aa r X i v : . [ m a t h . M G ] J a n ON THE COMMENSURABILITY OF HYPERBOLIC COXETERGROUPS
EDOARDO DOTTI
Abstract.
In this paper we study the commensurability of hyperbolic Coxeter groupsof finite covolume, providing three necessary conditions for commensurability. More-over we tackle different topics around the field of definition of a hyperbolic Coxetergroup: which possible fields can arise, how this field determines a range of possibledihedral angles of a Coxeter polyhedron and we provide two new sets of generatorsfor quasi-arithmetic groups. This work is a concise version of chapters 4 and 5 of theauthor’s Ph.D. thesis [8].
Keywords : Hyperbolic space, commensurability, Coxeter group, field of definition.
MSC : 20F55, 22E40, 11R21. 1.
Introduction
For n ≥
2, let H n be the real hyperbolic space of dimension n and denote by Isom( H n )its isometry group. Consider a space form H n / Γ, where Γ is a discrete subgroup ofIsom( H n ). Two such space forms are commensurable if they admit a common finite-sheeted cover. We are interested in distinguishing hyperbolic space forms up to commen-surability.The situation is well understood in dimensions two and three. For n = 3 the groupIsom + ( H ) of orientation preserving isometries can be identified with the group PSL(2 , C ).Due to the work of Maclachlan and Reid [26] there are two powerful commensurabilityinvariants for Kleinian groups in PSL(2 , C ), the invariant trace field and invariant quater-nion algebra , which form a complete set of invariants for arithmetic Kleinian groups.In higher dimensions the situation needs more investigation. When dealing with arith-metic (of the simplest type) hyperbolic lattices, Gromov and Piatetski-Shapiro [12] pro-vide a complete commensurability criterion. Consider an arithmetic lattice with its asso-ciated totally real field and quadratic form. Then, their commensurability criterion statesthat two such lattices are commensurable if and only if the two associated fields coincideand the two forms are similar over their field.However, in the non-arithmetic context, no general commensurability criterion is knownup to date.In this paper we study the problem of commensurability of hyperbolic Coxeter groupsof finite covolume. These are discrete subgroups of Isom( H n ) generated by finitely manyreflections in the bounding hyperplanes of Coxeter polyhedra, which are polyhedra whose angles are integral submultiples of π . We always suppose that the volume of the Coxeterpolyhedra are finite. Following the work of Vinberg in [35], we shall associate a field ofcycles and a quadratic form to every hyperbolic Coxeter group: the Vinberg field andthe
Vinberg form . Inspired by the result of Gromov and Piatetski-Shapiro, we prove thefollowing.
Theorem.
Let Γ and Γ be two commensurable cofinite hyperbolic Coxeter groupsacting on H n , n ≥
2. Then their Vinberg fields coincide and the two associated Vinbergforms are similar over this field.We are then able to refine the previous theorem by associating to a Coxeter groupas above a ring, the
Vinberg ring , and show that this ring is also a commensurabilityinvariant.After that we focus on which field can arise as the Vinberg field of a quasi-arithmetichyperbolic Coxeter group and then we study how the extension degree of the Vinbergfield effects the possible angles of its Coxeter polyhedron. The paper concludes with twonew sets of generators for the Vinberg field of a quasi-arithmetic Coxeter group. Specif-ically, we shall see that the Vinberg field of such a Coxeter group is generated by thecoefficients of the characteristic polynomial of its Gram matrix on one side and by the co-efficients of the characteristic polynomial of any Coxeter transformation on the other side.The paper is structured as follows. In Section 2 we present all the theory needed forthe rest of the paper such as hyperbolic Coxeter groups and commensurability. In Section3 we prove the Theorem above, the commensurability property of the Vinberg ring andwe discuss the similarity classification of Vinberg forms. In the last section we study theVinberg field and provide new sets of generators as mentioned above. All the results willbe supported by examples.This work is a concise version of chapters 4 and 5 of the author’s Ph.D. thesis [8], andthe proofs in this work are a direct adaptation from those in [8].
Acknowledgments.
The author would like to thank Prof. Dr. Ruth Kellerhals for thesupport, valuable discussions and helpful suggestions. This work was supported by theSwiss National Science Foundation, projects 200020 156104 and 200021 172583.2.
Preliminaries
Hyperbolic space, Coxeter polyhedra and Coxeter groups.
Let n ≥ H n the real hyperbolic space of dimension n . We use here the vector spacemodel , or hyperboloid model . Equip R n +1 with the Lorentzian product defined as h x, y i = n X i =1 x i y i − x n +1 y n +1 . The hyperboloid model for H n is then given by the set H n := { x ∈ R n +1 | || x || = h x, x i = − , x n +1 > } N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 3 with metric d H n ( x, y ) = arcosh( −h x, y i ) for all x, y ∈ H n . The space R n +1 equipped with the Lorentzian product is denoted by R n, . Its associ-ated quadratic form, the Lorentzian form , will be denoted by q − ( x ) := h x, x i .The group of isometries Isom( H n ) is the Lie group of positive Lorentzian matrices (1) O + ( n,
1) = (cid:8) A ∈ Mat( n + 1 , R ) | A T JA = J, [ A ] n +1 ,n +1 > (cid:9) , where J = diag(1 , . . . , , −
1) is the diagonal matrix which represents the Lorentzian form.
Remark 2.1.
