Integral binary Hamiltonian forms and their waterworlds
IIntegral binary Hamiltonian forms and their waterworlds
Jouni Parkkonen Frédéric PaulinDecember 23, 2019
Abstract
We give a graphical theory of integral indefinite binary Hamiltonian forms f analo-gous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin forbinary Hermitian forms. Given a maximal order O in a definite quaternion algebra over Q , we define the waterworld of f , analogous to Conway’s river and Bestvina-Savin’s ocean , and use it to give a combinatorial description of the values of f on O × O .We use an appropriate normalisation of Busemann distances to the cusps (with analgebraic description given in an independent appendix), and the SL ( O ) -equivariantFord-Voronoi cellulation of the real hyperbolic -space. In the beautiful little book [Con] (see also [Wei, Hat]), Conway uses Serre’s tree X Z of themodular lattice SL ( Z ) in SL ( R ) (see [Ser2]), considered as an equivariant deformationretract of the upper halfplane model of the hyperbolic plane H R , in order to give a graphicaltheory of binary quadratic forms f . The components C of H R − X Z consist of points closerto a given cusp p/q ∈ P ( Q ) of SL ( Z ) than to all the other ones. When f is indefinite,anisotropic and integral over Z , Conway constructs a line R ( f ) in X Z , called the river of f , separating the components C of H R − X Z such that f ( p, q ) > from the ones with f ( p, q ) < . This allows a combinatorial description of the values taken by f on integralpoints.Bestvina and Savin in [BeS] have given an analogous construction when R is replacedby C , Z by the ring of integers O K of a quadratic imaginary extension K of Q , H R by H R and X Z by Mendoza’s spine X O K in H R for the Bianchi lattice SL ( O K ) in SL ( C ) (see [Men]). They construct a subcomplex O ( f ) of X O K , called the ocean of f , for anyindefinite anisotropic integral binary Hermitian form f over O K , separating the componentsof H R − X O K on whose point at infinity f is positive from the negative ones, and provethat it is homeomorphic to a -plane.In this paper, we give analogs of these constructions and results for Hamilton’s quater-nions and maximal orders in definite quaternion algebras over Q .Let H be the standard Hamilton quaternion algebra over R , with conjugation x (cid:55)→ x ,reduced norm n and reduced trace tr . Let O be a maximal order in a quaternion algebra A over Q , which is definite (that is, A ⊗ Q R = H ), with class number h A and discriminant D A . An example is given by the Hurwitz order O = Z + Z i + Z j + Z i + j + k , in which Keywords: binary Hamiltonian form, rational quaternion algebra, maximal order, Hamilton-Bianchigroup, reduction theory, waterworld, hyperbolic -space. AMS codes: a r X i v : . [ m a t h . N T ] D ec ase h A = 1 and D A = 2 . We refer for more information to [Vig] and Subsection 2.1.The Hamilton-Bianchi group SL ( O ) , which is defined using Dieudonné determinant, is alattice in SL ( H ) . It acts discretely on the real hyperbolic -space H R with finite volumequotient. The number of cusps of the hyperbolic orbifold SL ( O ) \ H R is h A by [KO, Satz2.1, 2.2], see also [PP2, §3].Analogously to [Men] in the complex case, we give in Section 3 an appropriate nor-malisation of the Busemann distance to the cusps, and we construct the Ford-Voronoi celldecomposition of H R for SL ( O ) , so that the interior of the Ford-Voronoi cell H α consistsof the points in H R closer to a given cusp α ∈ P r ( A ) of SL ( O ) than to all the others.If X O is the codimension skeleton of the Ford-Voronoi cellulation, called the spine of SL ( O ) , then the hyperbolic -orbifold SL ( O ) \ H R retracts by strong deformations ontothe finite -dimensional orbihedron SL ( O ) \ X O .Using uniform -, - and -polytopes, we give in Example 4.4 when D A = 2 and inExample 4.5 when D A = 3 , a complete description of the quotient SL ( O ) \ X O and of thelink of its vertex. For instance, if O is the Hurwitz order, then SL ( O ) \ X O is obtained byidentifying opposite faces and taking the quotient of any -dimensional cell of X O by itsstabilizer. In this case, a -dimensional cell of X O identifies with the -cell (the self-dualconvex regular Euclidean -polytope with Schläfli symbol { , , } ), and its stabilizer isisomorphic with an index subgroup of the Coxeter group [3 , , .Following H. Weyl [Wey], we will call Hamiltonian form a Hermitian form over H withanti-involution the conjugation. We refer to Subsection 2.3 and for instance to [PP2] forbackground. See [PP2] also for a sharp asymptotic result on the average number of theirintegral representations. Let f : H × H → R be a binary Hamiltonian form, with f ( u, v ) = a n ( u ) + tr ( u b v ) + c n ( v ) , which is integral over O (its coefficients a, b, c satisfy a, c ∈ Z and b ∈ O ) and indefinite(its discriminant ∆( f ) = n ( b ) − ac is positive). We choose this definition of integrality forsimplicity as in [PP2], in order to avoid half-integral coefficient in the matrix of the form.The group of automorphs of f is the arithmetic lattice SU f ( O ) = { g ∈ SL ( O ) : f ◦ g = f } . If C is a Ford-Voronoi cell for SL ( O ) , let F ( C ) = f ( a,b ) n ( O a + O b ) where ab − ∈ P r ( A ) is thecusp of C . We will say that C is respectively positive, negative or flooded if F ( C ) > , F ( C ) < or F ( C ) = 0 . Contrarily to the real and complex cases, there are always floodedFord-Voronoi cells, since by taking a Z -basis of O , the Hamiltonian form f becomes anintegral binary quadratic form over Z with ≥ variables, hence always represents .Our countably many flooded Ford-Voronoi cells are thus the analogues of Conway’s two lakes for an indefinite isotropic integral binary quadratic form over Z . On the componentsof H R − X Z along the lakes, Conway proved that the values of such a form consist in aninfinite arithmetic progression. An analogous result holds in our case, that we only statewhen the class number is one in this introduction in order to simplify the statement (seeProposition 5.3 for the general result.) Proposition 1.1. If h A = 1 , given a flooded Ford-Voronoi cell C , there exists a finite setof nonconstant affine maps { ϕ i : H → R : i ∈ F } defined over Q such that the set ofvalues of F on the Ford-Voronoi cells meeting C is (cid:83) i ∈ F ϕ i ( O ) . waterworld W ( f ) of f as the subcomplex of the spine separating positive Ford-Voronoi cells from negative ones,that is, W ( f ) is the union of the cells of X O contained in (the boundary of) both a positiveand a negative Ford-Voronoi cell. The coned-off waterworld C W ( f ) is the union of W ( f ) and, for all cells σ of W ( f ) contained in a flooded Ford-Voronoi cell H α , of the cone withbase σ and vertex at infinity α . The following result (see Section 5) in particular saysthat C W ( f ) is a piecewise hyperbolic polyhedral -plane contained in the spine of SL ( O ) except for its ideal cells.Let C ( f ) be the hyperbolic hyperplane of H R whose boundary is the projective set ofzeros { [ u : v ] ∈ P r ( H ) : f ( u, v ) = 0 } of f . Theorem 1.2.
The closest point mapping from the coned-off waterworld
C W ( f ) to C ( f ) is an SU f ( O ) -equivariant homeomorphism. Section 2 recalls the necessary information on the definite quaternion algebras over Q , the Hamilton-Bianchi groups, and the binary Hamiltonian forms. Section 3 gives theconstruction of the normalized Busemann distance to the cusp, and uses it to give aquantitative reduction theory à la Hermite (see for instance [Bor2]) for the arithmeticgroup SL ( O ) . We describe the Ford-Voronoi cellulation for SL ( O ) and its spine X O inSection 4. We define the waterworlds and prove their main properties in Section 5. Thenoncommutativity of H and the isotropic property of f require at various point of this texta different approach than the one in [BeS].Recall (see for instance [PP2, §7] and Section 3) that there is a correspondence betweenpositive definite binary Hamiltonian forms with discriminant − and the upper halfspacemodel of the real hyperbolic -space. In the independent Appendix A, we give an algebraicformula for the Busemann distance of a point x ∈ H R to a cusp α ∈ P r ( A ) in terms ofthe covolume of the O -flag associated with α , with respect to the volume of the positivedefinite binary Hamiltonian form associated with x , analogous to the one of Mendoza inthe complex case. Furthermore, in the proof of Theorem 3.5, we use the upper bound onthe minima of positive definite binary Hamiltonian forms given in [ChP]: If γ ( O ) is theupper bound, on all such forms f with discriminant − , of the lower bound of f ( u, v ) onall nonzero ( u, v ) ∈ O × O , then γ ( O ) ≤ (cid:112) D A . (1) Acknowledgements:
This work was supported by the French-Finnish CNRS grant PICS № 6950.The second author greatly acknowledges the financial support of Warwick University for a onemonth stay, decisive for the writing of this paper. We warmly thank the referee of a previousversion of this paper for a long list of useful comments which have improved its content andpresentation.
We refer to [PP2] for more informations on the objects considered in this paper, and weonly recall what is strictly needed.3 23/12/2019 .1 Background on definite quaternion algebras over Q A quaternion algebra over a field F is a four-dimensional central simple algebra over F .We refer to [Vig] for generalities on quaternion algebras. A real quaternion algebra isisomorphic either to M ( R ) or to Hamilton’s quaternion algebra H over R , with basiselements , i, j, k as a R -vector space, with unit element and i = j = − , ij = − ji = k .We define the conjugate of x = x + x i + x j + x k in H by x = x − x i − x j − x k ,its reduced trace by tr ( x ) = x + x , and its reduced norm by n ( x ) = x x = x x . Note that n ( xy ) = n ( x ) n ( y ) , tr ( x ) = tr ( x ) and tr ( xy ) = tr ( yx ) for all x, y ∈ H . For every matrix X = ( x i,j ) ≤ i ≤ p, ≤ j ≤ q ∈ M p,q ( H ) , we denote by X ∗ = ( x j,i ) ≤ i ≤ q, ≤ j ≤ p ∈ M q,p ( H ) itsadjoint matrix. We endow H with the Euclidean norm x (cid:55)→ (cid:112) n ( x ) , making the R -basis , i, j, k orthonormal.Let A be a quaternion algebra over Q . We say that A is definite (or ramified over R )if the real quaternion algebra A ⊗ Q R is isomorphic to H , and we then fix an identificationbetween A and a Q -subalgebra of H . The reduced discriminant D A of A is the product ofthe primes p ∈ N such that the quaternion algebra A ⊗ Q Q p over Q p is a division algebra.Two definite quaternion algebras over Q are isomorphic if and only if they have the samereduced discriminant, which can be any product of an odd number of primes (see [Vig,page 74]).A Z -lattice I in A is a finitely generated Z -module generating A as a Q -vector space. An order in A is a unitary subring O of A which is a Z -lattice. In particular, A = Q O = O Q .Each order of A is contained in a maximal order. For instance O = Z + Z i + Z j + Z i + j + k is a maximal order, called the Hurwitz order , in A = Q + Q i + Q j + Q k with D A = 2 . Let O be an order in A . The reduced norm n and the reduced trace tr take integral values on O . The invertible elements of O are its elements of reduced norm . Since x = tr ( x ) − x ,any order is invariant under conjugation.The left order O (cid:96) ( I ) of a Z -lattice I is { x ∈ A : xI ⊂ I } . A left fractional ideal of O is a Z -lattice of A whose left order is O . A left ideal of O is a left fractional ideal of O contained in O . A (left) ideal class of O is an equivalence class of nonzero left fractionalideals of O for the equivalence relation m ∼ m (cid:48) if m (cid:48) = m c for some c ∈ A × . The classnumber h A of A is the number of ideal classes of a maximal order O of A . It is finite andindependent of the maximal order O , and we have h A = 1 if and only if D A = 2 , , , , (see for instance [Vig]).The reduced norm n ( m ) of a nonzero left ideal m of O is the greatest common divisorof the norms of the nonzero elements of m . In particular, n ( O ) = 1 . By [Rei, p. 59], wehave n ( m ) = [ O : m ] . (2)The reduced norm of a nonzero left fractional ideal m of O is n ( c m ) n ( c ) for any c ∈ N − { } such that c m ⊂ O . By Equation (2), if m , m (cid:48) are nonzero left fractional ideals of O with m (cid:48) ⊂ m , we have n ( m (cid:48) ) n ( m ) = [ m : m (cid:48) ] . (3)For K = H or K = A , we consider K × K as a right module over K and we denoteby P r ( K ) = ( K × K − { } ) /K × the right projective line of K , identified as usual with theAlexandrov compactification K ∪ {∞} where [1 : 0] = ∞ and [ x : y ] = xy − if y (cid:54) = 0 .4 23/12/2019 .2 Background on Hamilton-Bianchi groups Refering to [Fue, Die, Asl], the
Dieudonné determinant
Det is the group morphism fromthe group GL ( H ) of invertible × matrices with coefficients in H to R ∗ + , defined by (cid:0) Det (cid:16) a bc d (cid:17)(cid:1) = n ( a d ) + n ( b c ) − tr ( a c d b ) . If c (cid:54) = 0 , we have (see loc. cit.) (cid:0) Det (cid:16) a bc d (cid:17)(cid:1) = n ( ac − dc − bc ) . (4)It is invariant under the adjoint map g (cid:55)→ g ∗ . Let SL ( H ) be the group of × matriceswith coefficients in H and Dieudonné determinant . We refer for instance to [Kel] formore information on SL ( H ) .The group SL ( H ) acts linearly on the left on the right H -module H × H . The projec-tive action of SL ( H ) on P r ( H ) , induced by its linear action on H × H , is the action byhomographies on H ∪ {∞} defined by (cid:16) a bc d (cid:17) · z = ( az + b )( cz + d ) − if z (cid:54) = ∞ , − c − dac − if z = ∞ , c (cid:54) = 0 ∞ otherwise . We use the upper halfspace model { ( z, r ) : z ∈ H , r > } with Riemannian metric ds ( z, r ) = ds H ( z )+ dr r for the real hyperbolic space H R with dimension . Its space atinfinity ∂ ∞ H R is hence H ∪ {∞} . The action of SL ( H ) by homographies on ∂ ∞ H R extendsto a left action on H R by (cid:16) a bc d (cid:17) · ( z, r ) = (cid:16) ( az + b ) ( cz + d ) + a c r n ( cz + d ) + r n ( c ) , r n ( cz + d ) + r n ( c ) (cid:17) . (5)In this way, the group PSL ( H ) is identified with the group of orientation preservingisometries of H R .For any order O in a definite quaternion algebra A over Q , the Hamilton-Bianchi group Γ O = SL ( O ) = SL ( H ) ∩ M ( O ) is a nonuniform arithmetic lattice in the connected real Lie group SL ( H ) (see for instance[PP1, page 1104] for details). In particular, the quotient real hyperbolic orbifold Γ O \ H R has finite volume. Remark.
