is, facets contained in the polar hyperplanes πv of the ultra-ideal vertices n v of P (k ≥ Ë). Furthermore, rk P has dihedral angles αk formed by intersecting facets π of P, dihedral angles between each polar facet in a πv and the facets of P containing n the pole v, as well as dihedral angles γk attached to the intersection of a polar facet n π πv P not v γ ∈] ] in with a facet of containing . By Lemma q. , the angle k ¨, can be identiûed as follows: ⎪⎧ k = n ⎪¨ ¼ for Ë, (q.Ë) γk = ⎨ ⎪ k+Ë k > ⎩arccos k for Ë. Û V̂ Each of the polyhedra ¨ and ¨ is part of a Napier cycle of type (see Section .). ̂ ̂ n Furthermore, for i ≥ Ë, we can form doubly-truncated orthoschemes Ui = Ui (αk ) ̂ ̂ n n and Vi = Vi (αk ) in H with graphs and additional weights as indicated in Figure ËË and in Figure Ë, respectively, each deûning a Napier cycle of type q. i ̂ i ̂ he additional weights ch(Ui ) and ch(Vi ), where h = n +Ë, n +, ¨, as indicated in Figures ËË and Ë are determined by formula (. ) upon passing from angular weights ω π π αn c = − ω such as q , or k to cos (see (.)). Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups Ë

Figure ËË Û (αn) ≤ i ≤ n : he doubly-truncated orthoschemes i k , Ë .

Figure Ë V̂ (αn) ≤ i ≤ n − : he doubly-truncated orthoschemes i k , Ë Ë.

3.5 Dissecting Ideal Rectified Regular Simplices and Orthoplexes

With the above preparations, we can formulate and prove our ûrst main result as stated in heorem Ë.Ë. Denote by δik ∈ {¨, Ë} the Kronecker-Delta function deûned for elements i, k in an index set I.

heorem q.B Let n ≥ q and Ë ≤ k ≤ n − be integers. For a regular polyhedron P n with Schläi symbol p p , the ideal k-rectiûed regular n-polyhedron ⊂ E { Ë, ... , n−Ë} ̂ n P = rk P ⊂ H admits the following dissections. If P is a simplex S with p p , then r S admits for i n a (i) re g Ë = ⋅ ⋅ ⋅ = n−Ë = q k re g ¨ ≤ ≤ dissection into i n i of δ -truncated orthoschemes isometric to Û αn , !( + Ë − )! ( − ¨i ) i ( k ) and each of these splits into n+Ë simply-truncated orthoschemes isometric to Û (αn), ¼ i ¨ k where αn = n−k . k arccos (n−k−Ë) If P is an orthoplex O with p p and p , then r O (ii) re g Ë = ⋅ ⋅ ⋅ = n− = q n−Ë = k re g admits for i n a dissection into n i n i of δ -truncated orthoschemes ¨ ≤ ≤ −Ë !( − )! (− ¨i ) isometric to V̂ αn , and each of these splits into n simply-truncated orthoschemes i ( k ) i n n isometric to V̂ α , where α √ Ë . ¨( k ) k = arccos n−k−Ë

