Hyperbolic manifolds and pseudo-arithmeticity

We introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in $\mathrm{PO}(n,1)$ with $n>3$. We further show that under an additional assumption (satisfied in all known cases), the covolumes of these lattices correspond to rational linear combinations of special values of $L$-functions.


Introduction
The study of hyperbolic 3-manifolds of finite volume has many relations with number theory, with a central role in this context being played by the notion of the invariant trace field (see [17]). The work of Vinberg [31] allows to define a comparable invariant (the adjoint trace field ) for any locally symmetric space. This paper studies finite volume hyperbolic manifolds of dimensions n > 3 from the perspective of the adjoint trace field, and the algebraic groups that are naturally associated with them (see Theorem 1.2).
Our main purpose is to introduce and motivate a notion of pseudo-arithmeticity, which to the best of our knowledge applies to all currently known lattices in PO(n, 1) with n > 3. In this introduction we state two main results: Theorem 1.5, where we show that the classical hyperbolic manifolds obtained by gluing are pseudoarithmetic; and Theorem 1.12, in which it is proved that under some mild assumption (defined in Sect. 1.5) the covolumes of pseudo-arithmetic lattices correspond to rational linear combinations of covolumes of arithmetic lattices.
1.1. Hyperbolic isometries and algebraic groups. We denote by H n the hyperbolic n-space, and we identify its group of isometries with the Lie group PO(n, 1). Let PO denote the real algebraic group such that PO(R) = PO(n, 1), and PO + its identity component. Note the following dichotomy: • for n even PO is connected, so that PO + (R) = PO(n, 1); • for n odd PO has two connected components, and we have PO + (R) = PO(n, 1) • . Recall that any complete hyperbolic n-manifold M can be written as a quotient M = Γ\H n , where Γ ⊂ PO(n, 1) is a torsion-free discrete subgroup (uniquely determined up to conjugacy). The manifold M has finite volume exactly when Γ is a lattice in PO(n, 1). We will usually assume that M is orientable, i.e., Γ ⊂ PO(n, 1) • . Then by Borel's density theorem Γ is Zariski-dense in PO + .
1.3. The case of glued manifolds. To date for n > 3 there are two sources of nonarithmetic lattices in PO(n, 1): (1) Hyperbolic reflection groups, 1 some of which can be proved to be nonarithmetic by using Vinberg's criterion [30]. (2) Hyperbolic manifolds constructed by gluing together "pieces" of arithmetic manifolds along pairwise isometric totally geodesic hypersurfaces. There are several constructions that fit the description in (2) (see [2,9,27]), all of which can be thought as (clever) variations of the original method 2 by Gromov and Piateski-Shapiro [10]. Their building blocks are always arithmetic pieces M i , i.e., hyperbolic n-manifolds with totally geodesic boundary taken inside arithmetic manifolds Γ i \H n (see Sect. 2.4-2.5). We will see in Corollary 2.6 that two such pieces M i and M j cannot be glued together unless Γ i and Γ j have the same field of definition (equivalently, the same trace field).
The following theorem applies to all nonarithmetic lattices constructed in the sense of (2). By a multiquadratic extension K/k we mean a (possibly trivial) field extension of the form K = k( √ a 1 , . . . , √ a s ) (a i ∈ k, s ≥ 0). Theorem 1.5. For n > 3, let M = Γ\H n be a hyperbolic n-manifold obtained by gluing together a finite number of arithmetic pieces M i ⊂ Γ i \H n , each of the arithmetic lattices Γ i being defined over k. Then: (1) The trace field K of Γ is a totally real multiquadratic extension of k.
(2) The ambient group of Γ is G K , where G is the ambient group of any of the arithmetic lattices Γ i .
The exact description of the trace field K -and in particular the degree of K/kdepends on the pieces M i and the way they are glued together. A precise treatment is the subject of a separate article by the second author [20]. See also [21].
1.4. Pseudo-arithmetic lattices. Theorem 1.5 motivates the following definitions: Definition 1.6 (Pseudo-admissible). We call pseudo-admissible (over K/k) an algebraic group of the form G K , where K/k is a finite (possibly trivial) totally real multiquadratic extension and G is an admissible k-group. Definition 1.7 (Pseudo-arithmetic). Let n > 3. A lattice Γ ⊂ PO(n, 1) will be called pseudo-arithmetic if its ambient group is pseudo-admissible.
It follows from the definition that (quasi-)arithmetic lattices are pseudo-arithmetic. Theorem 1.5 shows that lattices obtained by gluing arithmetic pieces are pseudo-arithmetic.
In Sect. 4 we will present a method to test if a given reflection group is pseudoarithmetic or not; this can be thought as an extension of Vinberg's (quasi-)arithmeticity criterion [30]. We have applied our method on the full list of groups provided with the software CoxIter (see [11]), which contains about a hundred non-quasi-arithmetic Coxeter groups for n > 3: all of them turn out to be pseudoarithmetic. A particularly interesting example is the Coxeter group Δ 5 presented in Figure 1; it has recently been shown (combining work of Fisher et. al. [8,Sect. 6.2] and of the second author [20]) that Δ 5 is not commensurable with any lattice obtained by gluing arithmetic pieces. At this point it is natural to ask: 4 Figure 1. A pseudo-arithmetic Coxeter 5-simplex group: Δ 5 Question 1.8. Let Γ ⊂ PO(n, 1) be a lattice, with n > 3. Is Γ necessarily pseudoarithmetic? Remark 1.9. In dimension n = 3 the abundance of hyperbolic manifolds obtained by Dehn surgeries makes very unlikely a positive answer to an analogue of this question.