The group O + ( n,
1) is not an algebraic group. However, one can project H n to the open unit ball and consider its projective model K n . A very important aspectof K n is that its isometries form an algebraic group. Consider the group of all matriceswhich preserve the Lorentzian form (cid:8) A ∈ Mat( n + 1 , R ) | A T JA = J (cid:9) = O( n, K n ) ∼ = O( n, / {± I } =: PO( n, . The fact that Isom( K n ) is an algebraic group will be exploited in Section 3.1 (see Remark3.4).Each hyperbolic hyperplane H e = e ⊥ is given as the orthogonal complement of a vector e ∈ R n +1 of Lorentzian norm 1, that is, H e = { x ∈ R n +1 | h x, e i = 0 } .A hyperplane H e divides H n into two half-spaces H − e = { x ∈ H n | h x, e i ≤ } and H + e = { x ∈ H n | h x, e i ≥ } such that H − e ∩ H + e = H e . A (convex) polyhedron P ⊂ H n is the intersection with non-empty interior of finitely many half-spaces, that is, P = N \ i =1 H − e i ,N ≥ n + 1, where the unit vector e i normal to the hyperplane H e i is pointing outwardsof P . If N = n + 1, then P is called an n -simplex .Particularly, a Coxeter polyhedron is a polyhedron all of whose angles between itsbounding hyperplanes are either zero or sub-multiples of π , hence of the form πk for k ∈ N , k ≥ H e in H n . A reflection with respect to the hyperplane H e is theapplication s e = s H e : H n → H n defined as s e ( x ) = x − h x, e i e and satisfying s e = 1.Let Γ = h s e , . . . , s e N i < Isom( H n ) be the discrete group generated by the reflectionsin the hyperplanes bounding a Coxeter polyhedron P = T Ni =1 H − e i in H n . If two hyper-planes H e i and H e j in the boundary of P intersect under an angle π/m ij , m ij ≥
2, then( s e i s e j ) m ij = 1 in Γ. If H e i and H e j are parallel or ultraparallel, s e i s e j is of infiniteorder in Γ. In this way, Γ represents an abstract Coxeter group. The group Γ is called a hyperbolic Coxeter group . The number of its generating reflections N is called the rankof Γ.In the sequel hyperbolic Coxeter groups will always be assumed to be cofinite , that is,the associated Coxeter polyhedron P has finite volume. A hyperbolic Coxeter group Γ issaid to be cocompact if P is compact. E. DOTTI
A hyperbolic Coxeter group and its Coxeter polyhedron can be most convenientlydescribed by means of its Gram matrix and its Coxeter graph as follows.For Γ < Isom( H n ) a Coxeter group of rank N with Coxeter polyhedron P = T Ni =1 H − e i , N ≥ n + 1, the Gram matrix associated to P and to Γ is the real symmetric matrix G := G ( P ) = G (Γ) = ( g ij ) ≤ i,j ≤ N with coefficients g ij = h e i , e j i . The Gram matrix G of a Coxeter group Γ is unique up to enumeration of the hyper-planes and has signature ( n,
1) (see [2, Chapter 6]). Moreover, a cycle (or cyclic product )of G is defined as g i i g i i . . . g i l − i l g i l i for any { i , i , . . . , i l } ⊂ { , , . . . , m } . A cycle is called simple if the indices i j in thecycle are all distinct.The Coxeter graph of Γ is the graph with N vertices for which the vertex i correspondsto the hyperplane H e i . Between two vertices i and j we have:i) an edge if the angle between H e i and H e j is π/k , k ≥
3. If k ≥ k ; if k = 3 the label is omitted;ii) an edge labelled with ∞ if H e i and H e j are parallel;iii) a dotted edge if H e i and H e j are ultraparallel. The dotted edge is labelled with thehyperbolic cosine of the length l = d H n ( H e i , H e j ) of their common perpendicular.2.2. Commensurability and arithmeticity.
Let H be a group. Two subgroups H , H ⊂ H are commensurable (in the wide sense) if and only if there exists an element h ∈ H such that H ∩ h − H h has finite index in both H and h − H h .This notion defines an equivalence relation. In our context, the group H will beIsom( H n ). Stable under commensurability are some properties of subgroups of Isom( H n )such as discreteness, cofiniteness, cocompactness and arithmeticity. This latter notioncan be further refined by splitting discrete subgroups in Isom( H n ) into three categories: arithmetic , quasi-arithmetic and nq-arithmetic .More precisely, let K ⊂ R be a totally real number field and let V be a vector space ofdimension n + 1 over K endowed with a quadratic form q of signature ( n, V, q ). Moreover for every non-trivial embedding σ : K ֒ → R we assume that the quadratic space ( V, q σ ) is positive definite, where q σ denotes thequadratic form obtained by applying σ to each coefficient of q . LetO( V, q ) := { U ∈ GL( n + 1 , R ) | q ( U x ) = q ( x ) ∀ x ∈ V ⊗ K R } . Notice that since q has the same signature and rank as the Lorentzian form q − , thereexists a real invertible matrix S such that S − O + ( n, S = O + ( V, q ).Let O K be the ring of integers of K . Consider a O K -lattice L and denote by O( L ) < O + ( V, q ) ∩ GL( n + 1 , K ) the group of linear transformations with coefficients in K thatpreserve the lattice L . The group O( L ) is discrete and of finite covolume [12, § Let R be a ring with field of fraction K and V a vector space of dimension n + 1 over K . An R -lattice L in V is an R -module in V of rank n + 1 for which Span K { L } = V . N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 5
Then, a discrete subgroup Γ in Isom( H n ) is called arithmetic of the simplest type if there exist K , q and L as above such that S − Γ S is commensurable with O( L ) inO + ( V, q ) ∩ GL( n + 1 , K ). In this case one says that Γ is defined over K with quadraticspace ( V, q ). Remark 2.2.
There is a more general definition of arithmetic group (see [2, Chapter6]). However, if a hyperbolic Coxeter group is arithmetic, then it is of the simplest type[35, Lemma 7]. Since we will be working only with hyperbolic Coxeter groups, we willalways refer to arithmetic groups of the simplest type as just arithmetic groups.More generally, a discrete subgroup Γ in Isom( H n ) is called quasi-arithmetic if thereexist K and q as above such that S − Γ S ⊂ O + ( V, q ) ∩ GL( n + 1 , K ) . In this case one says that Γ is defined over K with quadratic space ( V, q ).Finally, a discrete subgroup Γ in Isom( H n ) is called non-quasi-arithmetic , nq-arithmetic from now on, if it is neither arithmetic nor quasi-arithmetic.We conclude this section by stating Vinberg’s criterion to decide whether a hyperbolicCoxeter group is arithmetic, quasi-arithmetic or nq-arithmetic (see [35, Theorem 2]). Theorem 2.3 (Vinberg’s arithmeticity criterion) . Let Γ < Isom( H n ) be a Coxeter groupof rank N and denote by G = ( g ij ) ≤ i,j ≤ N its Gram matrix. Let e K be the field generatedby the entries of G , and let K (Γ) be the field generated by all the possible cycles of G .Then Γ is quasi-arithmetic if and only if:i) e K is totally real;ii) for any embedding σ : e K ֒ → R which is not the identity on K (Γ) , the matrix G σ ,obtained by applying σ to all the coefficients of G , is positive semidefinite.Moreover, a quasi-arithmetic group Γ is arithmetic if and only ifiii) the cycles of G are algebraic integers in K (Γ) .In both cases, Γ is defined over K (Γ) . Commensurability of hyperbolic Coxeter groups
In this section we prove the theorem stated in the Introduction and we show thecommensurability property of the Vinberg ring. An important role will be played by thetheory of fields of definition, which will be recalled in this section.3.1.
The Vinberg construction and fields of definition.