Proof We follow roughly the strategy of Debrunner’s proof of heorem q.Ë and re- capitulate the most important ingredients. P v v b Ad (i): Suppose that =∶ ⟨ ¨, ... , n⟩ is a Euclidean regular simplex with center n v v P̂ r S n n and with vertices ¨, ... , n so that = k re g ⊂ H . We interpret H in the projective Kn k v i n P Q model . Since ≥ Ë, each vertex i , ¨ ≤ ≤ , of is outside of the quadric n,Ë and therefore pole of its polar hyperplane represented by the hyperbolic hyperplane Ei , say. In order to describe the dissections of P, we adopt Debrunner’s elegant notation as follows.Let [i, k] ∶= {i, i + Ë, ... , k − Ë, k} for integers ¨ ≤ i ≤ k ≤ n. For a set I ⊂ [¨, n], denote by FI the face of P with vertices vi , i ∈ I, and let F∅ ∶= ∅ and F P F b F B [¨,n] ∶= . Each I is a regular simplex with (bary-)center ( I) =∶ I. In particular, i n B B v B b for ¨ ≤ ≤ , one has {i} = [i,i] = i , as well as [¨,n] = n. he Euclidean ¨ R.Kellerhals n U B B P -simplex ¨ ∶= ⟨ [¨,n] ... [n,n]⟩ is a characteristic simplex of and fundamental Sym P S P n domain for the action of ( ) ≅ n+Ë. herefore, dissects into ( + Ë)! isometric U U B b copies of ¨. Furthermore, ¨ is an orthoscheme whose vertex ûgure at [¨,n] = n n A U Kn is a spherical ( − Ë)-orthoscheme of type n−Ë. Among the vertices of ¨ not in , B B l n k n we see that [l ,n], ... , [n,n] are ultra-ideal points precisely for = − + Ë, ... , , B ∂Kn while the vertex [n−k,n] is an ideal point on . Next, consider the Euclidean n-simplices U B B B B i n (q.) i ∶= ⟨ [¨,¨] ... [¨,i−Ë] [i,n] ... [n,n]⟩, ¨ ≤ ≤ . Each simplex Ui arises from the following construction. For ¨ ≤ i ≤ n, consider the partition of [¨, n] by I = [¨, i − Ë] and J = [i, n]. Dissect both, the regular simplex σ (i − Ë)-face FI into its i! characteristic simplices σ(RI) =∶ RI (σ ∈ Si ), and the regular τ simplex (n + Ë − i)-face FJ into its (n + Ë − i)! characteristic simplices τ(RJ) =∶ RI τ S P S F F P i n i ( ∈ n+Ë−i ). Since = re g is the join I ○ J, splits into !( + Ë − )! simplices Rσ Rτ S F F I ○ I , which are permuted by the elements of n+Ë that stabilise I (and J). One of these simplices is Ui for a suitable ordering of the vertices. By heorem q.Ë, we know U n+Ë U that each i is an orthoscheme admitting a dissection into i copies of ¨. ̂ Let us pass to the k-rectiûed polyhedron P associated with P = Sre g. he hyper- E E v v bolic polar hyperplanes ¨, ... , n associated with the ultra-ideal vertices ¨, ... , n induce a (simple or double) truncation of Ui (¨ ≤ i ≤ n) as given by (q.) and its isometric copies in the decomposition of P as described above, making a bridge to the ̂ ̂ n polyhedra Ui = Ui (αk ). Indeed, we will show that ̂ ̂ Ui = P ∩ Ui for ¨ ≤ i ≤ n, which will ûnish the proof of (i). We consider the following two cases. i U B B (a)Let = ¨, and consider the characteristic simplex ¨ = ⟨ [¨,n] ... [n,n]⟩ B v P E E with vertex [n,n] = n of that is simply-truncated by ¨. he hyperplane ¨ meets B U B the ideal vertex [n−k,n] of ¨ at inûnity, and the vertex [¨,n], being the in-center of P S A l n H U = re g, is of type n−Ë. For ¨ ≤ ≤ , denote by n−l the hyperplane bounding ¨, B H E H which is opposite to [l ,n], and write n+Ë ∶= ¨. By Property (iii), n+Ë is orthogonal H H H H π j n to the hyperplanes ¨, ... , n−Ë, while ∡( j, j+Ë) = for Ë ≤ ≤ − Ë. Since B H H αn q [n−k,n] is ideal, we get ∡( ¨, Ë) = k as above. Furthermore, by Lemma q. and H H γn its proof, ∡( n , n+Ë) = k according to (q.Ë). Hence, the truncated orthoscheme U P̂ Û αn ¨ ∩ coincides with the polyhedron ¨( k ) given by Figure Ë¨. Notice that the dihedral angle α formed by two facets of P = Sre g is identical to the dihedral angle formed by the corresponding pair of (truncated) facets of P̂, and αn U this angle is therefore equal to k .Hence, the orthoscheme ¨ can be described by n π (U ) = n(α ) the graph Σ ¨ Σ k , q according to Figure . Furthermore, the orthoschemes Ui (α) arising in the dissection of P = Sre g according to Debrunner’s heorem q.Ë(i) n are given by Ui (αk ) for ¨ ≤ i ≤ n (see FigureÇ ). ̂ n (b)Let i > ¨.It remains to show that the polyhedra Ui ∩ P, where Ui = Ui (αk ), ̂ n coincide with the polyhedra Ui (αk ) as given by Figure ËË. To this end, observe that U i B v B v each of the orthoschemes i , > ¨, shares the vertices [¨,¨] = ¨ and [n,n] = n with P = Sre g (see (q.)). Both vertices of the Ui are truncated by the hyperbolic H E H E P U polar hyperplanes n+Ë = ¨ and n+ ∶= n of . Fix such a i , and denote Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups Ë