Note that by definition the trace field K of a pseudo-arithmetic lattice is totally real. The following question -thus weaker than Question 1.8 -seems easier to answer, yet we do not know its current status. Question 1.10. Let n > 3. Is the trace field of a lattice Γ ⊂ PO(n, 1) necessarily totally real? 1.5. First type lattices. A lattice Γ ⊂ PO(n, 1), is said to be of the first type if its ambient group is of the form G = PO q (resp., G = PO + q ), where PO q denotes the projective orthogonal group (resp., its identity component) of a quadratic form q defined over the trace field K; see Sect. 2.1 for details. This extends the terminology sometimes used for arithmetic lattices. For n even, any K-form of PO is of the type PO q , and thus for those dimensions all lattices are trivially of the first type. It follows from part (2) of Theorem 1.5 that lattices obtained by gluing arithmetic pieces are of the first type, since such pieces contain totally geodesic hypersurfaces (see Prop. 2.2). Moreover, every Coxeter group is of the first type (the corresponding quadratic form can be read off the Gram matrix, see [31, §4]). To summarize, all currently known lattices in PO(n, 1) with n > 3 are either: • arithmetic, or • pseudo-arithmetic of the first type.
A lattice simultaneously belongs to both categories if and only if it is arithmetic of the first type (sometimes also called "standard" arithmetic lattices).
In the rest of the paper we will often make implicit use of the following. Lemma 1.11. Let G = PO q be pseudo-admissible over K/k, for some quadratic form q over K. Then G ∼ = PO f,K for some quadratic form f over k.
Proof. By definition G = G 0,K for some admissible k-group G 0 . Note that the extension K/k is multiquadratic, whence Galois. For any σ ∈ Gal(K/k), we thus have POσ q ∼ = PO q . This implies that σ q = a σ q • A σ for some a σ ∈ K × and A σ ∈ GL n (K). The maps σ → a σ and σ → A σ are easily seen to be 1-cocycles, so that a σ = bσ(b −1 ) and A σ = Bσ(B −1 ) for some b ∈ K × and B ∈ GL n (K) respectively (Hilbert's Theorem 90). It follows that f = bq • B is invariant by Gal(K/k) and thus defined over k, with G ∼ = PO f,K .
1.6. Volumes. Let G = PO f,K (or PO + f,K ) be a pseudo-admissible group over K/k. Then we may associate with G a set of arithmetic lattices in the following way. Let us assume that K is explicitly given by K = k( √ a 1 , . . . , √ a r ), i.e., the degree of K/k is s = 2 r . We will assume (as we may) that the quadratic form f is diagonal in the variables x 0 , . . . , x n , with negative coefficient in x 0 . For any multi- All the a i are totally positive since K is totally real by assumption. It follows that all f i are admissible quadratic forms over k. For each i ∈ {0, 1} r we choose an arithmetic subgroup Γ i ⊂ PO f i (k), i.e., a subgroup commensurable with PO f i (O k ). We say that the set of arithmetic lattices We will show that -in some precise sense -the homology over Q of the group G(K) is generated by classes associated with the Γ i ; see Theorem 3.5. A direct consequence for the volume is the following. Theorem 1.12. Let Γ ⊂ PO(n, 1) be a pseudo-arithmetic lattice of the first type (with n > 3), and {Γ i } a set of arithmetic lattices subordinated to the ambient group of Γ. Then For n even the result readily follows from the Gauss-Bonnet theorem, and Theorem 1.12 has no interest there (but Theorem 3.5 presumably has). For manifolds obtained by gluing the result was already known; see [7,Sect. 1.4]. Note also that for quasi-arithmetic lattices (which corresponds to the case r = 0 above) the result is proved in [7,Theorem 1.3], and the condition "of the first type" is superfluous.
The covolume of arithmetic lattices is well-understood: it is essentially expressible by means of special values of L-functions (see [24,25]). In particular the values of the summands vol(Γ i \H n ) can be determined up to rationals. The following example illustrates Theorem 1.12 with the case of the Coxeter group Δ 5 (which was introduced in Sect. 1.4). Example 1.13. It was already proved in [30,Sect. III] that the nonuniform Coxeter group Δ 5 ⊂ PO(5, 1) is nonarithmetic, with trace field K = Q( √ 2). It is the unique hyperbolic simplex group that is nonarithmetic for n > 3. Applying the procedure described in Sect. 4.1, one finds from its Gram matrix that Δ 5 ⊂ PO f (K), with f = f 0 given in (1.3). This shows that PO f,K is pseudo-admissible, and thus Δ 5 is pseudo-arithmetic of the first type. A subordinated set of arithmetic lattices is given by  [34,Sect. 11]. The covolume of Δ 5 has been studied by Kellerhals, who obtained a closed formula 3 in [12, Sect. 3] (where Δ 5 = AU 5 ). Based on this, the following approximation has been found in collaboration with Steve Tschantz. Its accuracy has been checked up to 160 digits.
Proving a sharp equality in (1.4) seems out of reach with the methods presented in this article. In particular, the method used to prove Theorem 1.12 does not provide any information about the β i (or even about their signs).