It would be very interesting to know if the image in
PSL ( H ) of SL ( O ) iscommensurable (up to conjugation) to one of the lattices in SO (1 , (cid:39) PSL ( H ) studiedby Vinberg [Vin], Allcock [ ? ], Everitt [Eve], Ratcliffe-Tschantz [RaS] and others.Recall that the maximal order O is left-Euclidean if for all a, b ∈ O with b (cid:54) = 0 , thereexists c, d ∈ O with a = cb + d and n ( d ) < n ( b ) , or, equivalently, if for every α ∈ A , thereexists c ∈ O such that n ( α − c ) < . By for instance [Vig, p. 156], O is left-Euclidean ifand only if D A ∈ { , , } . The following elementary lemma gives a nice set of generatorsfor SL ( O ) . For us, it will be useful in Section 4. See also [Spe, §4] and [JW, §8] for thefirst claim for the Hurwitz order.5 23/12/2019 emma 2.1. If O is left-Euclidean, then the group SL ( O ) is generated by J = (cid:18) (cid:19) , T w = (cid:18) w (cid:19) for w ∈ O and C u,v = (cid:18) u v (cid:19) for u, v ∈ O × . In particular, the anti-homography z (cid:55)→ z normalizes the action by homographies of SL ( O ) on H . Proof.
The last claim follows from the first one, since J − = J , T − w = T − w , C − u,v = C u − ,v − and for all z ∈ H , we have J · z = J · z, T w · z = T w · z, C u,v · z = C v − ,u − · z . Let G be the subgroup of SL ( O ) generated by the matrices J, T w , C u, v for w ∈ O and u, v ∈ O × (their Dieudonné determinant is indeed ). Let us prove that any M = (cid:18) a bc d (cid:19) ∈ SL ( O ) belongs to G , by induction on the integer n ( c ) . If c = 0 , then M = C a, d T a − b belongs to G . Otherwise, since O is left-Euclidean, there exists w, c (cid:48) ∈ O such that a = wc + c (cid:48) and n ( c (cid:48) ) < n ( c ) . Hence M = (cid:18) w (cid:19) (cid:18) (cid:19) (cid:18) c dc (cid:48) b − w d (cid:19) belongs to G by induction. (cid:3) Corollary 2.2. If O is left-Euclidean, if { w , w , w , w } is a Z -basis of O and if S is agenerating set of the group of units O × , then the set { J, T w , T w , T w , T w } ∪ { C u,v : u, v ∈ S } is a generating set for SL ( O ) . (cid:3) The action by homographies of the group Γ O = SL ( O ) preserves the right projectivespace P r ( A ) = A ∪ {∞} , which is the set of fixed points of the parabolic elements of Γ O acting on H R ∪ ∂ ∞ H R . In particular, the topological quotient space Γ O \ ( H R ∪ P r ( A )) isthe compactification of the finite volume hyperbolic orbifold Γ O \ H R by its (finite) spaceof ends. A binary Hamiltonian form f is a map H × H → R with f ( u, v ) = a n ( u ) + tr ( u b v ) + c n ( v ) whose coefficients a = a ( f ) , b = b ( f ) and c = c ( f ) satisfy a, c ∈ R , b ∈ H . Note that f (( u, v ) λ ) = n ( λ ) f ( u, v ) for all u, v, λ ∈ H .The matrix M ( f ) of f is the Hermitian matrix (cid:16) a bb c (cid:17) , so that f ( u, v ) = (cid:16) uv (cid:17) ∗ (cid:16) a bb c (cid:17) (cid:16) uv (cid:17) . discriminant of f is ∆( f ) = n ( b ) − ac. An easy computation shows that the Dieudonné determinant of M ( f ) is equal to | ∆( f ) | .A binary Hamiltonian form is indefinite if takes both positive and negative values. It iseasy to check that a form f is indefinite if and only if ∆( f ) is positive, see [PP2, §4].The linear action on the left on H × H of the group SL ( H ) induces an action on theright on the set of binary Hamiltonian forms f by precomposition. The matrix of f ◦ g is M ( f ◦ g ) = g ∗ M ( f ) g . For every g ∈ SL ( H ) , we have ∆( f ◦ g ) = ∆( f ) . (6)For every indefinite binary Hamiltonian form f , with a = a ( f ) , b = b ( f ) and ∆ = ∆( f ) ,let C ∞ ( f ) = { [ u : v ] ∈ P r ( H ) : f ( u, v ) = 0 } . In P r ( H ) = H ∪ {∞} , the set C ∞ ( f ) is the -sphere of center − ba and radius √ ∆ | a | if a (cid:54) = 0 ,and it is the union of {∞} with the real affine hyperplane { z ∈ H : tr ( zb ) + c = 0 } of H otherwise. The values of f are positive on (the representatives in H × H in) one of the twocomponents of P r ( H ) − C ∞ ( f ) and negative on the other one. The set C ( f ) = { ( z, r ) ∈ H × ]0 , + ∞ [ : f ( z,
1) + a r = 0 } is the ( -dimensional) hyperbolic hyperplane in H R with boundary at infinity C ∞ ( f ) . Forevery g ∈ SL ( H ) , we have C ∞ ( f ◦ g ) = g − C ∞ ( f ) and C ( f ◦ g ) = g − C ( f ) . (7)Given an order O in a definite quaternion algebra over Q , a binary Hamiltonian form f is integral over O if its coefficients belong to O . Note that such a form f takes integralvalues on O × O , but the converse might not be true. The lattice Γ O = SL ( O ) of SL ( H ) preserves the set of indefinite binary Hamiltonian forms f that are integral over O . Thestabilizer in Γ O of such a form f is its group of automorphs SU f ( O ) = { g ∈ Γ O : f ◦ g = f } . If f is integral over O , then SU f ( O ) \ C ( f ) is a finite volume hyperbolic -orbifold, since SU f ( O ) is arithmetic and by Borel-Harish-Chandra’s theorem (though it might have beenknown before this theorem). In this section, we study the geometric reduction theory of positive definite binary Hamil-tonian forms, as in Mendoza [Men] for the Hermitian case. The results will be usefulin Section 5. We start by recalling the correspondence between H R and positive definitebinary Hamiltonian forms with discriminant − .Let Q be the -dimensional real vector space of binary Hamiltonian forms, and Q + itsopen cone of positive definite ones. The multiplicative group R × + of positive real numbers7 23/12/2019cts on Q + by multiplication. We will denote by R × + f the orbit of f and by P + Q + thequotient space Q + / R × + . It identifies with the image of Q + in the projective space P ( Q ) of Q .Let (cid:104)· , ·(cid:105) Q be the symmetric R -bilinear form (with signature (4 , ) on Q such that forevery f ∈ Q , (cid:104) f, f (cid:105) Q = − f ) . That is, for all f, f (cid:48) ∈ Q , we have (cid:104) f, f (cid:48) (cid:105) Q = a ( f ) c ( f (cid:48) ) + c ( f ) a ( f (cid:48) ) − tr ( b ( f ) b ( f (cid:48) ) ) . (8)By Equation (6), we have, for all f, f (cid:48) ∈ Q and g ∈ SL ( H ) (cid:104) f ◦ g, f (cid:48) ◦ g (cid:105) Q = (cid:104) f, f (cid:48) (cid:105) Q . (9)Let Q +1 the submanifold of Q + consisting of the forms with discriminant − , and let Θ : H R → Q +1 be the homeomorphism such that, for every ( z, r ) ∈ H R , M (Θ( z, r )) = 1 r (cid:18) − z − z n ( z ) + r (cid:19) . The fact that this map is well defined and is a homeomorphism follows by checking that itscomposition by the canonical projection Q + → P + Q + is the inverse of the homeomorphismdenoted by Φ : R × + f (cid:55)→ (cid:16) − b ( f ) a ( f ) , (cid:112) − ∆( f ) a ( f ) (cid:17) in [PP2, Prop. 22]. By loc. cit., the map Θ is hence (anti-)equivariant under the actionsof SL ( H ) : For all x ∈ H R and g ∈ SL ( H ) , we have Θ( gx ) = Θ( x ) ◦ g − . (10)Let O be a maximal order in a definite quaternion algebra A over Q . For every α ∈ A ,let I α = O α + O , which is a left fractional ideal of O . Let f α be the binary Hamiltonian form with matrix M ( f α ) = 1 n ( I α ) (cid:18) − α − α n ( α ) (cid:19) . Note that f α is a positive scalar multiple of the norm form associated with α : for all z ∈ H , f α ( u, v ) = (cid:0) u v (cid:1) M ( f α ) (cid:18) uv (cid:19) = 1 n ( I α ) n ( u − αv ) . Besides depending on α , the form f α does depend on the choice of the maximal order O .But its homothety class R × f α depends only on α .Let f ∞ be the binary Hamiltonian form whose matrix is M ( f ∞ ) = (cid:18) (cid:19) , that is, f ∞ : ( u, v ) (cid:55)→ n ( v ) . Note that for every α ∈ P r ( A ) = A ∪ {∞} , the form f α is nonzero anddegenerate (its discriminant is equal to ), and R × f α belongs to the boundary of P + Q + P ( Q ) . The map Φ − : H R → P ( Q ) given by x (cid:55)→ R × + Θ( x ) extends continuously to a SL ( A ) -(anti-)equivariant homeomorphism between H R ∪ P r ( A ) and its image in P ( Q ) bysending α to R × f α for every α ∈ P r ( A ) . Proposition 3.2 below makes precise the scalingfactor for the action of SL ( A ) on the forms f α for α ∈ P r ( A ) . Its proof will use thefollowing beautiful (and probably well-known) formula. Lemma 3.1.
For all g = (cid:18) a bc d (cid:19) ∈ SL ( H ) and z, w ∈ H such that g · z, g · w (cid:54) = ∞ , wehave n ( g · z − g · w ) = 1 n ( cz + d ) n ( cw + d ) n ( z − w ) . Proof.
Since (cid:18) az + b aw + bcz + d cw + d (cid:19) = g (cid:18) z w (cid:19) and by taking the square of the Dieudonné determinant (see Equation (4)), we have n ( g · z − g · w ) = n (( az + b )( cz + d ) − − ( aw + b )( cw + d ) − )= 1 n ( cw + d ) n (( az + b )( cz + d ) − ( cw + d ) − ( aw + b ))= 1 n ( cz + d ) n ( cw + d ) n (( az + b )( cz + d ) − ( cw + d )( cz + d ) − ( aw + b )( cz + d ))= 1 n ( cz + d ) n ( cw + d ) n ( z − w ) . (cid:3) Proposition 3.2.
For all g = (cid:18) a bc d (cid:19) ∈ SL ( A ) and α = [ x : y ] ∈ P r ( A ) , we have f g · α ◦ g = n ( O x + O y ) n ( O ( ax + by ) + O ( cx + dy )) f α . Note that this implies that f g · α ◦ g = f α if g ∈ SL ( O ) . Proof.
The result is left to the reader when α = ∞ or g · α = ∞ , hence we assume that α, g · α (cid:54) = ∞ . By Lemma 3.1, for all z ∈ H such that g · z (cid:54) = ∞ , we have f g · α ◦ g ( z,
1) = n ( cz + d ) f g · α ( g · z,
1) = n ( cz + d ) n ( I g · α ) n ( g · z − g · α )= 1 n ( I g · α ) n ( cα + d ) n ( z − α ) = n ( I α ) n ( I g · α ) n ( cα + d ) f α ( z, . The result easily follows. (cid:3)
For all α ∈ P r ( A ) = A ∪ {∞} and x ∈ H R , let us define the distance from x to the pointat infinity α by d α ( x ) = (cid:104) f α , Θ( x ) (cid:105) Q . See Appendix A for an alternate description of the map d α : H R → R .The next result gives a few computations and properties of these maps d α (which dependon the choice of maximal order O ). We will see afterwards that ln d α is an appropriatelynormalised Busemann function for the point at infinity α .9 23/12/2019 roposition 3.3. (1) For all ( z, r ) ∈ H R and α ∈ A , we have d α ( z, r ) = 1 r n ( I α ) (cid:0) n ( z − α ) + r (cid:1) , and d ∞ ( z, r ) = r .(2) For all x ∈ H R and α = [ u : v ] ∈ P r ( A ) , we have d α ( x ) = Θ( x )( u, v ) n ( O u + O v ) . (3) For all g = (cid:18) a bc d (cid:19) ∈ SL ( A ) and α = [ x : y ] ∈ P r ( A ) , we have d g · α ◦ g = n ( O x + O y ) n ( O ( ax + by ) + O ( cx + dy )) d α . In particular, if g ∈ SL ( O ) and α ∈ P r ( A ) , then d g · α ◦ g = d α . Proof. (1) Since M ( f α ) = n ( I α ) (cid:18) − α − α n ( α ) (cid:19) and M (Θ( z, r )) = r (cid:18) − z − z n ( z ) + r (cid:19) , wehave, by Equation (8), d α ( z, r ) = (cid:104) f α , Θ( z, r ) (cid:105) Q = 1 r n ( I α ) (cid:0) ( n ( z ) + r ) + n ( α ) − tr ( α z ) (cid:1) = n ( z − α ) + r r n ( I α ) . The computation of d ∞ is similar and easier.(2) Let x = ( z, r ) ∈ H R and f = Θ( x ) . If v (cid:54) = 0 , then α = uv − , and by the definition of Θ and Assertion (1), f ( u, v ) n ( O u + O v ) = f ( α, n ( I α ) = 1 n ( I α ) (cid:0) α (cid:1) M ( f ) (cid:18) α (cid:19) = n ( α ) − α z − z α + n ( z ) + r r n ( I α ) = n ( z − α ) + r r n ( I α ) = d α ( x ) . Similarly, if v = 0 , then f ( u,v ) n ( O u + O v ) = f (1 ,
0) = r = d α ( x ) .(3) For every w ∈ H R , using the (anti-)equivariance property of Θ , Equation (9) andProposition 3.2, we have d g · α ◦ g ( w ) = (cid:104) f g · α , Θ( gw ) (cid:105) Q = (cid:104) f g · α , Θ( w ) ◦ g − (cid:105) Q = (cid:104) f g · α ◦ g, Θ( w ) (cid:105) Q = n ( O x + O y ) n ( O ( ax + by ) + O ( cx + dy )) (cid:104) f α , Θ( w ) (cid:105) Q = n ( O x + O y ) n ( O ( ax + by ) + O ( cx + dy )) d α ( w ) . (cid:3) Since SL ( O ) is a noncocompact lattice with cofinite volume in SL ( H ) and set ofparabolic fixed points at infinity P r ( A ) , there exists (see for instance [Bow]) a Γ -equivariantfamily of horoballs in H R centered at the points of P r ( A ) , with pairwise disjoint interiors.10 23/12/2019ince SL ( O ) \ H R may have several cusps as mentioned in the introduction, there arevarious choices for such a family, and we now use the normalized distance to the points of P r ( A ) in order to define a canonical such family, and we give consequences on the structureof the orbifold SL ( O ) \ H R .For all α ∈ P r ( A ) and s > , we define the normalized horoball centered at α withradius s as B α ( s ) = { x ∈ H R : d α ( x ) ≤ s } . The terminology is justified by the following result, which proves in particular that B α ( s ) is indeed a (closed) horoball. Recall that the Busemann function β : ∂ ∞ H R × H R × H R → R is defined, with t (cid:55)→ ξ t any geodesic ray with point at infinity ξ ∈ ∂ ∞ H R , by ( ξ, x, y ) (cid:55)→ β ξ ( x, y ) = lim t → + ∞ d ( x, ξ t ) − d ( y, ξ t ) . Proposition 3.4.