⊥ by el , ¨ ≤ l ≤ n + , the unit normal space-like vector with Hl = el directed out- U e e n wards of i .It follows that ¨, ... , n+ (indices modulo + q) form a Napier cycle l n c e e N of type q (see Section .). For Ë ≤ ≤ , the weights l = ⟨ l−Ë, l ⟩n,Ë are given n n π by − cos ωl where ωl ∈ {αk , αk , } denote the angular weights of the orthoscheme U U αn q c c c i = i ( k ) according to FigureÇ . he remaining weights n+Ë, n+, ¨ of the cy- H H cle N are given by the formulas (. ). his proves that the hyperplanes ¨, ... , n+ ̂ ̂ bounding Ui ∩P are described by the cyclic graph of Figure ËË, and that, ûnally, Ui ∩P ̂ n coincides with Ui (αk ).

Ad (ii): he proof is similar to (i). Let us describe the dissection procedure of an ideal hyperbolic k-rectiûed regular orthoplex rk Ore g and the respective appearance of the ̂ n polyhedra Vi (αk ) as claimed. Consider a Euclidean regular n-orthoplex Ore g, and denote by vi , v−i the n pairs of vertices of Ore g such that the segments ⟨viv−i ⟩, Ë ≤ i ≤ n, meet orthogonally in their common midpoint (and in-center) z. he n symmetry hyperplanes generated by all vertices except one pair induce a dissection of the poly- O n S zv v v v hedron re g into simplices, all isometric to ∶= ⟨ Ë ... n⟩. he facet ⟨ Ë ... n⟩ O S n v v v v of re g (and of ) is a regular ( − Ë)-simplex whose faces ⟨ Ë ... i ⟩ and ⟨ i+Ë ... n⟩ can be dissected barycentrically into i! and (n − i)! characteristic orthoschemes R ′ ′ and R , respectively. Form the join Vi ∶= R ○ ⟨z⟩ ○ R for Ë ≤ i ≤ n, and denote by BI the barycenter of the face FI with vertices vi , i ∈ I, of Ore g, as above. In n this picture, for i = ¨, the polyhedron Ore g is cut into its n! characteristic or- V zB B S i n thoschemes all isometric to ¨ = ⟨ [Ë,n] ... [n,n]⟩ ⊂ , and for Ë ≤ ≤ , the simplex V B B zB B n i n i i = ⟨ [Ë,n] ... [Ë,i] [i+Ë,n] ... [n,n]⟩ is one of the !( − )! pairwise isometric simplices dissecting Ore g. Again, it can be shown that each such simplex is an orthoscheme. For details, see [ , pp. Ëþ¨–ËþË]. As in the proof of (i), one easily veriûes ̂ n that the truncated orthoschemes Vi (αk ) = rk Ore g ∩ Vi provide the decomposition of rk Ore g as claimed. ∎

As a ûrst application of heorem q.B, let us consider the ideal birectiûed regular r O orthoplex re g in hyperbolic 8-space. We can prove the following result.

Corollary q.8 he volume of the ideal birectiûed regular orthoplex r O 8 is re g ⊂ H given by √ r O Ëþq L vol8( re g) = q ( , −q) ≃ 8.q Ë, ËB where L s D D r−s is the Dirichlet L-series deûned by the Kronecker symbol ( , ) = ∑r≥Ë( r ) D . ( r )

r O Proof By Remark q.þ, the ideal birectiûed regular orthoplex re g has dihedral π π π α8 = γ8 = r Ore g angles q , q and , and part (ii) of the above theorem implies that can be cut into 88! = B þ, Ë¨ polyhedra isometric to the (truncated) characteristic V̂ þ simplex ¨ = [q, , q , B] with Coxeter graph given as follows. R.Kellerhals