Final remarks.
1.7.1. The definition of pseudo-arithmeticity can be transferred verbatim for lattices in PU(n, 1) (n ≥ 2). Nonarithmetic lattices in PU(n, 1) are known to exist for n = 2, 3, and none of the known examples is quasi-arithmetic. In PU(2, 1) there exist nonarithmetic lattices whose trace fields are not quadratic extensions (see [4, Table A.2]); in particular they cannot be pseudo-arithmetic. 1.7.2. In [10, Question 0.4] Gromov and Piateski-Shapiro famously asked whether for any lattice Γ ⊂ PO(n, 1) of sufficiently large dimension the quotient Γ\H n admits a nice partition into "subarithmetic pieces". In some weak sense (i.e., at the level of homology), Theorem 3.5 gives a positive answer to this question for the class of pseudo-arithmetic lattices of the first type. It is not clear however if a statement closer to their original formulation can be achieved for those lattices; in particular, if a positive answer to the following question could hold: Question 1.14. Let Γ ⊂ PO(n, 1) be a pseudo-arithmetic lattice of the first type (n > 3), and let {Γ i } be a set of arithmetic lattices subordinated to its ambient group. Does the collection of subgroups {Γ ∩ Γ i } generate the group Γ?

Pseudo-arithmeticity of gluings
The goal of this section is to prove Theorem 1.5.

Rational hyperplanes.
Let (V, f ) be an admissible quadratic space over a number field k, i.e., f : V → k is an admissible quadratic form. We denote by O(V, f ), or simply O f , the algebraic group of orthogonal transformations of (V, f ). We denote by PO(V, f ) = PO f its adjoint form (it corresponds to the group PGO(V, f ) in the notation of [13]). For a field extension K/k, the group of Kpoints PO f (K) can be concretely described as the quotient GO( [13,Sect. 12.A and Sect. 23.B]). In particular, PO(V, f ) acts k-rationally on the projective space PV C , and so does its identity component PO This is equivalent to saying that the projective space PW is k-closed in PV R . A useful characterization is the following.
Proof. Let B be the bilinear form associated with f . If W = e ⊥ for some e ∈ V = V k , it follows immediately from the formula ρ( B(e,e) x that ρ is k-rational. For the converse implication, note that PW corresponds to the set of fixed points of ρ, and thus ρ ∈ PO f (k) implies that PW (and thus H(W )) is k-rational since the action is defined over k.