We now associate a qua-dratic space to a hyperbolic Coxeter group following a construction due to Vinberg in[35].Let Γ be a hyperbolic Coxeter group of rank N and let e , . . . , e N ∈ R n, be the outernormal unit vectors of its Coxeter polyhedron. Let G = ( g ij ) ≤ i,j ≤ N be the Gram matrixof Γ. For any { i , i , . . . , i l } ⊂ { , , . . . , N } consider the cyclic product of 2 G (3) b i i ...i l := 2 l g i i g i i . . . g i l − i l g i l i . E. DOTTI
Define the field K (Γ) := Q ( { b i i ...i l } ) of all cycles of 2 G . It is obvious that K (Γ) isgenerated by the simple cycles.Next, for { i , i , . . . , i k } ⊂ { , , . . . , N } , define the vectors(4) v := 2 e and v i i ...i k := 2 k g i g i i . . . g i k − i k e i k , and consider the K (Γ)-vector space V spanned by the vectors { v i i ...i k } according to(4). By [10, Lemma 1], V is of dimension n + 1. Moreover, as shown in [25] for example, V is left invariant by Γ since(5) s e j ( v i i ...i k ) = v i i ...i k − v i i ...i k j , and(6) h v i i ...i k , v j j ...j l i ∈ K (Γ) . Since 2 G is of signature ( n, V yields aquadratic form q = q V of signature ( n,
1) on V .By combining the equations (5) and (6) a quick computation shows that(7) h s e j ( v i i ...i k ) , s e j ( v j j ...j l ) i = h v i i ...i k , v j j ...j l i . By the construction of the K (Γ)-vector space V in terms of the vectors (4) and theform q V , we obtain a natural embedding Γ ֒ → O( V, q ). Observe that this construction isindependent of the arithmetic nature of Γ.Therefore any hyperbolic Coxeter group has an associated field and quadratic formwhich justifies the following definition.
Definition 3.1.
Let Γ be a hyperbolic Coxeter group. Theni) the field K (Γ) = Q ( { b i i ...i l } ) is called the Vinberg field of Γ;ii) the quadratic form q = q V is called the Vinberg form of Γ;iii) the quadratic space (
V, q ) is called the
Vinberg space of Γ.The next objective is to show that the Vinberg field and the similarity class of the Vin-berg form are two commensurability invariants. Before that, we need more terminology.Let ( V , q ), ( V , q ) be two quadratic spaces of dimension m ≥ K . Then( V , q ) and ( V , q ) are isometric (denoted by ∼ =) if and only if there is an isomorphism S : V → V such that q ( x ) = q ( Sx ) ∀ x ∈ V . They are similar (denoted by ∽ ) if there exist a λ ∈ K ∗ such that ( V , q ) and ( V , λq )are isometric. The scalar λ is called similarity factor .Isometry and similarity induce equivalence relations. In the sequel, we often abbre-viate and speak about isometric ( similar ) quadratic forms instead of isometric (similar)quadratic spaces. Furthermore, if one represents two quadratic forms by two m × m matrices Q and Q over K , then being isometric means that there exists an invertiblematrix S ∈ GL( m, K ) such that Q = S T Q S . N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 7
We conclude this part with some aspects about fields of definition, which will playan essential role for the upcoming proofs. All the theory presented here is taken fromVinberg’s paper [36].Let H be a Lie group. The adjoint trace field Q (Tr Ad H ) of H is defined as the fieldgenerated by the traces of the adjoint representation of the elements of H , namely Q (Tr Ad H ) = Q (Tr Ad( h ) | h ∈ H ) . Let U be a finite dimensional vector space over a field F , and let R ⊂ F be an integrallyclosed Noetherian ring. Denote by ∆ a family of linear transformations of U . The ring R is said to be a ring of definition for ∆ if U contains an R -lattice which is invariant under∆. When R is a field we call R a field of definition .If a principal ideal domain R is a ring of definition for ∆, then we can find a basis of U such that every element of ∆ can be written as a matrix having entries in R .Let us specialise the context and consider a Coxeter group Γ < Isom( H n ). As we haveseen in Section 3.1, the space R n, contains the K (Γ)-module V which is invariant underΓ. That is, the Vinberg field K (Γ) is a field of definition for Γ. The next lemma impliesthat the Vinberg field K (Γ) is actually the smallest field of definition associated to Γ. Lemma 3.2 ([36], Lemma 11 and Lemma 12) . Let Γ be a hyperbolic Coxeter group withGram matrix G and let F be a field of characteristic . An integrally closed Noetherianring R ⊂ F is a ring of definition for Γ if and only if R contains all the simple cycles of G . Lastly, for the following proofs we need the result [36, Theorem 5] of Vinberg. Werecapitulate here a more specific version suitable to our context.
Theorem 3.3.
Let Γ be a cofinite hyperbolic Coxeter group with Vinberg space ( V, q ) andGram matrix G . Let R be an integrally closed Noetherian ring. Then the following isequivalent:i) R is a ring of definition of Γ ,ii) R is a ring of definition of Ad Γ ,iii) R contains all the simple cyclic products of G . Remark 3.4.
It is important to notice that in [36] Vinberg considers Zariski densegroups generated by reflections of a quadratic space defined over an algebraically closed field. This hypothesis does not apply directly to our situation since the isometry groupPO( n,
1) of Klein’s projective model K n is defined over the reals.Our version of the theorem can be retrieved from the original one as follows. Pass tothe complexified space R n +1 ⊗ R C endowed with the standard (real) Lorentzian form q − .Let O C ( n,
1) be the group of complex ( n + 1) × ( n + 1) matrices which preserve q − , andform the projective group PO C ( n,
1) = O C ( n, / {± I } .Recall that a cofinite hyperbolic Coxeter group is Zariski dense (over R ) in PO( n, C ( n, C . We can now apply the original Theorem 5 of [36] which implies Theorem 3.3. E. DOTTI
Commensurability conditions for hyperbolic Coxeter groups.
We are nowable to prove the theorem stated in the Introduction.
Theorem 3.5.
Let Γ and Γ be two commensurable cofinite hyperbolic Coxeter groupsacting on H n , n ≥ . Then their Vinberg fields coincide and the two associated Vinbergforms are similar over this field. We start the proof by showing that two commensurable Coxeter groups have the sameVinberg field.
Proposition 3.6.
Let Γ < Isom( H n ) be a cofinite Coxeter group, n ≥ . Then theassociated Vinberg field and the adjoint trace field coincide, that is (8) K (Γ) = Q (Tr Ad Γ) . Proof.