In [8, Section .], we showed that the (arithmetic) hyperbolic Coxeter group Γ¨ with V̂ graph Σ( ¨) is commensurable to the Coxeter group ΓË with graph

by identifying ΓË as a subgroup of index q in the group Γ¨. he group ΓË itself is the f 8 (maximal) reection subgroup of the group of units of the quadratic form q whose covolume has been determined by Ratcliòe and Tschantz (see Remark .8). More precisely, according to [ , Table Ë], one has that √ þË q L −þ covolþ(ΓË) = ( , −q) ≃ 8. ¨q ⋅ Ë¨ . Ëþ ⋅ þ ⋅ 8 √ 8 √ (r Ore g) = 8! ⋅ þË q L( − ) = Ëþq L( − ) ∎ As a consequence, vol8 q Ëþ⋅þ⋅8 , q ËB q , q as asserted. 4 Quaternionic (pseudo-)modular Groups and their Covolumes

In the sequel, we interpret the orientation-preserving isometries of hyperbolic - and þ-spaces by means of certain quaternionic × matrices. In this way, and following Johnson [ËË], we can relate the quaternionic (pseudo)-modular groups to certain arithmetic hyperbolic Coxeter groups. By combining our previous results and applying them to the ideal birectiûed B-cell, we will be able to prove our second main result about the covolume of the hybrid modular group as given by heorem .Ë.

4.1 Quaternionic × Matrices and Hyperbolic Isometries

n n−Ë Consider hyperbolic n-space in the upper half space U = E × R+. In this model, + n the group Isom (U ) of orientation preserving or direct hyperbolic isometries is iso- + n n morphic to the group M (U ) of direct Möbius transformations of E , which leave n U invariant. By Poincaré extension, the latter group is isomorphic to the group of ̂n−Ë n−Ë direct Möbius transformations of the extended ground space E = E ∪ {∞}. As + + n+ in the classical case of Isom (U ) ≅ PSL(, R), the group Isom (U Ë), n ≥ Ë, can be identiûed with a projective group of × matrices according to Vahlen, Maass, n and Ahlfors (see [Ë, ]). To this end, interpret the real vector space R as the set of Cliòord vectors x x x i x i C Cliòord algebra C = ¨ + Ë Ë + ⋅ ⋅ ⋅ + n−Ë n−Ë ∈ n of the n, which n i i is the associative real algebra generated by − Ë elements Ë, ... , n−Ë subject to the relations ik il = −il ik (k =~ l) and ik = −Ë. A typical element a ∈ Cn can written a = aI I aI ∈ I ik ⋅ ⋅ ⋅ ik in the form ∑I , R, where runs through all products Ë r with ≤ k < ⋅ ⋅ ⋅ < k < n i = i ∶= ¨ Ë r , where we include the empty product in the form k¨ ¨ Ë. S a a scalar part a C We call ( ) ∶= ¨ the of . Accordingly, n is a real vector space of di- n−Ë mension that can be equipped with a Euclidean norm deûned by SaS = ∑I aI . In C C C particular, Ë = R, = C and q = H. ∗ On Cn, there are three important involutions. he mapping a ↦ a is deûned ∗ ′ I = ik ⋅ ⋅ ⋅ ik I ∶= ik ⋅ ⋅ ⋅ ik a ↦ a by sending each Ë r to r Ë , while is given by replacing ′∗ each factor ik by −ik . he conjugation a ↦ a is the composition a ∶= a .Obviously, ∗ Cliòord vectors x ∈ Cn satisfy x = x , and a non-zero Cliòord vector x is invertible Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups q

with inverse x−Ë = x′~SxS. he non-zero Cliòord vectors form a multiplicative group, which is termed the Cliòord group Gn. A Cliòord matrix is an element of the set

SL(, Cn) ∶= a b T a b c d G ab∗ cd∗ c∗a d∗b n ad∗ bc∗ = c d U , , , ∈ n ∪ {¨}; , , , ∈ R ; − = Ë¡.