Sharp hypersurfaces.
Gluings are realized along totally geodesic embedded hypersurfaces of finite volume. To simplify the following discussion, we will call sharp such a hypersurface embedded in a hyperbolic manifold M of finite volume. Note that if M is compact then any totally geodesic hypersurface embedded in M is sharp.
Proof. The first statement is already known; see for instance [19,Sect. 10]. For be a hyperplane lift of N , and let ρ ∈ PO f (R) be the associated reflection. Let Λ be the stabilizer of H(W ) in Γ, so that N = Λ\H(W ). The latter being sharp, we have that Λ is Zariski-dense in the group of isometries of H(W ). It follows that the centralizer Z of Λ in PO f consists of two elements: 1 and ρ (since this holds over C for the centralizer of PO n (C) in PO n+1 (C)). But in a k-group of order 2, the nontrivial element must be a k-point (for it is easily seen to be invariant under the Galois group). That is, ρ ∈ PO f (k), and the result follows from Lemma 2.1.

Corollary 2.3.
Let M be an arithmetic hyperbolic n-manifold defined over k, with n > 3. Then any sharp hypersurface in M is arithmetic of the first type, and defined over k.
Proof. Let M = Γ\H(V, f ), and N ⊂ M be a sharp hypersurface, i.e., N = Λ\H(W ) for Λ the stabilizer in Γ of some k-rational hyperplane H(W ). The lattice Λ is known to be arithmetic; see [19,Theorem 3.2]. Let f 0 be the restriction of f to W . Then PO + (W, f 0 ) is a k-group admissible for PO(n − 1, 1), whose k-points contain the arithmetic subgroup Λ. We conclude that PO + (W, f 0 ) is the ambient group of Λ.