Let α ∈ P r ( A ) and s > .(1) There exists c α ∈ R such that ln d α ( x ) = β α ( x, (0 , c α for every x ∈ H R .(2) If α ∈ A , then B α ( s ) is the Euclidean ball of center (cid:0) α, s n ( I α )2 (cid:1) and radius s n ( I α )2 . If α = ∞ , then B α ( s ) is the Euclidean halfspace consisting of all ( z, r ) with r ≥ s .(3) For all g ∈ SL ( O ) , we have g ( B α ( s )) = B g · α ( s ) . Proof. (1) If α = ∞ , then for every ( z, r ) ∈ H R , we have d α ( z, r ) = r and β ∞ (( z, r ) , (0 , β ∞ ((0 , r ) , (0 , − ln r , hence the result holds with c ∞ = 0 .If α ∈ A , since SL ( A ) acts transitively on P r ( A ) , let g = (cid:18) a bc d (cid:19) ∈ SL ( A ) be suchthat α = g · ∞ . Recall that the Busemann function is invariant under the diagonal actionof SL ( H ) on ∂ ∞ H R × H R × H R and is an additive cocycle in its two variables in H R . ByProposition 3.3 (3) since ∞ = [1 : 0] , we hence have for every x ∈ H R ln d α ( x ) = ln d g ·∞ ( g ( g − x )) = ln d ∞ ( g − x ) n ( O a + O c )= β ∞ ( g − x, (0 , − ln n ( O a + O c ) = β g ·∞ ( x, g (0 , − ln n ( O a + O c )= β α ( x, (0 , β α ((0 , , g (0 , − ln n ( O a + O c ) . Hence the result holds, and taking x = (0 , , we have by Proposition 3.3 (1) c α = ln n ( α ) + 1 n ( I α ) . (2) If α ∈ A , for every ( z, r ) ∈ H R , by Proposition 3.3 (1), we have d α ( z, r ) ≤ s if and onlyif n ( z − α ) + r ≤ s r n ( I α ) , that is, if and only if n ( z − α ) + ( r − s n ( I α )2 ) ≤ (cid:0) s n ( I α )2 (cid:1) . Thesecond claim of Assertion (2) is immediate.(3) This follows from Proposition 3.3 (3). (cid:3) The following result extends and generalizes a result for D A = 2 of [Spe, §5].11 23/12/2019 heorem 3.5. Let O be a maximal order in a definite quaternion algebra A over Q .(1) For all distinct α, β ∈ P r ( A ) , the normalized horoballs B α (1) and B β (1) have disjointinterior. Furthermore, their intersection is nonempty if and only if α = ∞ and β ∈ O , or β = ∞ and α ∈ O , or α, β (cid:54) = ∞ and I α I β = O ( α − β ) , in which case they meet in one andonly one point.(2) We have H R = (cid:91) α ∈ P r ( A ) B α (cid:0) (cid:112) D A (cid:1) . Before proving this result, let us make two remarks.(i) Note that B (1) and B ∞ (1) intersect (exactly at their common boundary point (0 , ) whatever the definite quaternion algebra A over Q is. Thus the constant s = 1 inAssertion (1) is optimal. The family ( B α (1)) α ∈ P r ( A ) is a (canonical) family of maximal(closed) horoballs centered at the parabolic fixed points of SL ( O ) with pairwise disjointinteriors. Since SL ( O ) is a lattice (hence is geometrically finite with convex hull of itslimit set equal to the whole H R ), the quotient SL ( O ) \ ( H R − (cid:83) α ∈ P r ( A ) B α (1) (cid:1) is compact(see for instance [Bow]).(ii) Assertion (2) is a quantitative version of the standard geometric reduction theory(see for instance [GR, Bor1, Leu]) for the structure of the arithmetic orbifold SL ( O ) \ H R . Itindeed implies that if R is a finite subset of SL ( A ) such that R ·∞ is a set of representativesof SL ( O ) \ P r ( A ) , and if D γ is a fundamental domain for the action on H of the stabilizerof ∞ in γ − SL ( O ) γ for every γ ∈ R , then a weak fundamental domain for the action of SL ( O ) on H R is the finite union (cid:83) γ ∈ R γ S γ where S γ is the Siegel set S γ = ( D γ × ]0 , + ∞ [ ) ∩ γ − B γ ·∞ ( (cid:112) D A ) . Proof. (1) Note that two horoballs centered at distinct points at infinity, which are notdisjoint but have disjoint interior, meet at one and only one common boundary point.Hence the last claim of Assertion (1) follows from the first two ones.First assume that α = ∞ , so that β ∈ A . By Proposition 3.4 (2), we have B ∞ (1) = { ( z, r ) ∈ H R : r ≥ } and B β (1) is the horoball centered at β with Euclidean diameter n ( I β ) (see Figure 1). They hence meet if and only if n ( I β ) ≥ , and their interiors meet ifand only if n ( I β ) > . But since O ⊂ I β , by Equation (3), we have n ( I β ) ≤ n ( O ) = 1 withequality if and only if I β = O , that is, β ∈ O . The result follows. H βα n ( I β )2 n ( I β ) B ∞ (1) n ( I α )2 Figure 1: Disjointness of normalized horoballs B α (cid:48) (1) for α (cid:48) ∈ P r ( A ) .Up to permuting α and β and applying the above argument, we may now assume that α, β (cid:54) = ∞ . The Euclidean balls B α (1) and B β (1) meet if and only if the distance d αβ
12 23/12/2019etween their Euclidean center is less than or equal to the sum of their radii r α and r β ,and their interior meet if and only if d αβ < r α + r β . By Proposition 3.4 (2) and by themultiplicativity of the reduced norms (see [Rei, Thm. 24.11 and p. 181]), we have (see theabove picture) d αβ − ( r α + r β ) = (cid:16) n ( α − β ) + (cid:0) n ( I α )2 − n ( I β )2 (cid:1) (cid:17) − (cid:0) n ( I α )2 + n ( I β )2 (cid:1) = n ( α − β ) − n ( I α ) n ( I β ) = n ( α − β ) − n ( I α I β ) . Since α − β ∈ I α I β and again by Equation (3), we have n ( α − β ) ≥ n ( I α I β ) , with equalityif and only if I α I β = O ( α − β ) . The result follows.(2) For every x ∈ H R , let ( u, v ) in O × O − { } realizing the minimum on O × O − { } of thepositive definite binary Hamiltonian form Θ( x ) , whose discriminant is − . Let α = [ u : v ] .Then by Proposition 3.3 (2) and by Equation (1), we have, since the norm of an integralleft ideal is at least , d α ( x ) = Θ( x )( u, v ) n ( O u + O v ) ≤ (cid:112) D A . This proves the result. (cid:3)
The following observation, which is closely related with the proof of Assertion (1) ofTheorem 3.5, will be useful later on.
Lemma 3.6.
For all α (cid:54) = β ∈ A , the hyperbolic distance between B α (1) and B β (1) is d ( B α (1) , B β (1)) = ln n ( α − β ) n ( I α I β ) . Proof.
This follows from the easy exercise in real hyperbolic geometry saying that the dis-tance in the upper halfspace model of the real hyperbolic n -space between two horospheres H , H (cid:48) with Euclidean radius r, r (cid:48) , and with Euclidean distance between their points atinfinity equal to λ , is d ( H , H (cid:48) ) = ln λ rr (cid:48) .This exercice uses the facts that the commonperpendicular between two disjoint horoballs isthe geodesic line through their points at infin-ity and that the (signed) hyperbolic length ofan arc of Euclidean circle centered at a point atinfinity with angles with the horizontal hyper-plane between α and π/ is − ln tan α . (cid:3) λ r (cid:48) H (cid:48) r r r (cid:48) λ/ α α (cid:48) − ln tan α − ln tan α (cid:48) H Figure 2 SL ( O ) Let A be a definite quaternion algebra over Q and let O be a maximal order in A . In thissection, we describe a canonical SL ( O ) -invariant cell decomposition of the -dimensionalreal hyperbolic space H R . We follow [Men, BeS] when the field H is replaced by C , theorder O by the ring of integers of a quadratic imaginary extension of Q , and H R by H R .13 23/12/2019or every α ∈ P r ( A ) , the Ford-Voronoi cell of α for the action of SL ( O ) on H R is theset H α of points not farther from α than from any other element of P r ( A ) : H α = { x ∈ H R : ∀ β ∈ P r ( A ) , d α ( x ) ≤ d β ( x ) } . In the complex case, this set is called the minimal set of α , see [Men]. Proposition 4.1.
Let α ∈ P r ( A ) .(1) For all g ∈ SL ( O ) , we have g ( H α ) = H g · α .(2) We have B α (1) ⊂ H α ⊂ B α ( √ D A ) .(3) The Ford-Voronoi cell H α is a noncompact -dimensional convex hyperbolic polytope,whose proper cells are compact, and the stabilizer of α in SL ( O ) acts cocompactly on itsboundary ∂ H α .(4) For every β ∈ P r ( A ) − { α } , let S α, β = { x ∈ H R : d α ( x ) = d β ( x ) } . Then S α, β is a hyperbolic hyperplane, that intersects perpendicularly the geodesic line with points atinfinity α and β . Furthermore, the Ford-Voronoi cells H α and H β have disjoint interiorand their (possibly empty) intersection is contained in S α, β . Thus H R = (cid:91) α ∈ P r ( A ) H α is a SL ( O ) -invariant cell decomposition of H R , whose codimension skeleton will bestudied in the remainder of this section. We will see in Examples 4.4 and 4.5 that theinclusions in Assertion (2) of this proposition, as well as the one of Theorem 3.5, are sharpwhen D A = 2 , . Proof. (1) This follows from Proposition 3.3 (3).(2) The inclusion on the left hand side follows from Theorem 3.5 (1): If x ∈ B α (1) and x / ∈ H α , then there exists β ∈ P r ( A ) − { α } such that d β ( x ) < d α ( x ) ≤ , thus the interiorsof B α (1) and B β (1) have nonempty intersection, a contradiction. If x / ∈ B α ( √ D A ) , thenby Theorem 3.5 (2), there exists β ∈ P r ( A ) − { α } such that x ∈ B β ( √ D A ) . Hence d β ( x ) ≤ √ D A < d α ( x ) , so that x / ∈ H α .(3) and (4) Since ln d α is a Busemann function with respect to the point at infinity α byProposition 3.4 (1), for every β ∈ P r ( A ) − { α } , the set H α,β = { x ∈ H R : d α ( x ) ≤ d β ( x ) } is a (closed) hyperbolic halfspace. Its boundary is S α, β , which is hence a hyperbolic hyper-plane that intersects perpendicularly the geodesic line with points at infinity α and β . Beingthe intersection of the locally finite family of hyperbolic halfspaces ( H α,β ) β ∈ P r ( A ) −{ α } , andcontaining the horoball B α (1) , the Ford-Voronoi cell H α is a noncompact -dimensionalconvex hyperbolic polytope. Since α is a bounded parabolic fixed point of the lattice SL ( O ) and by Assertion (2), the stabilizer of α in SL ( O ) acts cocompactly on ∂ H α , andhence the boundary cells of H α are compact. (cid:3) The horoballs B (1) and B ∞ (1) with disjoint interiors meet at (0 , ∈ H R , and atmost two horoballs with disjoint interior can meet at a given point of H R . Thus, the Ford-Voronoi cells at and at ∞ have nonempty intersection, which is a compact -dimensionalhyperbolic polytope. This intersection Σ O = H ∩ H ∞
14 23/12/2019s called the fundamental cell of the spine of SL ( O ) . We will describe it in Example 4.4when D A = 2 and in Example 4.5 when D A = 3 . Lemma 4.2.
Let α ∈ P r ( A ) be such that e = H ∞ ∩ H ∩ H α is a -dimensional cell inthe boundary of Σ O . Then min { n ( I α ) , n ( I α − ) } ≥ D A , and the horizontal projection of e to H is contained in the Euclidean hyperplane { z ∈ H : tr ( α z ) = 1 + n ( α ) − n ( I α ) } . Proof.