he quantity ad∗ − bc∗ is called the pseudo-determinant of T and is such that the set SL(, Cn) becomes a group under matrix multiplication (see [Ë, p. Ë]). Its associated projective group PSL C SL C λI λ ∗ (, n) = (, n)~{ S ∈ R } ̂n −Ë acts bijectively on E by fractional linear transformations T(x) = (ax +b)(cx +d) , −Ë −Ë ̂n+Ë T(¨) = bd , T(∞) = ac , and this action appropriately extended to E preserves n+ + n+ U Ë. As a consequence, the group PSL(, Cn) is isomorphic to Isom (U Ë). + In particular, in the quaternionic case, we have that PSL(, H) ≅ Isom (U ) (for some geometric properties of its discrete subgroups, see [Ë]). here is another matrix group over the quaternions closely related to hyperbolic isometries. Following Wilker [Ç], consider a × matrix M with coeõcients in the quaternion algebra H given by a b M a b c d = c d , , , , ∈ H.

Since quaternions can be interpreted by means of complex × matrices, M can be identiûed with a block matrix M ∈ Mat( , C) whose (ordinary) determinant is a non- negative real number that can be written according to − det M = SadS + SbcS − S(acdb) = Sad − aca ËbS for a =~ ¨. Based on this, the Dieudonné determinant of M is deûned by ¼ −Ë ∆ = ∆(M) ∶=+ SadS + SbcS − S(acdb) = Sad − aca bS, and ∆ satisûes all required properties of a determinant function. Moreover, the set a b S L( ) = T = ∈ ( )S (T) = ¡ ∆ , H c d Mat , H ∆ Ë , ̂ is a group whose elements act on H by fractional linear transformations. Finally, it can be shown that the projective analogue PS L S L I ∆ (, H) = ∆ (, H)~{± } + is isomorphic to the group Isom (U þ) (for some geometric properties of its discrete subgroups, see [¨]).

4.2 Basic Systems of Quaternionic Integers q q q i q j Consider the normed real associative algebra H of quaternions = ¨ + Ë + + q k i i j i k i j N q T q q ∈ H where = Ë, = , and = , as usual. he norm ( ) and the trace ( ) of q are given by N(q) = q + q + q + q T(q) = S(q) = q ¨ Ë q , ¨, R.Kellerhals

which appear naturally in the quadratic equation ( .Ë) q − T(q)q + N(q) = ¨. Quaternions of norm Ë are called units and form a group that is isomorphic to the special unitary group SU(). According to Johnson–Weiss [Ëq], [ËË,Chapter Ëþ], a basic system of elements in H is a set S such that (a) each element in S is a (quadratic) algebraic integer; (b) S is a subring of H; the elements of norm Ë in S form a subgroup of the group of unit quaternions; (c) in the real vector space H, the elements of S are the points of a four-dimensional lattice spanned by the units. q S(q) = q ∈ ∪ Ë By means of ( .Ë), condition (a) holds for a quaternion if ¨ Z Z and N(q) ∈ Z. By [Ëq, heorem .Ë], there are precisely three basic systems of integral quaternions which can be described—briey—as follows. he ûrst basic system is the ring H am = Z[i, j] whose four units Ë, i, j, and k = i j span the lattice of Hamilton integers. he ring H am can be regarded as a quaternionic analogue of the ring of Gaussian integers Z[i]. he second basic system is the ring Hur= Z[u, v] of Hurwitz integers where the u v u = Ë − Ë i − Ë j + Ë k v = Ë + Ë i − Ë j + Ë k quaternions , are deûned by and . One veriûes that each Hurwitz integer is an integral combination of Ë, u, v, w where u, v, w satisfy the relations u − u = v − v = w − w = uvw = Ë. he ring H ur has units consisting of the Ç Hamilton units ±Ë, ±i, ±j, ±k together ± Ë ± Ë i ± Ë j ± Ë k with the ËB units of the form . he ring H ur contains H am as a√sub- [ω] ω = − Ë + Ë i ring and can be viewed as a quaternionic analogue of the ring Z , q , of Eisenstein integers since (uv)−Ë = ω. ω j hybrid integers ω he√third basic system is the ring H yb = Z[ , ] of where = − Ë + Ë i j q and satisfy the relations ω + ω = j = (ω j) = (ω j) = −Ë. here are Ë hybrid units, given by Ë, ω, ω, j, ω j, ω j and their negatives.