Extending similitudes. For a similitude
i is of dimension 1. Thus S 0 and S 1 are similar; let α ∈ k be such that S 0 is isomorphic to αS 1 . Further let λ ∈ k be such that ϕ : 2.4. Hyperbolic pieces. By a hyperbolic piece (of dimension n) we mean a complete orientable hyperbolic n-manifold of finite volume with a (possibly empty) boundary consisting of finitely many sharp hypersurfaces. We say that a piece is singular if it has nonempty boundary, and regular otherwise (in which case it is just a hyperbolic manifold in the sense of Sect. Proof. Let k be the field of definition of M 0 , and K the trace field of M . Clearly we have K ⊂ k. Now M being singular, it contains a sharp hypersurface N in its boundary, whose field of definition must be k by Corollary 2.3. Using Remark 1.3 it is easily checked that the field of definition of N (seen as a (n − 1)-manifold) corresponds to the trace field of π 1 (N ), seen as a subgroup of PO(n, 1). We conclude that k ⊂ K, and thus M and M 0 (resp. their fundamental groups Γ and Γ 0 ) have the same trace field. Let G be the ambient group of Γ 0 . From the inclusion Γ ⊂ Γ 0 we have Γ ⊂ G(k), and it follows that G is the ambient group of Γ as well (by uniqueness).
We will say that an arithmetic piece M (resp. its fundamental group) is defined over k if its trace field is k; this extends the terminology used for regular pieces (see Remark 1.3). where M 1 is an arithmetic piece, and for i ≥ 1 the piece M i+1 is obtained from M i by one of the two following operations: (I) gluing an arithmetic piece M 0 to M i using an isometry φ between two hypersurfaces in their respective boundary, i.e., (II) "closing up" two hypersurfaces in the boundary of M i , i.e., where φ is an isometry between two hypersurfaces in the boundary of M i . Theorem 1.5 is a specialization of the following. Proof. We proceed inductively. If M is an arithmetic piece over k then (1) and (2) hold trivially. Suppose then that M 1 is a piece constructed by gluing arithmetic pieces over k, and that it respects the conditions (1) and (2) (recurrence assumption). We will glue M 1 using a sharp hypersurface N 1 ⊂ M 1 . Let K 1 be the trace field of M 1 , which is multiquadratic over k. Let G = PO + (V, f ) be the ambient group of an arithmetic piece containing N 1 , so that G K 1 is the ambient group of M 1 . Thus we may write M 1 = Γ 1 \ M 1 , for M 1 ⊂ H(V, f ) and Γ 1 a discrete subgroup of G(K 1 ) stabilizing M . Let us first examine the case of the gluing of type (I), i.e., M is obtained by gluing M 1 to an arithmetic piece M 0 along a hypersurface N 0 ⊂ M 0 isometric to N 1 . Abstractly we have π 1 (M ) = π 1 (M 0 ), π 1 (M 1 ) . By Corollary 2.6 we have that M 0 is defined over k. Moreover, it is of the first type (Prop. 2.2), and we can write M 0 = Γ 0 \ M 0 , where M 0 ⊂ H(V 0 , f 0 ) for some admissible quadratic space (V 0 , f 0 ) over k, and Γ 0 ⊂ PO + f 0 (k). We fix a hyperplane lift H(W 0 ) of N 0 in H(V 0 , f 0 ), which is necessarily k-rational by Prop. 2.2. We fix as well a hyperplane lift H(W 1 ) ⊂ H(V, f ) of N 1 ; it is k-rational for the same reason. Since N 1 and N 0 are isometric, the quadratic spaces W 1 and W 0 must be similar over k (see [10, 2.6]). More precisely, there exists a similitude ϕ : W 0 → W 1 whose extension to W 0 ⊗ k R lifts the gluing isometry between N 0 and N 1 . By Lemma 2.4 there exists an extension F/k, at most quadratic, over which ϕ extends to a similitude φ : V 0 ⊗ k F → V ⊗ k F (and note that the scaling factor can be chosen totally positive here, so that F is totally real). In particular, conj(φ)(Γ 0 ) ⊂ G(F ). The map φ induces an isometry between H(V 0 , f 0 ) and H(V, f ) that matches the lift hyperplanes of N 0 and N 1 . Possibly after composing with the reflection through H(W 1 ) (which is k-rational by Lemma 2.1), we can hence see the respective universal covers M 0 and M 1 of M 0 and M 1 as lying on both sides of H(W ) in H(V, f ). We can thus write M = Γ\H(V, f ), where Γ = conj(φ)(Γ 0 ), Γ 1 . Hence Γ ⊂ G(F 1 ), where F 1 is the composite of F and K 1 . In particular, the trace field K of Γ is either K 1 or F 1 , which in either case is totally real and multiquadratic over k. It follows that G K is the ambient group of Γ.
Finally note that the inductive proof can be started with any arithmetic piece. Thus the group G can be the ambient group of any of these pieces, as stated in (2).
For a gluing of type (II), we consider two isometric sharp hypersurfaces N 0 , N 1 ⊂ M 1 . If M 0 ⊂ M 1 is an arithmetic piece containing N 0 and (V 0 , f 0 ) and W 0 are defined as above, the same argument gives a similitude φ : V 0 ⊗ k F → V ⊗ k F , where F/k is a totally real extension which is at most quadratic. Now by the induction hypothesis, PO + (V 0 , f 0 ) K 1 is an ambient group for Γ 1 . It follows that we have an isometry ψ : H(V 0 , f 0 ) → H(V, f ) such that conj(ψ) is defined over K 1 and g = φ • ψ −1 (seen as an element of G(R)) lifts an isometry N 1 ∼ = N 0 . Now by construction, there exists g ∈ G(K) such that Γ = Γ 1 , g , and g induces the isometry N 1 ∼ = N 0 . It follows that g and g coincide on H(W 1 ), and thus g ∈ {g , g ρ} where ρ denotes reflection through H(W 1 ). As before, this shows that Γ ⊂ G(F 1 ), where F 1 is the composite of F and K 1 . Thus the trace field K of Γ is multiquadratic over k, and G K is the ambient group of Γ.