Note that α (cid:54) = 0 , ∞ . By Proposition 4.1 (2), the intersection B ∞ ( √ D A ) ∩ B ( √ D A ) ∩ B α ( √ D A ) contains e , hence the intersections B ∞ ( √ D A ) ∩ B α ( √ D A ) and B ( √ D A ) ∩ B α ( √ D A ) are nonempty. Since B ∞ ( √ D A ) is the Euclidean halfspace of points ( z, r ) with r ≥ √ D A and B α ( √ D A ) is a Euclidean ball tangent to the horizontal planewith diameter √ D A n ( I α ) by Proposition 3.4 (2), this implies that √ D A n ( I α ) ≥ √ D A , sothat D A n ( I α ) ≥ . Since g = (cid:18) (cid:19) belongs to SL ( O ) and maps to ∞ and α to α − ,and by Proposition 3.4 (3), the intersection B ∞ ( √ D A ) ∩ B α − ( √ D A ) is nonempty, hencesimilarly D A n ( I α − ) ≥ . B ∞ (1) H λα n ( I α )0 (cid:112) n ( I α ) α Figure 3The set of points equidistant to and ∞ is the open Euclidean upper hemisphere ofradius centered at , and the set of points equidistant to α and ∞ is the open Euclideanupper hemisphere of radius (cid:112) n ( I α ) centered at α . The projection to H of the intersectionof these hemispheres is contained in the affine Euclidean hyperplane of H perpendicular tothe real vector line containing α that passes through the projection λα with λ > to thatline of any point at Euclidean distance from and at Euclidean distance (cid:112) n ( I α ) from α . An easy computation (considering the two cases when n ( α ) > as in Figure 3 or when n ( α ) ≤ ) using right angled triangles gives that λ = n ( α ) − n ( I α )2 n ( α ) . Since ( u, v ) (cid:55)→ tr ( u v ) is the standard Euclidean scalar product on H , this gives the result. (cid:3) The spine of SL ( O ) is the codimension skeleton of the cell decomposition into Ford-Voronoi cells of H R , that is X O = (cid:91) α (cid:54) = β ∈ P r ( A ) H α ∩ H β = (cid:91) α ∈ P r ( A ) ∂ H α .
15 23/12/2019t is an SL ( O ) -invariant piecewise hyperbolic polyhedral complex of dimension . We referfor instance to [BrH] for the definitions related to polyhedral complexes, CAT(0) spacesand orbihedra. Note that the stabilizers in SL ( O ) of the cells of X O may be nontrivial.The spine is called the minimal incidence set in the complex case in [Men] and [SV], andthe cut locus of the cusp in [HP, §5] when the class number is one.For every hyperbolic cell C of X O and every α ∈ P r ( A ) such that C ⊂ ∂ H α , theradial projection along geodesic rays with point at infinity α from C to the horosphere ∂B α (1) is a homeomorphism onto its image, and the pull-back of the flat induced lengthmetric on this horosphere endows C with a structure of a compact Euclidean polytope.This Euclidean structure does not depend on the choice of α , since the (possibly empty)intersection H α ∩ H β is equidistant to B α (1) and B β (1) for all distinct α, β in P r ( A ) . Itis well known (see for instance [Ait]) that these Euclidean structures on the cells of X O endow X O with the structure of a CAT(0) piecewise Euclidean polyhedral complex.Furthermore, X O is a SL ( O ) -invariant deformation retract of H R along the geodesicrays with points at infinity the points in P r ( A ) and since the quotient orbifold with bound-ary SL ( O ) \ (cid:0) H R − (cid:83) α ∈ P r ( A ) B α (1) (cid:1) is compact, the quotient space SL ( O ) \ X O is a finitelocally CAT(0) piecewise Euclidean orbihedral complex.The following result gives a description of the cell structure of SL ( O ) \ X O when O isleft-Euclidean. See Examples 4.4 and 4.5 for a more detailed study when D A = 2 , . Proposition 4.3.
The Hamilton-Bianchi group SL ( O ) acts transitively on the set of -dimensional cells of its spine X O if and only if D A ∈ { , , } . In these cases, the horizontalprojection of the fundamental cell Σ O to H is the Euclidean Voronoi cell of for the Z -lattice O in the Euclidean space H . Proof. If SL ( O ) acts transitively on the -dimensional cells of X O , then X O = SL ( O ) Σ O ,and the stabilizer of ∞ in SL ( O ) acts transitively on the set of -dimensional cells in ∂ H ∞ ,since (cid:18) (cid:19) ∈ SL ( O ) preserves Σ O = H ∞ ∩ H and exchanges H ∞ and H . This sta-bilizer consists of the upper triangular matrices with coefficients in O , hence with diagonalcoefficients in O × . The orbit of ∈ H under this stabilizer is exactly O . Since Σ O iscompact and contained in the open Euclidean upper hemisphere centered at with radius , by horizontal projection on H , this proves that H is covered by the open balls of radius centered at the points of O . Hence O is left-Euclidean.Conversely, if O is left-Euclidean, then the class number of A is , and SL ( O ) actstransitively on the Ford-Voronoi cells. In order to prove that SL ( O ) acts transitively onthe -dimensional cells of X O , we hence only have to prove that the stabilizer of ∞ in SL ( O ) acts transitively on the -dimensional cells of ∂ H ∞ . For this, let α ∈ A be suchthat H ∞ ∩ H α is a -dimensional cell in ∂ H ∞ . Let us prove that α ∈ O , which givesthe result. Due to problems caused by the noncommutativity of H , the proof of [BeS,Prop. 4.3] does not seem to extend exactly. We will use instead Lemma 2.1.16 23/12/2019 α α n ( α ) S α n ( α ) , ∞ S , ∞ S α, ∞ Figure 4Assume for a contradiction that α / ∈ O . Since O is left-Euclidean, there exists c ∈ O such that n ( α − c ) < . Up to replacing α by α − c , since translations by O preserve H ∞ , we may assume that < n ( α ) < . For every β ∈ A and β (cid:48) ∈ P r ( A ) − { β } , letus denote by S β,β (cid:48) the Euclidean upper hemisphere centered at β equidistant from thepoints at infinity β and β (cid:48) . In particular, S , ∞ has radius . The inversion with respectto the sphere containing S , ∞ acts by an orientation-reversing isometry on H R , and actson the boundary at infinity P r ( H ) = H ∪ {∞} by z (cid:55)→ z n ( z ) = z . By Lemma 2.1, it hencenormalizes SL ( O ) and, in particular, sends S α, ∞ to S α n ( α ) , , and fixes S , ∞ (see Figure4). Since n ( α ) < , the hemisphere S α, ∞ is therefore below the union of S , ∞ and S α n ( α ) , ,which contradicts the fact that H ∞ ∩ H α , which is contained in S α, ∞ , is a -dimensionalcell in ∂ H ∞ .In order to prove the last claim of Proposition 4.3, note that n ( I α ) = 1 if α ∈ O , andthat the above proof shows that the -dimensional cells contained in ∂ H ∞ and meetingthe fundamental cell along a -dimensional cell are contained in spheres centered at pointsin O . Therefore, by Lemma 4.2, the horizontal projection of Σ O is the intersection ofthe halfspaces containing and bounded by the Euclidean hyperplanes with equation tr ( α z ) = n ( α ) for all α ∈ O . Since this hyperplane is the set of points z in the Euclideanspace H equidistant to and α , this proves that the horizontal projection of Σ O is indeedthe Voronoi cell at of the Z -lattice O . (cid:3) Example 4.4.
Let A = Q + Q i + Q j + Q k ⊂ H be the definite quaternion algebra over Q with D A = 2 , and let O = Z + Z i + Z j + Z i + j + k be the (maximal) Hurwitz orderin A . The Hurwitz order O is the lattice of type F = D ∗ . The group of unit Hurwitzquaternions has elements O × = (cid:110) ± , ± i, ± j, ± k, ± ± i ± j ± k (cid:111) . The Voronoi cell Σ H O of for the lattice O in H is (up to homothety) the -cell , whichis the (unique) self-dual, regular, convex Euclidean -polytope, whose Schläfli symbol is { , , } . The vertices of Σ H O are the quaternions i O × = (cid:110) ± ± i , ± ± j , ± ± k , ± i ± j , ± i ± k , ± j ± k (cid:111) . See for instance [CS, p. 119] for more details and references.Let H × be the subgroup of H × that consists of the quaternions of norm . The groupmorphism H × × H × → SO(4) that associates to ( u, v ) ∈ H × × H × the orthogonal trans-formation z (cid:55)→ uzv − is surjective, see for instance [Ber2, Thm. 8.9.8]. The group of17 23/12/2019uclidean symmetries of the -cell is the Coxeter group [3 , , . It consists of the elements z (cid:55)→ uzv − , z (cid:55)→ u z v − of O(4) , where either u and v are unit Hurwitz integersor u/ √ and v/ √ are in i O × .By Proposition 4.3, the fundamental cell of SL ( O ) is Σ O = { ( z, t ) ∈ H R : z ∈ Σ H O , n ( z ) + t = 1 } . With the notation of Lemma 2.1, the stabilizer of Σ O in SL ( O ) consists of the matrices C a,d = (cid:18) a d (cid:19) and J C a,d = (cid:18) da (cid:19) with a, d ∈ O × . When Σ O is identifiedwith Σ H O by the horizontal projection, the diagonal matrices induce by Equation (5) rotational symmetries of Σ H O , and the antidiagonal ones induce another orientation-reversing symmetries, together forming a subgroup of index in the Coxeter group [3 , , .The quotient SL ( O ) \ X O is obtained by identifying the opposite -dimensional cells of Σ O (which are regular octahedra) by translations by elements of O , and by forming thequotient by the stabilizer of Σ O . In particular, all the vertices of X O are in the same orbitunder SL ( O ) . i
11 + iS ∞ , i + j + k S ∞ ,i S ∞ , i S ∞ , S ∞ , S ,i S ,i S , S , i Figure 5: Boundary of equidistant hemispheres and halfplanes in C ⊂ H .Speiser [Spe, §5] observed that the estimate of Proposition 4.1 (2) is sharp in this ex-ample: H R is indeed completely contained in (cid:83) α ∈ P r ( A ) B α ( √ , and the orbit that containsall the vertices of Σ O is not contained in the union of the interiors of the horoballs B α ( √ .Furthermore, Speiser proved that the point v = (cid:16) i , √ (cid:17) belongs to the boundary of exactly horoballs B α ( √ , the ones with α in E = (cid:110) ∞ , , , i, i, i ± j ± k , − i = 1 + i (cid:111) . In particular, v is a vertex of the spine X O , contained in the boundary of exactly Ford-Voronoi cells H α for α in this set.18 23/12/2019he set E contains exactly pairs { α, β } of distinct elements such that the interiorsof the horoballs B α ( √ and B β ( √ are disjoint, these pairs being {∞ , − i } , { , i } , { , i } , { i + j + k , i − j − k } and { i + j − k , i − j + k } . If { α, β } is one of these pairs, the Ford-Voronoi cells H α and H β intersect only at v . For all other pairs in E , the intersection isa higher-dimensional cell.As , , i, i, i ± j ± k are in O and − i is not in O , there are Ford-Voronoi cellsincident to v that intersect H ∞ in a -dimensional -cell (see Figure 5, which representsthe intersection with the plane in H containing , , i of the closures of the equidistantspheres and planes between some pairs of elements in {∞ , , , i, i, i + j + k } , so thatthe horizontal projection of v is the common intersection points of the straight lines). Asimilar property holds for all the other Ford-Voronoi cells incident to v : For example, H intersects in a -dimensional cell the Ford-Voronoi cells H ∞ , H , H i , H i ± j ± k , H i , butnot H i +1 by Theorem 3.5 (1), since I I i +1 = O (cid:54) = O (1 + i ) . Thus the pattern of pairwiseintersections into -dimensional cells of these Ford-Voronoi cells is given by Figure 6and the number of -cells containing v is exactly
40 = (10 × / , one for each edge ofthis intersection pattern. + i + j + k + i + j - k ∞ + i i1 + i - j - k + i - j + k + i Figure 6: Pattern of intersections into -dimensional cells of Ford-Voronoi cells centeredat {∞ , , , i, i + 1 , i ± j ± k , i } .The boundary of each H α is tiled by -cells, combinatorially forming the -cell hon-eycomb. The dual of this honeycomb is the -cell honeycomb. Therefore, the link of thevertex v in the tessellation of ∂ H α for all α ∈ E is the dual of the -cell, which is theboundary of the -cube, such that the intersection of the link with each of the eight -cellsis a -cube.Gluing together the ten boundaries of -cubes (that have been subdivided in eight -cubes each) according to the above intersection pattern proves that the link of v in thespine X O is the -skeleton of the -cube (which is the -dimensional regular polytope withSchläfli symbol { , , , } ). Example 4.5.