4.3 Covolumes of Some Quaternionic (Pseudo-)modular Groups

+ Consider the group PSL(, H) ≅ Isom (U ) represented by quaternionic Clif- ford matrices and restrict the coeõcient ring from H to one of the basic systems S of quadratic integers in H as described in Section .. Each of the three groups PSL(, S) is a particularly nice arithmetic discrete group of direct isometries acting on hyperbolic -space with ûnite covolume. In [Ëq] (see also [ËË, Section Ëþ.]), John- son and Weiss studied various properties of PSL(, S), which they call a quaternionic pseudo-modular group PS∗L (and denote it there by (S) referring to the underlying unit pseudo-determinant of Ahlfors). hey show that each group PSL(, S) can be identiûed with a ûnite index subgroup of a certain hyperbolic Coxeter group, and the identiûcation is made explicit in terms of suitable generators. Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups þ

In this way, the Hamilton pseudo-modular group PSL(, H am) = PSL(, Z[i, j]) turns out to be isomorphic to an index Ë subgroup of the hyperbolic Coxeter simplex group [q, , q, ] (see [Ëq, p. Ë8q]). By the Reduction Formula (.þ) of Section . , the volume of the Coxeter orthoscheme [q, , q, ] can be computed to be π~ÇB so that the following result holds:

π PSL covol ( (, H am)) = . 8 We refer to [Ë] for further and more algebraic, arithmetic, and geometric details about the group PSL(, Z[i, j]). he Hurwitz pseudo-modular group PSL(, H ur) = PSL(, Z[u, v]) can be iden- tiûed with a semidirect product of PSL(, H am) with a cyclic group of order q generated by an element transforming Hamilton integers into Hurwitz integers (see [Ë, p. 8þË] and [Ëq, p. Ë8 ]). As a consequence, one obtains that

π PSL covol ( (, H ur)) = . he hybrid pseudo-modular group PSL(, H yb) = PSL(, Z[ω, j]) is isomorphic to a certain index subgroup of the Coxeter pyramid group [B, q, q, q, ∞] of covolume π~þ ¨ (see Section . ), which in turn relates to the symmetry group of the ideal π r Sre g( ) rectiûed þ-cell Ë q (see Example .). It follows that π PSL covol ( (, H yb)) = . Ëqþ modular groups PS L + U þ Let us pass to the case of higher in ∆ (, H) ≅ Isom ( ). In PS L PS L particular, the modular groups ∆ (, H am) and ∆ (, H ur) are arithmetic discrete groups of ûnite covolume that are intimately related to one another. In fact, by work of Johnson and Weiss [Ëq, Section 8] (see also [ËË, Section Ëþ.q]), it is known that PS L the Hamilton modular group ∆ (, H am) is isomorphic to a certain subgroup of index Ë in the Coxeter simplex group [q, , q, q, ]. Furthermore, the Hurwitz mod- PS L ular group ∆ (, H ur) is closely related to the Coxeter simplex group [q, , q, q, q] PS L and contains the group ∆ (, H am) as a subgroup of index q¨ (see [Ëq, Section ] and [ËË, Section Ëþ. ]). By means of our volume expression (.Ë¨) for [q, , q, q, ], we are able to derive the following results: PS L 8 ζ ( .) covolþ ∆ (, H am) = (q), qÇ PS L 8 ζ covolþ ∆ (, H ur) = (q). ËË, þ¨