Lemma 3.2 implies that the Vinberg field K (Γ) is the smallest field of definition ofΓ. By point i ) of Theorem 3.3, K (Γ) is a field of definition of Ad Γ as well and by point iii ) K (Γ) is contained in every field of definition of Ad Γ. By [36, Corollary of Theorem1], Q (Tr Ad Γ) is the smallest field of definition of Ad Γ. Hence, the equality (8) follows. (cid:3) Corollary 3.7.
Let Γ , Γ < Isom( H n ) be two cofinite Coxeter groups, n ≥ . If Γ and Γ are commensurable, then their associated Vinberg fields coincide, that is, K (Γ ) = K (Γ ) . Proof.
By Proposition 3.6 we know that K (Γ ) = Q (Tr Ad Γ ) and K (Γ ) = Q (Tr Ad Γ ).The adjoint trace field of a hyperbolic lattice is a commensurability invariant (see [7,Proposition 12.2.1]). Therefore the claim follows. (cid:3) Remark 3.8. i) Reflections in O + ( n,
1) have traces equal to n −
1. Thus, by Corollary 1 of Theorem 4of [36], the smallest field of definition of Γ is Q (Tr Γ). Hence we also get the equality K (Γ) = Q (Tr Γ).ii) By the Local Rigidity Theorem [31, Chapter 1], the adjoint trace field Q (Tr Ad Γ) ofa Coxeter group in Isom( H n ) is a number field for n ≥
4. Therefore, by Proposition3.6, the Vinberg field K (Γ) is a number field. Moreover, K (Γ) is a number fieldfor n = 3 as well. This is a consequence of the connection between K (Γ) and theinvariant trace field K Γ (2) ([25, Theorem 3.1]) and the fact that K Γ (2) is a numberfield ([26, Theorem 3.1.2]). Example 3.9.
Consider the two non-cocompact nq-arithmetic Coxeter pyramid groupsΓ and Γ acting on H as shown in Figure 1.For Γ we have the cycle − √ Q or Q (cid:0) √ (cid:1) . For Γ , the triangular tail path gives the cycle (cid:0) √ (cid:1) . All other simple cycles are either in Q or Q (cid:0) √ (cid:1) . Therefore the Vinbergfields are K (Γ ) = Q (cid:0) √ (cid:1) and K (Γ ) = Q (cid:0) √ (cid:1) . Thus Γ and Γ are incommensurable. N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 9 ∞ Γ ∞ Γ Figure 1.
Two Coxeter pyramid groups Γ and Γ in Isom( H ).Let us return to the proof of Theorem 3.5 and show that two commensurable hyperbolicCoxeter groups have similar Vinberg forms. The proof will follow the same strategy asindicated by Gromov and Piateski-Shapiro in Theorem 2.6 of [12] for arithmetic groups,and which has been elaborated by Johnson, Kellerhals, Ratcliffe and Tschantz in Theorem1 of [19] for the special case of hyperbolic Coxeter simplex groups.Consider two commensurable hyperbolic Coxeter groups Γ and Γ represented inO + ( n,
1) and denote their Vinberg field by K . There is a matrix X ∈ O + ( n,
1) and thereare two subgroups H < Γ and H < Γ , each of finite index, such that H = X − H X .One can assume that H and H are contained in SO + ( n, + ( n,
1) of determinant one matrices. Let ( V , q ) be the Vinberg space over K associatedto Γ and equipped with the basis { v , . . . , v n +1 } according to (4). With respect to thisbasis, all elements of Γ are matrices over K , since K is a field of definition of Γ. Clearlythe forms q and q − are equivalent over R . The same reasoning applies to the Vinbergspace ( V , q ). Let Q and Q be the matrix representations of the Vinberg forms q and q in the relative bases. Let the real matrices T and T be the representations of theisometries between the Vinberg forms and the Lorentzian form q − . Then the matrix(9) S := T − XT represents an isometry between q and q , since Q = S T Q S . Moreover define the twogroups H ′ := T − H T and H ′ := T − H T .Consider the isomorphism between the orthogonal groups O( q ) and O( q ) given by(10) φ : A → SAS − . Lemma 3.10.
The map φ restricts to a K -linear map on Mat( n + 1 , K ) .Proof. Let i ∈ { , } . Denote by O + ( q i ) the group of q i -orthogonal maps which leaveeach sheet of the hyperboloid H n +1 i = { x ∈ R n +1 | q i ( x ) = − } invariant. The isometrybetween q i and q − gives a group isomorphism between O( q i ) and O( q − ). This iso-morphism maps O + ( q i ) onto O + ( n, + ( q i ) is mapped onto SO + ( n, H ′ i ⊂ SO + ( q i ). Now, SO + ( n,
1) is a non-compact connected simple Lie groupand thus the same can be said for SO + ( q i ). Since H ′ i has finite covolume, by the Boreldensity theorem [4] we get that Span R ( H ′ i ) = Span R (SO + ( q i )) in Mat( n + 1 , R ). Fur-thermore, the action of SO + ( n,
1) on C n +1 is irreducible , and hence that the action of See the Erratum to the paper “Commensurability classes of hyperbolic Coxeter groups”, due to J.Ratcliffe and S. Tschantz, presented in [8, Appendix D]. SO + ( q i ) is irreducible as well. By Burnside’s theorem [5] (see also [21]) we get the equalitySpan R (SO + ( q i )) = Mat( n + 1 , R ), which implies that Span R ( H ′ i ) = Mat( n + 1 , R ).Notice that for each α ∈ K and C ∈ Mat( n + 1 , K ) we have φ ( αC ) = αφ ( C ). Recallthat K is a field of definition for H i , thus H ′ i ⊂ Mat( n + 1 , K ). By the same argumentsas before, we have that Span K ( H ′ i ) = Mat( n + 1 , K ). Moreover, by (9), φ ( H ′ ) = φ ( T − H T ) = T − XH X − T = T − H T = H ′ . We deduce that φ (Span K ( H ′ )) = Span K ( H ′ ). Therefore φ restricts to a K -linear mapon Mat( n + 1 , K ). (cid:3) Based on Lemma 3.10 we are finally ready to prove the last step as given by thefollowing proposition. Its proof is a direct adaptation of the corresponding step in theproof of [19, Theorem 1].
Proposition 3.11.