The maximal order of the definite quaternion algebra (cid:0) − , − Q (cid:1) of discrim-inant D A = 3 is Z [1 , i, i + j , k ] , see [Vig, p. 98]. Using the unique Q -linear map from (cid:0) − , − Q (cid:1) to H sending to , i to j , j to k √ and k to − i √ , we identify (cid:0) − , − Q (cid:1) with19 23/12/2019he Q -subalgebra A of H generated by , i √ , j and k √ , and the maximal order is thenidentified with O = Z [1 , ρ, j, ρj ] , where ρ = 1 + i √ . The group of units of O is the dicyclic group of order O × = {± , ± j, ± ρ, ± ρ , ± ρj, ± ρ j } . The elements of the maximal order O = Z [1 , ρ ] + Z [1 , ρ ] j of A are the vertices of the - duoprism honeycomb in the -dimensional Euclidean space H . The elements of theset V , = { , , j, j, ρ, ρj, ρj, j + ρ, ρ (1 + j ) } , contained in O , are the vertices of its fundamental - duoprism , which is a uniform -polytope with Schläfli symbol { }×{ } (the Cartesian product of two equilateral triangles,whose -skeleton is given in Figure 7). We refer to Coxeter’s three papers [Cox1, Cox, Cox3]for notation and references about uniform polytopes and their Coxeter groups, with thehelp of the numerous and beautiful Wikipedia articles. ρ j j ρ ρ + ρ j ρ + j11 + ρ j1 + j Figure 7: The -skeleton of the - duoprism.The Voronoi cell Σ H O of for the lattice O in H is the - duoprism whose Schläflisymbol is { } × { } . It is the Cartesian product of two copies of the Voronoi cell of for the hexagonal lattice of the Eisenstein integers in C whose set of vertices is V = {± i √ , ± ± i √ } . Thus, the set of vertices of Σ H O is V + jV . These vertices, including z = 12 + i √ j k √ j + ρ )(1 + ρ ) − , belong to A and all have reduced norm .By Proposition 4.3, the fundamental cell Σ O of SL ( O ) is the subset of the Euclideanunit sphere in H n C whose horizontal projection is Σ H O . In particular, all the vertices of Σ O have Euclidean height √ . Let u and v be either both in O × or both in ρ O × . The
20 23/12/2019appings z (cid:55)→ uzv − and z (cid:55)→ u z v − are Euclidean symmetries of Σ H O , and they form theCoxeter group [[6 , , of the symmetries of the - duoprism Σ H O .With the notation of Lemma 2.1, the stabilizer of Σ O in SL ( O ) consists of the matrices C a,d = (cid:18) a d (cid:19) and J C a,d = (cid:18) da (cid:19) with a, d ∈ O × . When Σ O is identifiedwith Σ H O by the horizontal projection, the diagonal matrices induce by Equation (5) rotational symmetries of Σ H O , and the antidiagonal ones induce another orientation-reversing symmetries, together forming a subgroup of index in [[6 , , .It is straightforward but tedious to check that the subgroup of [[6 , , that arisesfrom the diagonal matrices in SL ( O ) acts transitively on the vertices of Σ H O . Thus, thissubgroup acts transitively on the vertices of the fundamental cell Σ O , which implies thatall vertices of the spine X O are in the same orbit.We will now turn to a study of the link of a vertex in X O . Let v = (cid:16) z , √ (cid:17) , which is the vertex of Σ O whose projection to H is z . Let g : H ∪ {∞} → H ∪ {∞} be thehomography z (cid:55)→ ( z − z ) − + z induced by M = (cid:18) z (cid:19) (cid:18) − z (cid:19) = (cid:18) z − z − z (cid:19) ∈ GL ( A ) . Lemma 4.6.
The element M belongs to the normalizer of SL ( O ) in SL ( H ) . Proof.
Computations (using Mathematica and SAGE) show that M conjugates all thegenerators of SL ( O ) given in Corollary 2.2 to elements of SL ( O ) , as follows. We have M J M − = (cid:18) ρ + j + ρj − ρ − j − ρj − j − ρ − ρj − − ρ − j − ρj (cid:19) ,M T M − = (cid:18) ρ + j + ρj − ρ − j − ρj − ρ − j − ρj (cid:19) ,M T j M − = (cid:18) − ρ + j + ρj ρ − j + ρj j − ρ − j + ρj (cid:19) ,M T ρ M − = (cid:18) ρ + 2 j − ρj − ρ ρ − ρ + j − ρj (cid:19) ,M T ρj M − = (cid:18) − ρ − j + 2 ρj ρ + j − ρj ρj ρ − j − ρj (cid:19) . Since C u,v C u (cid:48) ,v (cid:48) = C uu (cid:48) ,vv (cid:48) and J C u,v J = C v,u for all units u, v, u (cid:48) , v (cid:48) of O , it suffices tocheck the following elements: M C , − M − = (cid:18) − ρ + 2 j + 2 ρj
42 + 2 ρ + 2 j + 2 ρj − ρ − j − ρj (cid:19) ,M C ,j M − = (cid:18) ρ + 3 j − ρ − j ρ − ρ − ρj (cid:19) ,
21 23/12/2019nd
M C ,ρ M − = (cid:18) ρ − j + 2 ρj j − ρj − ρ − j + 2 ρj ρ − j − ρj (cid:19) . Thus, M belongs to the normalizer of SL ( O ) in SL ( H ) . (cid:3) Proposition 4.7. If D A = 3 , then the set of α ∈ A such that v belongs to the boundaryof B α ( √ is V = V , ∪ g ( V , ) ∪ {∞ , z } . For every α ∈ A , the point v of H R does not belong to the interior of B α ( √ . The second claim implies that when r < √ , the family (cid:0) B α ( r ) (cid:1) α ∈ A does not cover H R . In particular, the inclusions in Proposition 4.1 (2) are also sharp when D A = 3 . Proof.
First observe that v as well as all the vertices of Σ O are in the horizontal plane { ( z, t ) ∈ H R : t = √ } , which is the boundary of B ∞ ( √ .For every α ∈ A , recall from Proposition 3.4 (2) that the horoball B α ( √ is theEuclidean ball tangent to H at α with Euclidean radius √ n ( I α )2 . Writing α = pq − with p, q ∈ O relatively prime, we have n ( I α ) = n ( q ) − . Thus if v ∈ B α ( √ , then the Euclideandiameter √ n ( I α ) of B α ( √ is at least the Euclidean height √ of v , that is n ( I α ) ≥ .Equality is only possible if α is the vertical projection to H of v , that is α = z . Since z = ( j + ρ )(1 + ρ ) − and j + ρ , ρ are relatively prime (their norms are and ), wehave z ∈ A and n ( I z ) = . Hence the point v does belong to the boundary of B z ( √ ,and if α (cid:54) = z , then n ( I α ) = 1 or n ( I α ) = . B α ( √ √ B β ( √ B z ( √ √ α ∈ V , z β ∈ g ( V , ) √ v B ∞ ( √ √ (cid:113) Figure 8: Intersection pattern at v of the covering family of horoballs (cid:0) B α ( √ (cid:1) α ∈ A First assume that n ( q ) = 1 , or equivalently that α ∈ O . Then n ( I α ) = 1 , hence B α ( √ is the Euclidean ball of center ( α, √ ) and radius √ , that intersects the horizontal planeat height √ in a horizontal ball centered at ( α, √ ) and of radius (cid:113) . The verticesof the fundamental - duoprism of O are exactly at this distance from z , and all otherelements of O are at greater distance from z . Hence (see Figure 8 on its left), v belongsto the boundary of B α ( √ for every α ∈ V , and v / ∈ B α ( √ if α ∈ O − V , .22 23/12/2019e begin the treatment of the remaining case n ( q ) = 2 by geometric observations.The homography g defined before Lemma 4.6 maps ∞ to z , z to ∞ , and the spherein H of center z and radius r to the sphere in H of center z and radius r , for every r > . In particular, g maps the sphere in H of center z and radius √ to itself and thePoincaré extension of g to H R (again denoted by g ) fixes v . Thus, g ( B ∞ ( √ B z ( √ and Lemma 4.6 implies that g preserves the SL ( O ) -equivariant family ( B α ( √ α ∈ A ofhoroballs.Now let β = pq − ∈ A be such that v ∈ B β ( √ and n ( q ) = 2 . Note that the Euclideanperpendicular projection from H R to H does not increase the Euclidean distances, and thatthe projection of the Euclidean center of B β ( √ is β and the projection of v is z (seeFigure 8). Since the radius of B β ( √ is √ , we hence have d ( z , β ) ≤ √ < √ . Since g fixes v and gB β ( √
3) = B g ( β ) ( √ by the above lemma, the element α = g − ( β ) , whichsatisfies v ∈ B α ( √ and is outside the ball of center z and radius √ , hence cannot havea denominator of norm . Therefore α has denominator and by the previous case, itbelongs to V , and v lies in the boundary of B α ( √ . So that β = g ( α ) belongs to g ( V , ) and v lies in the boundary of B β ( √ . (cid:3) An easy computation gives g ( V , ) = (cid:110) j − j , ρj − j ¯ ρ , ρ + j ρ − j , ρ (1 + j )2 = 1(1 − j ) ¯ ρ , j + ρj − j ( ¯ ρ −
1) + j , ρ + j + ρj , ρ + j , ρ + ρj , ρ + j + ρj (cid:111) . As any element β in g ( V , ) is the sum of an element of O with the inverse of an element of O with reduced norm , we have n ( I β ) = and the horoball B β ( √ has Euclidean radius √ . This horoball intersects the horizontal plane { ( z, t ) ∈ H R : t = √ } in a horizontalball of Euclidean radius √ . In particular, the points in g ( V , ) are at Euclidean distance √ of z and the horoballs tangent to v are positioned as in Figure 8.By Proposition 4.7, the link of v in the cellulation of H R by the Ford-Voronoi cells of O has 20 -cells, which are the intersections of a small sphere centered at v with the Ford-Voronoi cells H α for α in V = V , ∪ g ( V , ) ∪ {∞ , z } . Furthermore, for all α (cid:54) = β in V , thehoroballs B α ( √ and B β ( √ are tangent at v if and only if { α, β } is one of the pairs {∞ , z } , { , ρ + j + ρj } , { , ρ + j + ρj } , { ρ, j + ρj } , { j, ρ + ρj } , { j, ρ + ρj } , { ρj, ρ + j } , { ρj, ρ + j } , { j + ρ, ρj } and { ρ + ρj, j } . By analyzing the intersections of the horoballs B α (1) contained in the Ford-Voronoi cells incident to v , we find that each Ford-Voronoicell containing v intersects others in -dimensional cells, that are images under SL ( O ) of the fundamental cell Σ O , combinatorially equal to the - duoprism { } × { } . Thegraph in Figure 9 shows the intersection pattern of the H α for α ∈ V .23 23/12/2019 + j1 + ρ j ρ ρ j j + ρ ρ + ρ j 1 + j + ρ j2 ρ + j + j + ρ j21 + ρ + j + ρ + ρ j2 ρ + ρ j21 + ρ + j + ρ j2 ρ + j + ρ j2 z Figure 9: Intersection pattern into -dimensional cells of Ford-Voronoi cells H α for α ∈ V .Thus the number of ( - duoprismatic) -dimensional cells of X O containing v isexactly
90 = (20 × / , one for each edge of this diagram.Consider the elements g ∞ , = (cid:18) ρ ρ − ρ − (cid:19) , g ∞ , = (cid:18) ρ jρ ρ (cid:19) and h ∞ = (cid:18) ρj j (cid:19) in SL ( O ) inducing respectively the homographies z (cid:55)→ ρzρ + 1 , z (cid:55)→ ρzρ − + j and h ∞ ( z ) = − ρjzj . Using the facts that z = j + ρ + ρj and ρj = jρ − , an easy computation shows that theyfix z and ∞ , hence fix v since they preserve the geodesic line between z and ∞ andthe horospheres centered at ∞ . Hence g ∞ , , g ∞ , and h ∞ belong to the stabilizer G v , ∞ of ∞ (or equivalently z ) in the stabilizer of v in SL ( O ) . These three elements actuallygenerate G v , ∞ . Similar computations give that • the group G generated by g ∞ , and g ∞ , is isomorphic to Z / Z × Z / Z , • h ∞ has order and conjugates g ∞ , and g ∞ , , hence each element of the abeliangroup G , to its inverse.Thus the group generated by g ∞ , , g ∞ , and h ∞ is a semidirect product ( Z / Z × Z / Z ) (cid:111)Z / Z with elements. The subgroup G v , ∞ acts transitively on V , : The graph inFigure 10 shows how the points of V , are mapped by g ∞ , (in continuous green) and g ∞ , (in dotted red).24 23/12/2019 j jρjρ + j
11 + ρjρ + ρj ρ Figure 10: Transitive action of G v , ∞ on V , .Since the inversion g conjugates g ∞ , and g ∞ , to their inverses, the group G v , ∞ alsoacts transitively on g ( V , ) . By easy computations, the element g ρ = (cid:18) − (cid:19) , inducingthe homography z (cid:55)→ (1 − z ) − , is an element of the stabilizer of v in SL ( O ) ; it fixes ρ ∈ V , and j + ρj ∈ g ( V , ) , maps ∞ to ∈ V , and ρj ∈ V , to ρj ∈ g ( V , ) , anddoes not fix z . Since G v , ∞ acts transitively on V , and on g ( V , ) , it follows that thestabilizer of v acts transitively on V = V , ∪ g ( V , ) ∪ {∞ , z } . The stabilizer of v in SL ( O ) coincides with the group generated by g ∞ , , g ∞ , , h ∞ and g ρ . It has ×
18 = 360 elements (the number of -cells of the link of v in the tesselation of H R times the order ofthe stabilizer of one -cell, the one corresponding to H ∞ ).The dual tiling of the - -duoprismatic tiling of H is the - -duoprismatic tiling. There-fore, the link of v in ∂ H ∞ (hence in all ∂ H α containing v ) is the -skeleton of the dualof the - duoprism, namely the - duopyramid, whose Schläfli symbol is { } + { } andwhose symmetry group has order × = 72 . The group generated by g ∞ , , g ∞ , and h ∞ is a subgroup of index in the full group of symmetries of the link of v in ∂ H ∞ . Thelink of v in H R is constructed of copies of the - duopyramid, that are glued togetheraccording to the intersection pattern described above, forming the -skeleton of the dual ofthe birectified -simplex . Since the birectified -simplex is called the dodecateron and hastwelve -faces, its dual, which has twenty -faces and does not seem to have a name in theliterature, could be called the icosateron . The full group of symmetries of the dual of theicosateron, whose Coxeter notation is [[4 ]] , has × elements. The stabilizerof v in SL ( O ) is naturally identified with a subgroup of index in [[4 ]] . This concludesthe study of Example 4.5. Remark.