4.4 The Ideal Birectified 6-cell and the Covolume of the Hybrid Modular Group PS L ∆ (, H yb) PS L( ) ⊂ Much less has been known about the hybrid modul√ar group ∆ , H yb + Isom (U þ) with coeõcient ring Z[ω, j], ω = Ë (−Ë + i q). Recently, in [Ë¨] and S L [ËË, Section Ëþ.þ], Johnson analysed the group ∆ (, H yb) in detail and determined B R.Kellerhals the generators as follows: ¨ Ë Ë ¨ Ë ¨ Ë ¨ A = , B = , M = , N = . −Ë ¨ ËË ¨ ω ¨ j Note that A = −I and that Mq = N = I. his analysis enabled him to relate the group PS L ∆ (, H yb) to the hyperbolic Coxeter prism group [B, q, q, q, q, B]. Based on this work, we are able to prove the following result (see heoremË. ). heorem .Ë For the hybrid modular group PS L PS L ω j , ∆ (, H yb) = ∆ (, Z[ , ]) PS L Ëq ζ covolþ( ∆ (, H yb)) = (q). ËÇ¨ PS L PS L ω j Proof By [ËË, Section Ëþ.þ], the group ∆ (, H yb) = ∆ (, Z[ , ]) is isomorphic to the semidirect product of a certain commutator subgroup of index Ç in [B, q, q, q, q, B] and the cyclic group of order generated by the N S L ω j PS L element ∈ ∆ (, Z[ , ]) above. his implies that covolþ( ∆ (, H yb)) = q ⋅ ([ ]) ([ ]) = Ëq ζ( ) covolþ B, q, q, q, q, B .It remains to show that covolþ B, q, q, q, q, B þ,8B¨ q as already announced in Remark .B. he Coxeter polyhedron [B, q, q, q, q, B] is associated with the ideal birectiûed reg- π π r Sre g ⊂ þ ular B-cell H of dihedral angles q and . In fact, it is the truncated char- Û π acteristic orthoscheme ¨ of the regular B-cell of dihedral angle q with ultra-ideal vertices all of whose triangles are replaced by an ideal point (see Section q.). Con- Û Û sider the truncated orthoschemes ¨ and i with graphs and weights given according to Figures Ë¨ and ËË( see also Lemma q. and formula (. )). In fact, the distinguished Û αþ = π = γþ weights for ¨ are B in view of (q.Ë). By heorem q.B(i), the polyhe- r S Û dron re g admits a dissection into B! simply-truncated orthoschemes ¨ isometric to [B, q, q, q, q, B] and, by taking i = q, a dissection into (q!) doubly-truncated or- Û Û B Û thoschemes q. Each of the polyhedra q can be dissected into q copies of ¨. hese dissection relations provide the volume identity Û Ë Û volþ([B, q, q, q, q, B]) = volþ( ¨) = volþ( q). ¨ Û Finally, it remains to identify the polyhedron q, as given by the graph in Figure ËË i = [( )[ ]] ( π ) for q, with the Coxeter polyhedron q, B of the cyclic graph Ω q whose Ëq ζ( ) volume is given by ÇÇ q according to Proposition .Ça nd (.Ëq). his can be done in two ways. Û [( cq cq cq)] he cyclic graph of q has symbol q, B, q, B, q, þ , B, ¨ where the squares of q the ingredients cl , l = þ, B, ¨, are computable by formula (. ). Without computation, cq cq cq the values þ , B, ¨ can be determined directly by using the Napier cycle property (see Section .) that the deletion of two non-adjacent nodes among ¨, ... , 8, representing vectors of the Napier cycle N, deûnes two Lorentz-orthogonal subspaces of Eþ,Ë. cq = cq = − π Hence, the deletion of the nodes , B and q, 8, respectively, yields ¨ þ cos B , cq = − π while the deletion of the pair ¨, shows that þ cos q . In fact, all corresponding subgraphs (aer deletion) are products of Euclidean type [q, B] × [q, B]. As a conse- Û [ ] quence, the polyhedron q is isometric to [(q, B) ], and the assertion follows. ∎ Ideal Uniform Polyhedra and Covolumes of Higher Dimensional Modular Groups 8

Remark . In [8], it was shown that the (arithmetic) Coxeter groups with Coxeter symbols [q, , q, q, ] and [B, q, q, q, q, B] are incommensurable. Although the quo- tient of their covolumes is a rational number by ( .) and heorem .Ë, there is no (orientable) hyperbolic þ-manifold covering both, the modular Hamilton (or Hurwitz) þ PS L þ PS L orbifold H ~ ∆ (, H am) and the modular hybrid orbifold H ~ ∆ (, H yb).

Acknowledgments he author would like to thank Matthieu Jacquemet for helpful discussions. he author was partially supported by Schweizerischer Nationalfonds ¨¨¨Ë–Ë8þÇq. References

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