Let Γ , Γ be two commensurable Coxeter groups in Isom( H n ) , n ≥ ,with Vinberg field K (Γ ) = K (Γ ) =: K . Then the two Vinberg forms q and q aresimilar over K . Moreover, the similarity factor is positive.Proof. Let 1 ≤ i, j ≤ n +1. Define the matrix I ij ∈ Mat( n +1 , K ) with coefficient [ I ] ij = 1and all the other coefficients equal to zero. Consider the isomorphism φ according to (10).Define M ij := φ ( I ij ) = SI ij S − , which is in Mat( n + 1 , K ) by Lemma 3.10. The matrix SI ij =: S ij has the j -th column which is equal to the i -th column of S and all the othercoefficients are equal to zero. Observe that [ M ij ] kl = [ S ] ki [ S − ] jl for all k, l, i, j . Thematrix S − is invertible, thus we can always find a pair { j, l } such that [ S − ] jl = 0. Let λ denote the inverse of the coefficient [ S − ] jl . In doing so, every coefficient of S can bewritten as λ multiplied with an entry of a matrix of the form M ij ∈ Mat( n + 1 , K ). Hencethere exists a matrix M ∈ Mat( n + 1 , K ) such that S = λM . Recall that Q = S T Q S holds, therefore Q = λ M T Q M , with Q , Q and M all in Mat( n + 1 , K ). Finally λ is a positive element belonging to K so that q is isometric to λ q , and the claimfollows (cid:3) Similarity classification of the Vinberg forms.
The study of similarity of qua-dratic forms heavily relies on isomorphisms of quaternion algebras and others elementsof the Brauer group. For a more detailed explanation on this topic we refer to [13]. Let K denotes a field of characteristic different from 2, and let q be a quadratic form ofdimension m over K , that is, q is defined on a vector space of dimension m over K .For two elements a , b ∈ K ∗ , we denote by ( a, b ) the quaternion algebra over K gener-ated by the elements 1, i , j , ij with the relations i = a , j = b and ij = − ji .Two quaternion algebras are said to be equivalent if and only if they are isomorphic.Equivalence classes of quaternion algebras form a group, which is a subgroup of the Brauergroup Br( K ). For some computational rules about the multiplication between quaternionalgebras, we refer to [22, Proposition 3.20]. If the field K is a number field, then twoquaternion algebras over K are isomorphic if and only if they have the same ramificationset (see [24, Theorem 4.1]). N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 11
For the definition of a ramification set Ram( A ) for a quaternion algebra A and itstheory we refer to [24]. In this paper, the computations of ramification sets are doneusing the package RamifiedPlaces of Magma © .The similarity classification of quadratic forms relies on two elements of the Brauergroup, which are closely related to one another. The first one the Hasse invariant s ( q )of a diagonal quadratic form q = h a , . . . , a m i . This is the element of the Brauer groupBr( K ) represented by the quaternion algebra s ( q ) = O i Let K be a number field and let q and q be two quadratic forms ofdimension m over K . For a λ ∈ K ∗ , q and λq are isometric if and only if the followingproperties are satisfied:i) dim( q ) = dim( λq ) ,ii) det( q ) ≡ det( λq ) in K ∗ mod ( K ∗ ) ,iii) s ( q ) = s ( λq ) ,iv) sgn( σ ( q )) = sgn( σ ( λq )) for all real embeddings σ : K ֒ → R . For n even, let Γ and Γ be two hyperbolic Coxeter groups with the same Vinbergfield K , and denote by q and q the associated Vinberg forms over K . Recall that dim( q ) = dim( q ) = n + 1 =: m , i.e. the dimension of both quadratic forms is odd.Then, condition ii ) of the Hasse-Minkowski Theorem 3.14 implies that det( q ) ≡ λ det( q )in K ∗ mod ( K ∗ ) . This means that λ can only be the value which balances the twodeterminants, that is, λ = det( q )det( q ) ∈ K ∗ / ( K ∗ ) (see also the proof of [28, Proposition5.4]). Using [28, Lemma 4.3], we can simplify the Hasse invariant s ( λq ) for λ = det( q )det( q ) ∈ K ∗ / ( K ∗ ) , and we obtain the complete set of similarity invariants for Vinberg forms asshown in Table 1. n Similarity criterion n ≡ s ( q ) = s ( q )sgn( σ ( q )) = sgn( σ ( λq )) n ≡ s ( q ) = ( λ, − · s ( q )sgn( σ ( q )) = sgn( σ ( λq )) Table 1. Similarity criterion for Vinberg forms of hyperbolic Coxetergroups.Notice that this similarity classification is compatible with the one provided by Maclach-lan for quasi-arithmetic groups. For these groups, the equality between signatures isalways satisfied.For n odd, let Γ and Γ be two hyperbolic Coxeter groups. If they are quasi-arithmetic,we refer to the similarity classification provided by Maclachlan (see Theorem 3.13). Oth-erwise, the similarity problem for their even-dimensional Vinberg forms q and q is moreinvolved. We present here a partial result, only.Applying condition ii ) of the Hasse-Minkowski Theorem 3.14 we get det( q ) ≡ det( λq )in K ∗ mod ( K ∗ ) which reduces to det( q ) ≡ det( q ) mod ( K ∗ ) . In contrast to theprevious case, we can not extract any information about λ . This fact can be stated inthe following lemma, sometimes referred to as the ratio-test . Lemma 3.15. Let Γ , Γ < Isom( H n ) , n odd, be two commensurable Coxeter groups withVinberg field K and Vinberg forms q and q , respectively. Then, det( q ) ≡ det( q ) ∈ K ∗ mod ( K ∗ ) . Example 3.16. As an incommensurability example using the Vinberg form, consider thetwo cocompact Coxeter groups Γ , Γ in Isom( H ) given in Figure 2. The groups Γ andΓ are so-called crystallographic Napier cycles (see [17]). Observe that both groups arequasi-arithmetic (but not arithmetic). N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 13 l l l 54 Γ l ′ l ′ l ′ 55 Γ Figure 2. The Coxeter groups Γ and Γ in Isom( H ).The weights l i and l ′ i of the dotted edges in the Coxeter graphs can be computed andare l = r (cid:16) 10 + 3 √ (cid:17) , l ′ = r (cid:16) √ (cid:17) ,l = 12 r(cid:16) √ (cid:17) , l ′ = r (cid:16) √ (cid:17) ,l = r (cid:16) 16 + 7 √ (cid:17) , l ′ = r (cid:16) 233 + 104 √ (cid:17) . The groups Γ and Γ have both K = Q (cid:0) √ (cid:1) as their Vinberg field. The diagonalisedassociated Vinberg forms over K are q = diag (cid:16) , , , − − √ , 20 + 8 √ (cid:17) ,q = diag (cid:18) , 52 + 12 √ , √ , − − √ , √ (cid:19) . These forms have the following Hasse invariants: c (Γ ) = (cid:16) − − √ , √ (cid:17) ,c (Γ ) = (cid:16) 10 + 2 √ , − (cid:17) · (cid:16) − − √ , √ (cid:17) . The ramification set Ram(Γ ) contains two prime ideals, one generated by 2, and theother generated by 5 and − √ 5. The ramification set Ram(Γ ) is empty. SinceRam(Γ ) = Ram(Γ ), the two quaternion algebras representing c (Γ ) and c (Γ ) are notisomorphic. Hence the Vinberg forms q and q are not similar, and the groups Γ andΓ are incommensurable.3.4. The Vinberg ring. In this section we are looking for additional commensurabilityinvariants for arbitrary hyperbolic Coxeter groups. Definition 3.17. Let Γ < Isom( H n ), n ≥ 2, be a cofinite Coxeter group with Grammatrix G . Consider all the cycles b i i ...i l = 2 l g i i g i i . . . g i l − i l g i l i of 2 G . The ring R (Γ) := O ( { b i i ...i l } ) is called the Vinberg ring of Γ.We show that the Vinberg ring is a ring of definition for certain groups and hence acommensurability invariant. Notice that the Vinberg ring as commensurability invariantis superfluous when considering arithmetic groups. Proposition 3.18. Let Γ < Isom( H n ) , n ≥ , be a cofinite Coxeter group with Grammatrix G . Assume that its Vinberg field K is a number field. Then the Vinberg ring R (Γ) is a commensurability invariant.Proof. For this proof we use some results of Davis about overrings ([6]) in the same wayas used by Mila in [29, Section 2.1]. Since by hypothesis the Vinberg field K is a numberfield, there exists a minimal ring of definition R for Γ ([36, Corollary to Theorem 1]) whichequals the integral closure of Z [Tr Ad Γ] in K . Clearly R is integrally closed and thereforecontains the ring of integers O of K . Thus R is the integral closure of O [Tr Ad Γ] =: R ′ in K . The ring R ′ is an overring of O . Since O is a Noetherian integral Dedekind domain, by[6, Theorem 1] its overring R ′ is integrally closed. This implies R = R ′ which means that O [Tr Ad Γ] is the smallest ring of definition for Γ. Moreover, the Vinberg ring R (Γ) is alsoan overring of O in such a way that it is integrally closed as well, and it is furthermoreNoetherian since it is a subring of the number field K (see [11, Theorem]). Hence R (Γ)is a ring of definition for Γ. By Theorem 3.3, R (Γ) ⊂ R ′ . Now, R ′ is the smallest ringof definition so that R (Γ) = O [Tr Ad Γ]. By Theorem 3 of [36], rings of definition arecommensurability invariants. Thus R (Γ) is a commensurability invariant. (cid:3) Example 3.19. As an example, consider the two non-cocompact quasi-arithmetic (butnot arithmetic) Coxeter cube groups Γ and Γ in Isom( H ) defined in Figure 3 (see[18]). They both have Q as Vinberg field and similar quadratic forms. Their Vinbergrings are given by R (Γ ) = Z [1 / 3] and R (Γ ) = Z [1 / and Γ are therefore incommensurable. √ Γ √ √ 36 Γ Figure 3. Two Coxeter cube groups Γ and Γ in Isom( H ). An overring of an integral domain is a subring of the quotient field containing that given ring. In ourcase, the integral domain is the ring of integers O of the Vinberg field K , which has the Vinberg field asits quotient field. N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 15 Caution 3.20. Two hyperbolic Coxeter groups having the same Vinberg field, the sameVinberg ring and similar Vinberg forms do not have to be commensurable. For exam-ple, consider the two non-cocompact quasi-arithmetic (but not arithmetic) Coxeter cubegroups Γ as above and Γ in Isom( H ) defined by the graph in Figure 4. Observe thatfor both groups, the Vinberg field is Q and the Vinberg ring is Z [1 / and Γ are quasi-arithmetic, we can apply Maclachlan’s criterion to decide whether the Vinbergspaces ( V , q ) and ( V , q ) are similar (see Theorem 3.13). √ Γ Figure 4. Quasi-arithmetic Coxeter cube group Γ acting on H .With Vinberg’s construction, we can compute the matrices representing q and q anddiagonalise them over Q . We obtain q = diag(4 , , , − 15) and q = diag (cid:18) , , − , (cid:19) . The quadratic forms q and q have both ( − , 3) as Hasse invariant and therefore theyhave identical Witt invariant represented by the quaternion algebra B = (1 , B ⊗ Q Q ( √− 1) over Q ( √− 1) is identical for both groups. Thisimplies that the Vinberg spaces are similar. However, as shown in [39] by means of ageometric argument, Γ and Γ are not commensurable.4. New generators for the Vinberg field In this last section we discuss various aspects of the Vinberg field of a Coxeter group,such as the possible Vinberg fields associated to quasi-arithmetic Coxeter groups and therange of the admissible dihedral angles of a Coxeter polynomial P in terms of the extensiondegree d of its Vinberg field. We conclude by providing two new sets of generators forthe Vinberg field of a quasi-arithmetic hyperbolic Coxeter group.4.1. The Vinberg field of a quasi-arithmetic hyperbolic Coxeter group. In thispart we present some results about possible Vinberg fields associated to quasi-arithmeticCoxeter groups in Isom( H n ), n ≥ 2. By Remark 1 of [35] a non-cocompact quasi-arithmetic Coxeter group has Vinberg field Q . Moreover there are no arithmetic Coxetergroups in Isom( H n ) for n ≥ 30 [38, Theorem 2.2]. We start by considering compact hyperbolic Coxeter n -simplices, which were classifiedby Lann´er and exist only for n ≤ 4. Their Coxeter graph are called Lann´er graph . Forthe complete list of Lann´er graphs see [34, Table 2.2], for example.Essential for the following is the fact [34, Corollary 2.1] that the Coxeter graph of acocompact hyperbolic Coxeter group contains a subgraph which is a Lann´er graph (calleda Lann´er subgraph ).Consider a cocompact quasi-arithmetic hyperbolic Coxeter group. Its Gram matrixsatisfies part ii ) of Theorem 2.3, which is crucial for the proof of the following result ofVinberg. Theorem 4.1 ([37], Proposition 17) . For a cocompact quasi-arithmetic hyperbolic Cox-eter group, the Vinberg field is generated by the determinant of any Lann´er subgraph ofthe Coxeter graph. In [37, Theorem 2 and Theorem 