When D A ∈ { , , } , let P O be the hyperbolic -polytope that consists of thepoints in the halfspace H ∞ whose horizontal projection to H is Σ H O . The quotient orbifold SL ( O ) \ H R is obtained from P O by gluing the vertical sides of P O by the translationsin the stabilizer of ∞ in SL ( O ) , and then folding by the action of the stabilizer of Σ O .The quotient space SL ( O ) \ X O obtained by making the above identifications in Σ O is a -dimensional cellular retract of SL ( O ) \ H R that could be used to study the homology of SL ( O ) and PSL ( O ) analogously to the study of the Bianchi groups in [Men] and [SV].25 23/12/2019 Waterworlds
Let A be a definite quaternion algebra over Q and let O be a maximal order in A . Let f be an indefinite integral binary Hamiltonian form over O .The form f defines a function F = F f : P r ( A ) → Q by F ([ x : y ]) = f ( x, y ) n ( O x + O y ) . This definition does not depend on the choice of representatives ( x, y ) ∈ A × A of [ x : y ] ∈ P r ( A ) , and f is uniquely determined by its associated function F . In particular, we maytake x, y ∈ O in order to compute F ([ x : y ]) , so that the numerator of the fraction defining F ([ x : y ]) belongs to Z . Note that SL( O ) acts with finitely many orbits on P r ( A ) , since thenumber of cusps is finite, and that the denominator defining F ([ x : y ]) is invariant under SL( O ) . Therefore there exists N ∈ N − { } such that F has values in N Z , hence the setof values of F is discrete.Note that for every g ∈ SL ( O ) , the function F f ◦ g associated to the form f ◦ g is F ◦ g (where we again denote by g the projective transformation of P r ( A ) induced by g ). Inparticular, F ◦ g = F if g ∈ SU f ( O ) .As in [Con] for integral indefinite binary quadratic forms, we will think of F as a mapwhich associates a rational number to (the interior of) any Ford-Voronoi cell. For instance,if D A = 2 and O is the Hurwitz order, then the values of F on the two Ford-Voronoi cells H ∞ , H containing the fundamental cell Σ O are f (1 , , f (0 , and the values of F on the Ford-Voronoi cells meeting Σ O in a -dimensional cell are f ( u, for u ∈ O × (see Figure11). f ( u, f ( − u, c ( f ) = f (0 , a ( f ) = f (1 , u − u ∞ Σ O Figure 11: Values of F on Ford-Voronoi cells meeting Σ O .Let m be a left fractional ideal of O . For every s ≥ , let ψ F, m ( s ) = Card SU f ( O ) \ (cid:8) ( u, v ) ∈ m × m : | F ( u, v ) | ≤ s, O u + O v = m (cid:9) , which is the number of nonequivalent m -primitive representations by F of rational numbersin N Z with absolute value at most s . We showed in [PP2, Theo. 1] and [PP3, Cor. 5.6]that there exists κ > such that, as s tends to + ∞ , ψ f, m ( s ) = 45 D A Covol(SU f ( O ))2 π ζ (3) ∆( f ) (cid:81) p | D A ( p − s (1 + O( s − κ )) .
26 23/12/2019 emma 5.1.
The function F takes all signs , + , − . Proof.
It takes positive and negative values since f is indefinite. The values of F areactually positive at the points in P r ( A ) in one of the two components of P r ( H ) − C ∞ ( f ) andnegative at the ones in the other component. But contrarily to the cases of integral binaryquadratic and Hermitian forms, all integral binary Hamiltonian forms f over O represent ,since by taking a Z -basis of O , the form f becomes an integral binary quadratic form over Z with variables and all integral binary quadratic forms over Z with at least variablesrepresent , see for instance [Ser1, p. 77] or [Cas, p. 75]. (cid:3) A Ford-Voronoi cell will be called flooded for f if the value of F on its point at infinity is . Lemma 5.1 says that there are always flooded Ford-Voronoi cells. See also [Vul, Cor. 4.8].The flooded Ford-Voronoi cells for f correspond to Conway’s lakes for an isotropic integralindefinite binary quadratic form over Z , see [Con, page 23]. There were only two lakes,whereas there are now countably infinitely many flooded Ford-Voronoi cells for f , one foreach parabolic fixed point of the group of automorphs of f . Example 5.2.
Consider the definite quaternion algebra A with D A = 2 , O the Hurwitzorder and a Hamiltonian form f with a ( f ) = 0 , b = b ( f ) , c = c ( f ) ∈ Z − { } suchthat b does not divide c nor c . Then H ∞ is flooded. Let α = xy − with x ∈ O and y ∈ O − { } relatively prime. If n ( y ) ≤ , then the Ford-Voronoi cell H α is not flooded,since otherwise the equation b tr (¯ x y ) + c n ( y ) = 0 would imply that b divides c or c .If n ( y ) > , then n ( I α ) = n ( O x + O y ) n ( y ) = n ( y ) < . Hence by Proposition 3.4 (2), we have B α ( √ ∩ B ∞ ( √
2) = ∅ . Therefore H α ∩ H ∞ = ∅ by Proposition 4.1 (2). This proves that H ∞ does not meet any other flooded Ford-Voronoi cell. Thus if the hyperbolic -orbifold SU f ( O ) \ C ( f ) has only one cusp, then the flooded Ford-Voronoi cells are pairwise disjoint.We actually do not know when SU f ( O ) \ C ( f ) has only one cusp.We have the following analog of the statement of Conway (loc. cit.) that the values ofthe binary quadratic form along a lake are in an infinite arithmetic progression. Proposition 5.3.
Let α ∈ P r ( A ) be such that the Ford-Voronoi cell H α is flooded for f .If α belongs to the SL ( O ) -orbit of ∞ , let Λ α = O . Otherwise, let Λ α = O ∩ α − O ∩ O α − ∩ α − O α − . Then there exists a finite set of nonconstant affine maps { ϕ j : H → R : j ∈ J (cid:48) } defined over Q such that the set of values of F on the Ford-Voronoi cells meeting H α is (cid:83) j ∈ J (cid:48) ϕ j (Λ α ) . Proof.
For every α ∈ P r ( A ) , let E α = { β ∈ P r ( A ) − { α } : H α ∩ H β (cid:54) = ∅} . Note that E g · α = g · E α for every g ∈ SL ( O ) , by Proposition 4.1 (1).First assume that α belongs to the SL ( O ) -orbit of ∞ . Then up to replacing f by f ◦ g for some g ∈ SL ( O ) such that g · ∞ = α , we may hence assume that α = ∞ .Let a = a ( f ) , b = b ( f ) and c = c ( f ) . Note that H ∞ is flooded for f if and onlyif f (0 ,
1) = 0 , that is, if and only if a = 0 . We then have b (cid:54) = 0 since f is indefinite.Hence F ( E ∞ ) = (cid:8) tr ( u b )+ c n ( I u ) : u ∈ E ∞ (cid:9) . Since the stabilizer of ∞ in SL ( O ) acts withfinitely many orbits on the cells of ∂ H ∞ , its finite index subgroup O acts by translationswith finitely many orbits on E ∞ . Hence there exists a finite subset J (cid:48) of A such that27 23/12/2019 ∞ = J (cid:48) + O . Since I α + o = I α for all α ∈ A and o ∈ O , the result follows with ϕ j : u (cid:55)→ tr ( b ( j + u ))+ c n ( I j ) for all j ∈ J (cid:48) .Assume now that α does not belong to the SL ( O ) -orbit of ∞ , so that in particular α ∈ A −{ } . Let Γ α be the stabilizer of α in SL ( O ) , which acts with finitely many orbitson E α . Let g = (cid:16) α −
11 0 (cid:17) , which belongs to SL ( A ) and whose inverse projectivelymaps α to ∞ . Then (see for instance [PP2, §5]), Λ α is a Z -lattice in H , such that thegroup of unipotent upper triangular matrices with coefficient - in Λ α is a finite indexsubgroup of g − Γ α g . A similar argument concludes. (cid:3) By a projective real hyperplane in ∂ ∞ H R = P r ( H ) = H ∪ {∞} , we mean in what followsthe boundary at infinity of a hyperbolic hyperplane in H R . The ones containing ∞ = [1 : 0] are the union of {∞} with the affine real hyperplanes in H . The ones not containing ∞ are the Euclidean spheres in the affine Euclidean space H . Lemma 5.4.
The form f is uniquely determined by the values of its associated function F at six points in P r ( A ) that do not lie in a projective real hyperplane. Proof.
Let a = a ( f ) , b = b ( f ) and c = c ( f ) . Let first prove that we may assume that thesix points in A ∪ {∞} are ∞ = [1 : 0] , , α = 1 and α , α , α ∈ A − { } .Note that for all x, y ∈ A and g ∈ GL ( A ) , if g , g are the components of the linearselfmap g of A × A , then F f ◦ g ([ x : y ]) = F f ◦ g ([ x : y ]) n ( O g ( x, y ) + O g ( x, y )) n ( O x + O y ) . (11)Given six points in P r ( A ) not in a projective real hyperplane of P r ( H ) , the first three ofthem constitute a projective frame of the projective line P r ( A ) . Hence by the existencepart of the fundamental theorem of projective geometry (see [Ber1, Prop. 4.5.10]), thereexists an element g ∈ GL ( A ) mapping them to ∞ , , . Note that this existence partdoes hold in the noncommutative setting, though the uniqueness part does not. The initialclaim follows by Equation (11).Now, the values of F at the points ∞ , , α , α , α , α give a system of six equationson the unknown a, b, c , of the form a = A , c = A , a + tr b + c = A , tr ( α i b ) = A i +3 for i ∈ { , , } . Thus a and c are uniquely determined, and b belongs to the intersection offour affine real hyperplanes in H orthogonal to α , α , α , α with equations tr ( α i b ) = A (cid:48) i for i ∈ { , , , } . The result follows since if α , α , α , α are linearly independent over R , then for all A (cid:48) , A (cid:48) , A (cid:48) , A (cid:48) ∈ R , such an intersection contains one and only one point of H . (cid:3) Proposition 5.5.
Let v be a vertex of the spine X O . The form f is uniquely determinedby the values of its associated function F on the Ford-Voronoi cells containing v , that is,on the points α ∈ P r ( A ) such that v ∈ H α . Proof.
A dimension count shows that there are at least six Ford-Voronoi cells meeting ateach vertex v of the spine. Their points at infinity cannot all be on the same projectivereal hyperplane P . Otherwise, the intersection of the equidistant hyperbolic hyperplanesbetween the pair of them yielding a -dimensional cell containing v would have dimensionat least : It would contain a germ of the orthogonal through v to the convex hull of P in H R . The result follows by Lemma 5.4. (cid:3)
28 23/12/2019he waterworld of f is W ( f ) = (cid:91) α (cid:54) = β ∈ P r ( A ) , F ( α ) F ( β ) < H α ∩ H β . As F ◦ g = F if g ∈ SU f ( O ) , the waterworld W ( f ) is invariant under the group ofautomorphs SU f ( O ) of f .Since f is always isotropic over A , the arguments of Conway and Bestvina-Savin forthe anisotropic case no longer apply, and the waterworld of f could be empty. We do notknow precisely when the waterworlds are nonempty, and we now study some examples. Example 5.6.
The binary Hamiltonian form f ( u, v ) = tr ( u v ) is indefinite with discrim-inant . The coefficients of f are rational integers so it is integral over any maximal order O of any definite quaternion algebra A over Q . Let us prove that the waterworld W ( f ) isnot empty.It is easy to check that C ∞ ( f ) = { z ∈ H : tr z = 0 } ∪ {∞} . Let a ∈ O be such that tr ( a ) = 1 (which does exist since O is maximal, hence tr : O → Z is onto, see for instancethe proof of Proposition 16 in [ChP]). In particular a (cid:54) = 0 , a (cid:54) = − ¯ a , and a, − ¯ a are in twodifferent components of ∂ ∞ H R − C ∞ ( f ) , so that F ( a ) F ( − ¯ a ) < . Let us prove that H a and H − ¯ a intersect in a -dimensional cell of X O , which thus belongs to W ( f ) . By Proposition4.1 (2), it is sufficient to prove that B a (1) and B − ¯ a (1) meet. By Theorem 3.5 (1), this isequivalent to proving that I a I ¯ a = O ( tr a ) . But this holds since tr a = 1 and I b = O when b ∈ O .Figure 12 illustrates the analogous case of the ocean in H R of the isotropic binaryHermitian form f ( u, v ) = tr ( u v ) considered as an integral form over the Eisenstein integers Z [ i √ ] . The blue hexagons are the components of the ocean of f in the hyperplane C ( f ) = { ( z, t ) ∈ H R : Im z = 0 } which is a copy of the (upper halfplane model of the) realhyperbolic plane.Figure 12: Ocean of Hermitian form f ( u, v ) = tr ( u v ) over Z [ i √ ] .We do not have an example of an empty waterworld and, in fact, it may be thatno such example exists. However, the ocean of the isotropic binary Hamiltonian form f ( u, v ) = tr ( u v ) considered over the Gaussian integers Z [ i ] is empty (see Figure 13). Inorder to prove this, let α ∈ Q ( i ) with tr α (cid:54) = 0 . Note that in the commutative case, n ( I α ) = n ( I − α ) , so that the Euclidean balls B α (1) and B − α (1) have the same radius. Bysymmetry, C ( f ) is the equidistant hyperbolic hyperplane of B α (1) and B − α (1) . Since Z [ i ] is Euclidean, the spine of SL ( Z [ i ]) has only one orbit of -cells (see [BeS]). Hence allthe intersections of the Ford-Voronoi cells are in the orbit of the fundamental cell, and29 23/12/2019herefore, H α and H − α intersect if and only if B α (1) and B − α (1) are tangent, that is, ifand only if B α (1) intersects C ( f ) .Since the hyperbolic -orbifold SL ( Z [ i ]) \ H R has only one cusp, there exists g = (cid:18) a bc d (cid:19) ∈ SL ( Z [ i ]) such that α = g · ∞ = ac − . Since g · ( − c − d ) = ∞ , the point g · ( − c − d,
1) = ( α, n ( c ) ) is the highest point in B α (1) = gB ∞ (1) . Thus the Euclideanradius of B α (1) is n ( c ) . As the Euclidean distance of α from C ∞ ( f ) is | tr α | , this impliesthat B α (1) intersects C ( f ) if and only if (cid:12)(cid:12) tr α (cid:12)(cid:12) ≤ n ( c ) , that is, if and only if tr a ¯ c = ± .This is impossible since the trace of any Gaussian integer is even.Figure 13: Empty ocean of Hermitian form f ( u, v ) = tr ( u v ) over Z [ i ] . Proposition 5.7.
If the union of the flooded Ford-Voronoi cells does not separate H R , andin particular if the flooded Ford-Voronoi cells are pairwise disjoint, then the waterworld of f is nonempty. Proof.
The assumption says that the topological space X = H R − (cid:91) α ∈ P r ( A ) , F ( α )=0 H α is connected. If W ( f ) = ∅ , then X = (cid:16) (cid:91) α ∈ P r ( A ) , F ( α ) < H α (cid:17) ∪ (cid:16) (cid:91) α ∈ P r ( A ) , F ( α ) > H α (cid:17) would be a partition into two nonempty (since f is indefinite) locally finite, hence closed,unions of closed polyhedra, contradicting the connectedness of X . (cid:3) Proposition 5.8.
The quotient SU f ( O ) \ W ( f ) is compact, and the set of flooded Ford-Voronoi cells consists of finitely many SU f ( O ) -orbits. Proof.
The points at infinity of the flooded Ford-Voronoi cells are the parabolic fixedpoints of SL ( O ) contained in C ∞ ( f ) , hence are the parabolic fixed points of the group ofautomorphs SU f ( O ) . Since SU f ( O ) is a lattice in the real hyperbolic -space C ( f ) , thequotient SU f ( O ) \ C ( f ) has only finitely many cusps. This proves the second claim.Let α, β ∈ P r ( A ) be such that F ( α ) F ( β ) < and the intersection H α ∩ H β is nonempty.Then the intersection B α ( √ D A ) ∩ B β ( √ D A ) is nonempty by Proposition 4.1 (2), hence thehyperbolic distance between the horoballs B α (1) and B β (1) is at most ln D A . By Lemma3.6, we hence have n ( α − β ) n ( I α I β ) ≤ D A .30 23/12/2019et a = a ( f ) , b = b ( f ) , c = c ( f ) and ∆ = ∆( f ) . Write α = [ x : y ] and β = [ u : v ] with x, y, u, v ∈ O and y, v ∈ Z . Note that (cid:18) x uy v (cid:19) ∗ (cid:18) a bb c (cid:19) (cid:18) x uy v (cid:19) = (cid:18) f ( x, y ) zz f ( u, v ) (cid:19) , for some z ∈ O . Since y, v ∈ R , an easy computation of Dieudonné determinants thusgives (cid:12)(cid:12) n ( z ) − f ( x, y ) f ( u, v ) (cid:12)(cid:12) = n ( xv − uy ) ∆ . Hence ≤ − f ( x, y ) f ( u, v ) ≤ n ( z ) − f ( x, y ) f ( u, v ) = n ( xv − uy ) ∆ and ≤ − F ( α ) F ( β ) = − f ( x, y ) f ( u, v ) n ( O x + O y ) n ( O u + O v ) ≤ n ( α − β ) n ( I α ) n ( I β ) ∆ ≤ D A ∆ . Since the set of values of F is discrete in R , this implies that F takes only finitely manyvalues on the Ford-Voronoi cells that intersect W ( f ) .Given any vertex v ∈ W ( f ) , for every g ∈ SL ( O ) , if F ( α ) = F ( g · α ) for all α ∈ A such that the Ford-Voronoi cell H α contains v , then f = f ◦ g by Proposition 5.5. Sincethere are only finitely many orbits of SL ( O ) on the vertices of the spine X O and since F takes only finitely many values on the Ford-Voronoi cells meeting the waterworld W ( f ) ,this implies that SU f ( O ) has only finitely many orbits of vertices in W ( f ) . The resultfollows. (cid:3) Note that there exist a positive constant and finitely many pairs { α, β } in A such that,for all indefinite integral binary Hamiltonian forms f over O up to the action of SL ( O ) ,the distance between H α and H β is at most this constant and F ( α ) F ( β ) < . Thisfollows, even if the waterworld W ( f ) could be empty, from the fact that the flooded Ford-Voronoi cells only have their points at infinity on the -sphere C ∞ ( f ) in P r ( H ) , and by thecocompactness of the action of SL ( O ) on its spine X O . The above arguments hence allowto give another proof of Corollary 25 in [PP2], saying that the number of SL ( O ) -orbits inthe set of indefinite integral binary Hamiltonian forms over O with given discriminant isfinite.We introduce two variants of W ( f ) . The sourced waterworld W + ( f ) of f is the unionof its waterworld and of its flooded Ford-Voronoi cells W + ( f ) = W ( f ) ∪ (cid:91) α ∈ P r ( A ) , F ( α )=0 H α . The coned-off waterworld
C W ( f ) of f is obtained from W ( f ) by adding geodesic rays fromits boundary points to the points at infinity of the corresponding flooded Ford-Voronoi cells C W ( f ) = W ( f ) ∪ (cid:91) α ∈ P r ( A ) , x ∈ W ( f ) ∩ H α : F ( α )=0 [ x, α [ . Both the sourced waterworld W + ( f ) and the coned-off waterworld C W ( f ) of f are invariantunder the group of automorphs SU f ( O ) of f .Before stating the main result of this paper, we give two lemmas and refer to Section6 of [BeS] for the proofs31 23/12/2019 emma 5.9. Let
P, P (cid:48) be hyperbolic hyperplanes in H n R that do not intersect perpendic-ularly. Then the closest point mapping from P to P (cid:48) is a homeomorphism onto a convexopen subset of P (cid:48) , which maps any hyperbolic polyhedron of P to a hyperbolic polyhedronof P (cid:48) . (cid:3) Lemma 5.10.
Let f be an indefinite integral binary Hamiltonian form over O . If (cid:96) is ageodesic line in H R that is perpendicular to the halfplane C ( f ) , oriented such that (cid:96) ( ±∞ ) ∈{ [ x : y ] ∈ P r ( H ) : ± f ( x, y ) > } , if (cid:96) meets transversally at a point z the interior of a -dimensional cell H α − ∩ H α + of X O with F ( α − ) ≤ and F ( α + ) ≥ and ( F ( α − ) , F ( α + )) (cid:54) =(0 , , then a germ of (cid:96) at z pointing towards (cid:96) ( ±∞ ) is contained in H α ± . Proof.
The proof of Claim 2 page 12 of [BeS] applies. (cid:3)
The following result implies Theorem 1.2 in the Introduction.
Theorem 5.11.
Let A be a definite quaternion algebra over Q and let O be a maximalorder in A . For every indefinite integral binary Hamiltonian form f over O , the closestpoint mapping π : W + ( f ) → C ( f ) is a proper SU f ( O ) -equivariant homotopy equivalence.If the flooded Ford-Voronoi cells for f are pairwise disjoint, then the closest point mapping π : C W ( f ) → C ( f ) is a SU f ( O ) -equivariant homeomorphism and its restriction to thewaterworld W ( f ) is a SU f ( O ) -equivariant homeomorphism onto a contractible -manifoldwith a polyhedral boundary component homeomorphic to R contained in every floodedFord-Voronoi cell. Proof.
The SU f ( O ) -equivariance properties are immediate. We will subdivide this proofinto several steps. Unless otherwise stated, polyhedra are compact and convex. Claim 1.
The closest point mapping π : W + ( f ) → C ( f ) has the following properties.(1) The restriction of π to any cell of W ( f ) is a homeomorphism onto its image, whichis a hyperbolic polyhedron in the hyperbolic hyperplane C ( f ) .(2) The restriction of π to any flooded Ford-Voronoi cell H α of f is a proper map ontoa noncompact convex hyperbolic polyhedron in C ( f ) containing B α (1) ∩ C ( f ) andcontained in B α ( √ D A ) ∩ C ( f ) .(3) If the flooded Ford-Voronoi cells for f are pairwise disjoint, then the restriction of π to any cell in the boundary of a flooded Ford-Voronoi cells for f is a homeomorphismonto its image, which is a hyperbolic polyhedron in the hyperbolic hyperplane C ( f ) . Proof. (1) Any -dimensional cell, hence any cell, of W ( f ) is a hyperbolic polyhedron inthe equidistant hyperbolic hyperplane S α, β = { x ∈ H R : d α ( x ) = d β ( x ) } for some α (cid:54) = β in P r ( A ) with F ( α ) F ( β ) < . Note that S α, β is not perpendicular to C ( f ) , otherwise α and β , which are the points at infinity of a geodesic line perpendicularto S α, β , would belong to the closure of the same component of ∂ ∞ H R − C ∞ ( f ) , whichcontradicts the fact that F ( α ) F ( β ) < . Hence Assertion (1) of Claim 1 follows fromLemma 5.9.(2) The closest point mapping from a horoball H to a hyperbolic hyperplane P passingthrough the point at infinity of H is a proper map (since the intersection of H with any32 23/12/2019eodesic line not passing through its point at infinity is compact), whose image is H ∩ P ,and which maps the geodesic segment between two points to the geodesic segment betweentheir images. Assertion (2) of Claim 1 hence follows from Proposition 4.1 (2).(3) If the flooded Ford-Voronoi cells for f are pairwise disjoint, any -dimensional cell, henceany cell, in the boundary of a flooded Ford-Voronoi cell for f is a hyperbolic polyhedronin the hyperbolic hyperplane S α, β for some α (cid:54) = β in P r ( A ) with F ( α ) = 0 and F ( β ) (cid:54) = 0 .Note that S α, β is again not perpendicular to C ( f ) , otherwise α and β would both belongto C ∞ ( f ) , and the Ford-Voronoi cells H α and H β would both be flooded for f and notdisjoint. The last assertion of Claim 1 follows. (cid:3) Claim 2.
We have the following parity properties.(1) Any -dimensional cell σ of W ( f ) not contained in a flooded Ford-Voronoi cell for f belongs to an even number of -dimensional cells of W ( f ) .(2) If the flooded Ford-Voronoi cells for f are pairwise disjoint, then any -dimensionalcell σ (cid:48) of W ( f ) contained in a flooded Ford-Voronoi cell for f belongs to an oddnumber of -dimensional cells of W ( f ) . Proof. (1) Since σ has codimension , the link of σ in the Ford-Voronoi cellulation ofthe manifold H R is a circle. Considering its intersection with the -dimensional cells, thiscircle subdivides into closed intervals with disjoint interiors, each one of them contained insome Ford-Voronoi cell. By the assumption on σ , these Ford-Voronoi cells are nonflooded.Hence the sign of F on each one of them is either + or − . In such a cyclic arrangement ofsigns, the number of sign changes is even. Assertion (1) follows.(2) Similarly, the link of σ (cid:48) is subdivided into at least closed intervals with disjointinteriors carrying a sign + , , − . By the assumptions, exactly one of them, denoted by I ,belongs to a flooded Ford-Voronoi cell H α for some α ∈ P r ( A ) , that is, carries the sign . Assume for a contradiction that the two intervals adjacent to I carry the same sign.Let β , β ∈ P r ( A ) be such that H α ∩ H β and H α ∩ H β are the -dimensional cellscorresponding to the endpoints of I . Note that the points at + ∞ of the geodesic linesstarting from a given point α of C ∞ ( f ) , passing through a geodesic line both of whoseendpoints β , β are contained in the same component C of ∂ ∞ H R − C ∞ ( f ) also belong to C . Hence all intervals in the link of σ (cid:48) carry the same sign, which contradicts the fact that σ (cid:48) belongs to W ( f ) . As for σ , this proves that the number of sign changes between + and − in the link of σ (cid:48) is odd. (cid:3) Claim 3. If σ and τ are distinct -dimensional cells of W ( f ) or flooded Ford-Voronoi cellsfor f , then π ( σ ) and π ( τ ) have disjoint interiors. Proof.
Note that no -dimensional cell of W ( f ) is contained in a flooded Ford-Voronoicell for f .For a contradiction, assume that a point p ∈ C ( f ) is contained in the interior of both π ( σ ) and π ( τ ) and, up to moving it a little bit, is not in the (measure ) image by π ofthe codimension skeleton of X O . Let (cid:96) be the geodesic line through p perpendicular to C ( f ) , meeting σ and τ at interior points x and y respectively. Since the cell complex X O is locally finite, we may assume that the geodesic segment [ x, y ] does not meet any other -dimensional cell of W ( f ) or flooded Ford-Voronoi cell for f than σ and τ .33 23/12/2019ssume for a contradiction that [ x, y ] is contained in σ ∪ τ . Then σ and τ are floodedFord-Voronoi cells, meeting in a -dimensional cell C , which is crossed transversally by [ x, y ] since (cid:96) does not meet the -skeleton of X O . Since σ, τ are flooded, their points at infinity α, β ∈ P r ( A ) belong to C ∞ ( f ) . Hence the hyperbolic hyperplane S α,β equidistant to α and β , which contains σ , is perpendicular to C ( f ) . In particular, (cid:96) , which is perpendicularto C ( f ) , is contained in the closure of one of the two connected component of H R − S α,β .This contradicts the fact that (cid:96) meets transversally C .Hence [ x, y ] is not contained in σ ∪ τ . Let ] x (cid:48) , y (cid:48) [ = [ x, y ] − ( σ ∪ τ ) ∩ [ x, y ] with x, x (cid:48) , y (cid:48) , y in this order on [ x, y ] , so that [ x (cid:48) , y (cid:48) ] is contained in a Ford-Voronoi cell H α forsome α ∈ P r ( A ) . Let σ (cid:48) and τ (cid:48) be the -dimensional cells of X O containing x (cid:48) and y (cid:48) respectively (note that for instance x = x (cid:48) and σ = σ (cid:48) if σ is a -dimensional cell of W ( f ) ,but x (cid:54) = x (cid:48) if σ is a flooded Ford-Voronoi cell).Now Lemma 5.10 implies that, since the two germs of the segment [ x (cid:48) , y (cid:48) ] at its end-points have opposite direction, the sign of F ( α ) should be both positive and negative, acontradiction. (cid:3) Claim 4.