3], Vinberg exploited the above result in order to showthat for n ≥ 14 the only possible Vinberg fields of a cocompact arithmetic Coxeter groupΓ are Q ( √ , Q ( √ , Q ( √ , Q ( √ , Q ( √ , √ , Q ( √ , √ , Q (cos 2 π/m )with m = 7 , , , , , 20. While, for n ≥ 22, the possible Vinberg fields are Q ( √ , Q ( √ , Q (cos 2 π/ . We exploit Theorem 4.1 by assuming that the Coxeter graph contains a Lann´er sub-graph of order at least three and present the following result. Proposition 4.2. Let Γ < Isom( H n ) be a cocompact quasi-arithmetic Coxeter group suchthat its Coxeter graph contains a Lann´er subgraph of order at least three. The possibleVinberg fields for Γ are Q , Q ( √ , Q ( √ , Q ( √ , Q ( √ , Q ( √ , √ , Q ( √ , √ , Q (cos 2 π/m ) for m = 7 , , , , , .Proof. By Theorem 4.1, the Vinberg field of Γ is the extension of Q by the determinantof any Lann´er subgraph. Since Γ is quasi-arithmetic, any such Lann´er subgraph of orderat least three describes an arithmetic Lann´er group. Indeed, the Gram matrices of thesubgroups corresponding to these Lann´er subgraphs must satisfy part ii ) of Theorem 2.3.Moreover, Lann´er graphs of order at least three do not have dotted edges. These twoconditions imply that the Lann´er subgraphs describe arithmetic groups (see [35, Remark3]). The arithmetic Lann´er groups of order three have been determined by Takeuchi[33]: there are finitely many examples. The Lann´er groups of orders four and five are allarithmetic with one exception (see [19], for example). Hence, we have to compute finitelymany determinants, and the result follows. (cid:3) The determinant of a Coxeter graph is the determinant of the corresponding Gram matrix. N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 17 Remark 4.3. When the Coxeter graph of Γ contains only Lann´er subgraphs of order two,other fields can arise. In his PhD thesis [9], Esselmann gives examples of such cocompactarithmetic hyperbolic Coxeter groups with Vinberg fields Q ( √ Q ( √ 17) and Q ( √ The Vinberg field and the dihedral angles of a hyperbolic Coxeter group. With the second part of Remark 3.8, we know that, for n > 2, the Vinberg field is anumber field. Thus the study of the extension degree of the Vinberg field is an interestingtopic. A bound on the extension degree on the Vinberg field for arithmetic hyperbolicCoxeter groups has been studied by many authors. A lot of work has been made byNikulin [30].In this part we see how the degree of the Vinberg field determines the range of admis-sible dihedral angles of a Coxeter polyhedron. Proposition 4.4. Let Γ < Isom( H n ) , n > , be a cofinite Coxeter group with Coxeterpolyhedron P and Vinberg field K of degree d . Then, for any dihedral angle πm of P onehas φ ( m ) ≤ d, where φ ( m ) is the Euler’s totient function.Proof. Let a = a m := cos (cid:0) πm (cid:1) . It is well-known that (see [23], for example)(11) [ Q ( a ) : Q ] = φ ( m ) / , where φ ( m ) is the Euler’s totient function φ ( m ), which is the number of positive integers,relatively coprime to m , between 1 and m , both included.Next, consider the Vinberg field K of Γ. To every dihedral angle πm of P correspondsa subgraph of the Coxeter graph of the form • – m –—– • which consists only of two nodes if m = 2. To this subgraph corresponds the cycle b i i = 4 cos (cid:0) πm (cid:1) ∈ K (see (3)). This forces [ Q (4 cos (cid:0) πm (cid:1) ) : Q ] ≤ d .By the angle doubling property of the cosine function we have Q (cid:0) (cid:0) πm (cid:1)(cid:1) = Q (cid:0) cos (cid:0) πm (cid:1)(cid:1) . Therefore, by (11), the weight m must satisfy the inequality[ Q (cid:18) cos (cid:18) πm (cid:19)(cid:19) : Q ] = φ ( m ) / ≤ d. (cid:3) Example 4.5. Let Γ be an arithmetic Coxeter group in Isom( H n ), n ≥ 14. By [37,Theorem 2 and Theorem 3], the degree d = [ K : Q ] is smaller than or equal to five. For d = 5, Proposition 4.4 yields φ ( m ) ≤ 10, for any dihedral angle πm . In general, for x notequal to 2 or 6, Euler’s function φ ( x ) satisfies φ ( x ) ≥ √ x. Thus, for m ≤ φ ( m ) ≤ 10. As a result, for n ≥ 14, all thepossible values for m are2 , , , , , , , , , , , , , , , , , , . The Gram field of a hyperbolic Coxeter group. For n ≥ 2, consider a Coxetergroup Γ in Isom( H n ) of rank N . Let G be its Gram matrix of signature ( n, 1) withcharacteristic polynomial χ G ( t ) = a + a t + · · · + a N t N , a N = 1 . The matrix G is uniquely defined by Γ up to simultaneous permutation of its lines andcolumns which would yield a similar matrix G ′ with identical characteristic polynomial.Notice that a N − = ( − N − Tr( G ) = ( − N − N . Moreover, each coefficient a r of χ G , r < N , can be expressed as the sum of all the principal minors of size N − r (see [14,p. 53], for example). In particular, a r vanishes for all r < N − ( n + 1). Definition 4.6. Let Γ be a hyperbolic Coxeter group of rank N . Let G be its Grammatrix with characteristic polynomial χ G ( t ) = a + a t + · · · + a N t N , a N = 1. The Gram field K ( G ) is the field generated by the coefficients of χ G ( t ) over Q , namely K ( G ) = Q ( a j | ≤ j ≤ N ) . Proposition 4.7. Let Γ be a cofinite quasi-arithmetic hyperbolic Coxeter group withVinberg field K . Then K = K ( G ) . Proof. We prove first the inclusion K ⊇ K ( G ). By [37, Proposition 11], the determinantof the Gram matrix G is given by a sum of cyclic products. The same result applies toevery principal submatrix of G . Since the coefficients of χ G can be expressed as the sumof principal minors of G (see [14, p. 53], for example), we get K ⊇ K ( G ).Assume that K ) K ( G ). Then there exists a non-trivial embedding σ : K ֒ → R whichis the identity on K ( G ). Let G σ be the matrix obtained by applying σ to every coefficientof G and let χ G = P Ni =0 a i x i be the characteristic polynomial of G . Since σ is a fieldhomomorphism, then χ G σ = P Ni =0 σ ( a i ) x i . The embedding σ fixes the coefficients of χ G ,thus χ G = χ G σ . In particular, G σ has signature ( n, 1) and is not positive semidefinite. This is a contra-diction to part ii ) of Theorem 2.3 and the claim follows. (cid:3) The