The -dimensional cells of the waterworld satisfy the following properties.(1) No -dimensional cell of W ( f ) is contained in two distinct flooded Ford-Voronoi cells.(2) Any -dimensional cell σ of W ( f ) not contained in a flooded Ford-Voronoi cell for f belongs to exactly two -dimensional cells τ and τ (cid:48) of W ( f ) , and π embeds theirunion.(3) Any -dimensional cell σ of W ( f ) contained in a flooded Ford-Voronoi cell H α for f belongs to exactly one -dimensional cell τ of W ( f ) , and π embeds the union of τ and τ (cid:48) = (cid:83) x ∈ σ [ x, α [ . Proof. (1) Assume for a contradiction that σ is a -dimensional cell of W ( f ) contained inthe flooded Ford-Voronoi cells H α and H β with α (cid:54) = β in P r ( A ) . Let τ be a -dimensionalcell of W ( f ) containing σ . Then the interiors of the images by π of τ and either H α or H β are not disjoint, which contradicts Claim 3.Let us prove Assertions (2) and (3). Three n -dimensional polytopes in H n R having acommon codimension face cannot have pairwise disjoint interiors, so that the claims onthe number of -dimensional cells of W ( f ) containing σ follows from Claim 3. Since thepolyhedra π ( τ ) and π ( τ (cid:48) ) are convex, the result follows. (cid:3) Claim 5.
The -dimensional cells of the waterworld satisfy the following properties.(1) For every -dimensional cell σ of W ( f ) not contained in a flooded Ford-Voronoi cellfor f , the link of σ in W ( f ) is a circle and the union of the -dimensional cells of W ( f ) containing σ embeds in C ( f ) by π .(2) If the flooded Ford-Voronoi cells for f are pairwise disjoint, for every -dimensionalcell σ (cid:48) of W ( f ) contained in a flooded Ford-Voronoi cell H α , the link of σ (cid:48) in W ( f ) isan interval and the union of the -dimensional cells of W ( f ) containing σ (cid:48) and of thegeodesic rays [ x, α [ for x in the two -dimensional cells of W ( f ) ∩ ∂ H α containing σ (cid:48) embeds in C ( f ) by π . Proof. (1) By Claim 4, the link Lk ( σ ) of σ in W ( f ) is a disjoint union of circles. Eachcomponent of Lk ( σ ) corresponds to a finite set of -dimensional cells cyclically arranged34 23/12/2019round σ . By Claim 4 again, their images by π are not folded, hence are cyclically arrangedaround π ( σ ) . If Lk ( σ ) was not connected, the image of two -dimensional cells of W ( f ) by π would have intersecting interiors, contradicting Claim 3.(2) An analogous proof gives that the link of σ (cid:48) in C W ( f ) is a circle. (cid:3) Claim 6.
The -dimensional cells of the waterworld satisfy the following properties.(1) For every -dimensional cell σ of W ( f ) not contained in a flooded Ford-Voronoi cellfor f , the link of σ in W ( f ) is a -sphere and the union of the -dimensional cells of W ( f ) containing σ embeds in C ( f ) by π .(2) If the flooded Ford-Voronoi cells for f are pairwise disjoint, for every -dimensionalcell σ (cid:48) of W ( f ) contained in a flooded Ford-Voronoi cell H α , the link of σ (cid:48) in W ( f ) is a -disc and the union of the -dimensional cells of W ( f ) containing σ (cid:48) and of thegeodesic rays [ x, α [ for x in any -cell of W ( f ) ∩ ∂ H α containing σ (cid:48) embeds in C ( f ) by π . Proof. (1) By Claim 5, the links of the vertices of the link Lk ( σ ) of σ in W ( f ) are circles,hence Lk ( σ ) is a compact surface, mapping locally homeomorphically to Lk ( π ( σ )) by π ,which is a -sphere. Hence Lk ( π ( σ )) is a union of -spheres, again with only one of themby Claim 3.(2) The proof that the link of σ (cid:48) in C W ( f ) is a -sphere is similar. (cid:3) Claim 7.
The vertices of the waterworld satisfy the following properties.(1) For every vertex v of W ( f ) not contained in a flooded Ford-Voronoi cell for f , thelink of v in W ( f ) is a -sphere and the union of the -dimensional cells of W ( f ) containing v embeds in C ( f ) by π .(2) If the flooded Ford-Voronoi cells for f are pairwise disjoint, for every vertex v (cid:48) of W ( f ) contained in a flooded Ford-Voronoi cell H α , the link of v (cid:48) in W ( f ) is a -discand the union of the -dimensional cells of W ( f ) containing v (cid:48) and of the geodesicrays [ x, α [ for x in any -cell of W ( f ) ∩ H α containing v (cid:48) embeds in C ( f ) by π . Proof.
The proof is similar to the previous one. (cid:3)
Now, the properness of π : W + ( f ) → C ( f ) follows from the fact that π is SU f ( O ) -equivariant, that SU f ( O ) acts cocompactly on W ( f ) and with finitely many orbits onthe set of flooded Ford-Voronoi cells by Proposition 5.8, and from its properness whenrestricted to each flooded Ford-Voronoi cell (see Claim 1).Claim 7 proves that when the flooded Ford-Voronoi cells for f are pairwise disjoint,the map π : C W ( f ) → C ( f ) is a proper local homeomorphism betwen locally compactspaces, hence is a covering map. Since C ( f ) is simply connected, π is hence a homeomor-phism on each of the connected components of C W ( f ) . But since π is injective outsidethe codimension skeleton by Claim 3, it follows that C W ( f ) is connected and π is ahomeomorphism. This concludes the proof of Theorem 5.11. (cid:3)
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An algebraic description of the distance to the cusps
Let A be a definite quaternion algebra over Q and let O be a maximal order in A . In thisindependent appendix, following Mendoza [Men] in the Hermitian case, we give an algebraicdescription of the distance functions d α to the rational points at infinity α ∈ P r ( A ) , definedjust before Proposition 3.3.An O -flag is a right O -submodule L of the right O -module O × O , with rank one (thatis, LA is a line in the A -vector space A × A ), such that the quotient ( O × O ) /L has notorsion. We denote by F O the set of O -flags.For all right O -submodules M of A × A and v ∈ A × A − { } , let us define M v = { x ∈ A : vx ∈ M } . Note that for every λ ∈ A − { } , we immediately have λM vλ = M v . (12) Example A.1.
Recall that the inverse I − of a left fractional ideal I of O is the rightfractional ideal of O I − = { x ∈ A : IxI ⊂ I } . It is well known and easy to check that for every a, b ∈ O , if ab (cid:54) = 0 , then ( O a + O b ) − = a − O ∩ b − O . (13)We claim that if v = ( a, b ) , then ( O × O ) v = ( O a + O b ) − . (14)Indeed, if a, b (cid:54) = 0 , then by Equation (13) ( O × O ) v = { x ∈ A : ( ax, bx ) ∈ O × O } = a − O ∩ b − O = ( O a + O b ) − . The result is immediate if a = 0 or b = 0 . Proposition A.2. (1) For all right O -submodule M of A × A and v ∈ A × A − { } , thesubset M v of A is a right fractional ideal of O .(2) For every v ∈ A × A − { } , the subset v ( O × O ) v of O × O is an O -flag.(3) For all O -flags L and all v ∈ L − { } , we have L = v ( O × O ) v . (4) The map SL ( A ) × F O → F O defined by ( g, L ) (cid:55)→ ( gv )( O × O ) gv for any v ∈ L − { } is an action on the set F O of O -flags of the group SL ( A ) .(5) The map Θ (cid:48) : P r ( A ) → F O defined by [ a : b ] (cid:55)→ ( a, b )( O × O ) ( a,b ) is a SL ( A ) -equivariantbijection.
36 23/12/2019 roof. (1) This follows immediately from the fact that M is stable by addition and bymultiplications on the right by the elements of O .(2) Let L = v ( O × O ) v ⊂ vA . Then L is contained in O × O by the definition of ( O × O ) v and is a right O -submodule of O × O by Assertion (1). Since v (cid:54) = 0 , note that ( O × O ) v is a nonzero right fractional ideal, so that L (cid:54) = 0 and L has rank one.Assume that w ∈ O × O has its image in ( O × O ) /L which is torsion. Then there exists y ∈ O − { } and x ∈ A such that wy = vx . Hence w = vxy − . Since w ∈ O × O , thisimplies that xy − ∈ ( O × O ) v , so that w ∈ L , and the image of w in ( O × O ) /L is zero.(3) As L has rank and v ∈ L − { } , we have L ⊂ vA ∩ ( O × O ) = v ( O × O ) v .Conversely, for every x ∈ ( O × O ) v so that vx ∈ O × O , let us prove that vx ∈ L . Since x ∈ A which is the field of fractions of O , there exists y ∈ O such that xy ∈ O . Hence ( vx ) y = v ( xy ) belongs to L , since v ∈ L and L is a right O -module. In particular, theimage of vx in ( O × O ) /L is torsion. Since L is an O -flag, this implies that this image iszero, as wanted. This proves that v ( O × O ) v is contained in L , hence is equal to L by theprevious inclusion.(4) Let us prove that this map is well defined. If v, w ∈ L − { } , since L has rank one,there exists x ∈ A − { } such that w = vx , thus, for every g ∈ SL ( A ) , by the linearity onthe right of g and by Equation (12), we have ( gw )( O × O ) gw = ( gv ) x ( O × O ) ( gv ) x = ( gv )( O × O ) gv . The fact that this map is an action is then immediate: for all g, g (cid:48) ∈ SL ( A ) and L ∈ F O ,let v ∈ L − { } and λ ∈ A be such that gvλ ∈ ( gv )( O × O ) gv − { } ; then using twiceEquation (12) and the linearity, we have g (cid:48) ( gL ) = g (cid:48) (cid:0) gv ( O × O ) gv (cid:1) = g (cid:48) (cid:0) gvλ ( O × O ) gvλ (cid:1) = g (cid:48) ( gvλ )( O × O ) g (cid:48) ( gvλ ) = ( g (cid:48) g ) vλ ( O × O ) ( g (cid:48) g ) vλ = ( g (cid:48) g ) v ( O × O ) v = ( g (cid:48) g ) L . (5) For every α = [ a : b ] ∈ P r ( A ) , the subset ( a, b )( O × O ) ( a,b ) , which is an O -flag byAssertion (2), does not depend on the choice of homogeneous coordinates of α by Equation(12). Hence the map Θ (cid:48) is well defined, and equivariant by the definition of the action of SL ( A ) on F O .The fact that Θ (cid:48) is onto follows from Assertion (3). Clearly, it is one-to-one since if ( a, b )( O × O ) ( a,b ) = ( c, d )( O × O ) ( c,d ) , then there is λ ∈ A − { } such that ( a, b ) = ( c, d ) λ . (cid:3) Let f : H × H → R be a positive definite binary Hamiltonian form and let L be a rankone right O -submodule of O × O . Then L is a rank free Z -submodule of H × H , and wedenote by (cid:104) L (cid:105) R the -dimensional real vector subspace of H × H generated by L , endowedwith the restriction of the scalar product (cid:104)· , ·(cid:105) f on H × H defined by f , hence with theinduced volume form. Recall that for all z, z (cid:48) ∈ H × H , we have (cid:104) z, z (cid:48) (cid:105) f = 12 (cid:0) f ( z + z (cid:48) ) − f ( z ) − f ( z (cid:48) ) (cid:1) . (15)We define the covolume of L for f as Covol f L = Vol( (cid:104) L (cid:105) R /L ) .
37 23/12/2019ecall that if G = ( (cid:104) e i , e j (cid:105) f ) ≤ i,j ≤ is the Gram matrix of a Z -basis ( e , e , e , e ) of L forthe scalar product (cid:104)· , ·(cid:105) f , then Covol f L = (det G ) . (16)See for instance [Ber2, Vol 2, prop. 8.11.6]. Theorem A.3.
For all x ∈ H R and α ∈ P r ( A ) , we have d α ( x ) = 2 √ D A (cid:0) Covol Θ( x ) Θ (cid:48) ( α ) (cid:1) . Proof.
Fix a, b ∈ O such that α = [ a : b ] . Let f = Θ( x ) , L = Θ (cid:48) ( α ) = ( a, b )( O × O ) ( a,b ) and L (cid:48) = ( a, b ) O . Since a, b ∈ O , we have O ⊂ ( O × O ) ( a,b ) , hence L (cid:48) is a finite index Z -submodule in L . Furthermore, by Equation (14) and the relation (see Equation (2))between the norm and reduced norm of a left integral ideal of O , we have [ L : L (cid:48) ] = [( O × O ) ( a,b ) : O ] = [( O a + O b ) − : O ] = [ O : O a + O b ]= n ( O a + O b ) . (17)Let ( x , x , x , x ) be a Z -basis of O , so that ( ( a, b ) x i ) ≤ i ≤ is a Z -basis of L (cid:48) . UsingEquation (15) and the fact that f (( u, v ) λ ) = n ( λ ) f ( u, v ) for all u, v, λ ∈ H , we have for ≤ i, j ≤ , (cid:104) ( a, b ) x i , ( a, b ) x j (cid:105) f = 12 (cid:0) f (cid:0) ( a, b )( x i + x j ) (cid:1) − f (( a, b ) x i ) − f (( a, b ) x j ) (cid:1) = f ( a, b )2 ( n ( x i + x j ) − n ( x i ) − n ( x j )) = f ( a, b )2 tr ( x i x j ) . Note that ( u, v ) (cid:55)→ tr ( u v ) is the standard Euclidean scalar product on H (makingthe standard basis (1 , i, j, k ) orthonormal), hence (cid:0) tr ( x i x j ) (cid:1) ≤ i,j ≤ is the Gram matrixof the Z -lattice O in the Euclidean space H . Therefore, by Equation (16) and by [KO,Lem. 5.5], we have (cid:0) det (cid:0) tr ( x i x j ) (cid:1) ≤ i,j ≤ (cid:1) = (2 ) Vol( H / O ) = 4 D A D A . (18)Thus using Equations (16), (17) and (18), we have Covol f ( L ) = 1[ L : L (cid:48) ] Covol f ( L (cid:48) ) = 1[ L : L (cid:48) ] (cid:0) det (cid:0) (cid:104) ( a, b ) x i , ( a, b ) x j (cid:105) f (cid:1) ≤ i,j ≤ (cid:1) = 1[ L : L (cid:48) ] (cid:0) f ( a, b )2 (cid:1) (cid:0) det (cid:0) tr ( x i x j ) (cid:1) ≤ i,j ≤ (cid:1) = D A f ( a, b ) n ( O a + O b ) . By Proposition 3.3 (2), this proves Theorem A.3. (cid:3)
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