Coxeter field of a hyperbolic Coxeter group. Let Γ < Isom( H n ), n ≥ { s , . . . , s N } . Consider aCoxeter transformation C = s · · · s N of Γ defined up to the ordering of the factors.With the real coefficients of the characteristic polynomial χ C ( t ) we define a new field,the Coxeter field , and prove that it coincides with the Vinberg field K (Γ) if Γ is quasi-arithmetic. N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 19 The proof is based on the work of Howlett [15] and the theory of M -matrices whichwe are going to review briefly.Let W = ( W, S ) be a Coxeter system with generating set S = { s , . . . , s N } satisfyingthe relations of a Coxeter group. By Tits’ theory, it is known that W can be representedas a subgroup of GL( V ) for a real vector space V of dimension N equipped with asuitable symmetric bilinear form B (see [16], for example). Denote by rad( V ) = { v ∈ V | B ( v, v ′ ) = 0 ∀ v ′ ∈ V } the radical of B which will play a role later on. A Coxeter element c ∈ W is the product of the N generators in S arranged in any order. The representative C T ∈ GL( V ) of c is called a Coxeter transformation of W .For a Coxeter element c = s · · · s N , the matrix of C T with respect to a basis { v , . . . , v N } of V , denoted again by C T , can be written according to (see [15], for example)(12) C T = − U − U T , where U ∈ GL( N, R ) is given by(13) U = ∗ . . . , with [ U ] st = 2 B ( v s , v t ) for t > s . Notice that(14) U + U T = 2 B. By means of the theory of M -matrices, Howlett ([15, Theorem 4.1], see also [1]) charac-terised abstract Coxeter groups in terms of a Coxeter transformation C T and its eigenval-ues. More concretely, an M-matrix is a real matrix with non-positive off-diagonal entriesall of whose principal minors are positive. For example, the matrix U given by (13) is an M -matrix.The proof of Howlett’s Theorem 4.1 in [15] is based on the following results. Lemma 4.8 ([15], Lemma 3.1) . Let U be a real matrix such that U + U T is positive defi-nite. Then U is invertible and − U − U T is diagonalisable over C with all of its eigenvalueshaving modulus one. Lemma 4.9 ([15], Lemma 3.2 and Corollary 3.3) . Let U be an M -matrix such that U + U T is not positive definite. Then − U − U T has a real eigenvalue λ ≥ . If U + U T is notpositive semidefinite, then λ > . If U + U T is positive semidefinite, all the eigenvaluesof − U − U T have modulus one and − U − U T is not diagonalisable. Later we will also need another lemma, which is stated in Howlett’s proof of Lemma4.9. Lemma 4.10. Let U be an invertible real matrix such that U + U T is positive semidefinite.Then the eigenvalues of − U − U T have all modulus one. Proof. For ǫ > U ǫ := U + ǫI . Since U + U T is positive semidefinite, U ǫ + ( U ǫ ) T is positive definite. By Lemma 4.8, all the eigenvalues of − ( U ǫ ) − ( U ǫ ) T have modulus one. The entries of U ǫ depend continuously on ǫ . The same can be said for − ( U ǫ ) − ( U ǫ ) T and the coefficients of its characteristic polynomial. Hence the eigenvaluesof − ( U ǫ ) − ( U ǫ ) T and their modulus depend continuously on ǫ , and the claim follows. (cid:3) Let Γ < Isom( H n ) be a hyperbolic Coxeter group with generating reflections s , . . . , s N .In this way Γ represents a geometric realisation of an abstract Coxeter group. Let P ∈ H n be its Coxeter polyhedron with outer unit normal vectors e , . . . , e N and associated Grammatrix G ∈ Mat( N, R ).Let C ∈ Γ be a Coxeter transformation of Γ. Our goal is to construct a new field K ( C )associated to C which we can identify later with the Vinberg field K (Γ). Our motivationcomes from [32, Theorem 1.8, (iv)], due to Reiner, Ripoll and Stump, relating Coxetertransformations of a finite complex reflection group to its field of definition (see alsoMalle in [27, Section 7A]).Inspired by this, we state the following definition. Definition 4.11. Let Γ be a hyperbolic Coxeter group. Let C ∈ Γ be a Coxeter trans-formation with characteristic polynomial χ C ( t ) = a + a t + · · · + a n +1 t n +1 , a n +1 = 1.The Coxeter field K ( C ) is the field generated by the coefficients of χ C ( t ) over Q , namely K ( C ) = Q ( a j | ≤ j ≤ n + 1) . It is not difficult to see that χ C ( t ) is palindromic ( a j = a n +1 − j ) if N = n + 1 + 2 k andit is pseudo-palindromic ( a j = − a n +1 − j ) if N = n + 1 + (2 k + 1), for some k ≥ N − ( n +1) is the dimension of the radical rad (cid:0) R N (cid:1) for the Tits represen-tation space (cid:0) R N , G (cid:1) . Clearly, every element in Γ viewed in GL (cid:0) R N (cid:1) acts as the identityon rad (cid:0) R N (cid:1) . Hence the same is true for every Coxeter transformation C T ∈ GL (cid:0) R N (cid:1) of Γ. Since dim (cid:0) R N / rad (cid:0) R N (cid:1)(cid:1) = n + 1, the characteristic polynomials χ C and χ C T arerelated by(15) ( t − ( N − ( n +1)) χ C ( t ) = χ C T ( t ) . In particular the field generated by the coefficients of χ C and the field generated by thecoefficients of χ C T coincide. With this preparation we are ready to prove the followingresult. Proposition 4.12. Let Γ be a cofinite quasi-arithmetic hyperbolic Coxeter group withVinberg field K , and let C be any Coxeter transformation of Γ . Then K = K ( C ) . Proof. We first show that K ⊇ K ( C ). The Vinberg field K is a field of definition (seeSection 3.1). Thus, by means of a suitable basis, the Coxeter transformation C can bewritten as a matrix with coefficients in K . Since the characteristic polynomial is invariantunder a basis change, we have that K ⊇ K ( C ). In [32], the field of definition of a Coxeter group W is the field generated by all the traces of the matricesrepresenting the elements of W . N THE COMMENSURABILITY OF HYPERBOLIC COXETER GROUPS 21 Assume that K ) K ( C ). Then there exists a non-trivial embedding σ : K ֒ → R that isthe identity on K ( C ). Consider the Coxeter transformation C T acting on (cid:0) R N , G (cid:1) whichcorresponds to C in the sense of Tits. By (12), we can express C T = − U − U T where U is an M -matrix. 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