Homology and volumes
The main result of this section is Theorem 3.5, of which Theorem 1.12 will be a direct consequence. Since these results hold up to commensurability, we will restrict ourselves to considering lattices that are torsion-free and orientation-preserving. The basic idea is, given an ambient group G K , to compare fundamental classes of lattices Γ ⊂ G(K) in a Q-vector space L(G); the latter will be defined in Sect. 3.1. Many of the proofs are generalizations of the arguments in [7], which deals with quasi-arithmetic lattices. Prop. 3.4 is new; it is the key result needed at the level of ambient groups. Its proof is given in Sect. 3. 3.2. The space L(G). Let K ⊂ R be a number field, and let G be a connected K-group such that G(R) • = PO(n, 1) • ; for the moment we are not assuming that G is admissible (nor pseudo-admissible). The inclusion G(K) ⊂ G(R) composed with the map δ in (3.1) induces a map H n (G(K); Q) → H n (G(R), Ω; Q), (3.2) whose image will be denoted by L(G). If G = G 0,K for some k-group G 0 and some extension K/k (so that G 0 (R) = G(R)), then L(G 0 ) ⊂ L(G) (where L(G 0 ) is the image of H n (G 0 (k); Q)).

Lemma 3.2. Let Λ ⊂ G(K) ∩ G(R) • be a torsion-free lattice. Then [Λ] lies in L(G).
Proof. The exact sequence 0 → JΩ → ZΩ → Z → 0 together with the inclusion Λ ⊂ G(K) give rise to the following diagram with exact rows, which appears in [7, Eq. (6.2)]: This last result permits to view the fundamental class of a lattice Γ as an element of L(G), where G is the ambient group of Γ. By Prop. 3.1 the fundamental class [Γ] determines the volume; the space L(G) can be thought as a replacement for the Bloch group, which appears in the case of dimension n = 3 (see for instance [17,Sect. 12.7]). In the rest of this section we will show that if Γ is pseudo-arithmetic of the first type, one can exhibit generators for L(G) corresponding to fundamental classes of arithmetic lattices.
Remark 3.3. The definition of L(G) is comparable to the "higher groups" P n in Neumann and Yang [22,Sect. 8], with the difference (crucial for our applications) that we work with K-points instead of R-points.

Subordinated fundamental classes.
Assume now that G = PO + f,K is pseudo-admissible over K/k with f a quadratic form over k. As in Sect. 1.6 we write f = f (x 0 , . . . , x n ) in a diagonal form with negative x 0 coefficient, and  Assuming the proposition, we are ready to state and prove the main theorem of this section. Together with Prop. 3.1 it directly implies Theorem 1.12. Proof. The algebraic groups G i are admissible over k, and therefore [7,Prop. 4.2] shows that H n (G(k); Q) has dimension one. The fact that the volume of Γ i \H n is non-zero combined with Prop. 3.1 and Lemma 3.2 implies that L(G i ) is of dimension one and generated by [Γ i ]. Now Prop. 3.4 shows that L(G) is generated by its subspaces L(G i ), and this finishes the proof.
Remark 3.6. We do not claim that the classes [Γ i ] are linearly independent. For n odd it might well be the case, since there are no expected congruences for the special values of L-functions that appears in the covolumes. But this justification certainly fails for n even, and it is not excluded that L(G) has dimension 1 there.
3.4. The homology of ambient groups. The goal of this section is to prove Prop. 3.4. We begin by relating the homology of an algebraic group to that of its simply-connected cover.
Recall (see [3,Sect. 6.3] and [7,Sect. 4]) that for the cohomology with coefficients in R we have two canonical isomorphisms from the continuous cohomology of G i (k ⊗ Q R): where X i is the compact dual space (or compact twin) associated with G i (k ⊗ Q R). For G i admissible, X i is the n-sphere; more precisely, X i is the symmetric space associated withf i , the positive definite version of f i (i.e., f i with the coefficient of x 0 made positive). It follows that the left hand side of (3.3) has dimension s = 2 r . The same argument shows that the dimension of H n ( G(K); R) equals that of H n (X; R), where the compact dual X associated with G(K ⊗ Q R) is a product S n ×· · ·×S n of s copies of the n-sphere; this dimension is s by the Künneth formula.
To prove that (3.3) is indeed an isomorphism it clearly suffices to show that the same map with coefficients extended to R is an isomorphism. This allows (after dualizing the cohomology) to use the canonical maps in (3.4) and their analogues for G. Observe that the first map in (3.4) is induced by the inclusion k ⊂ k ⊗ Q R and the second is just the natural isomorphism between the corresponding Lie algebras (combined with multiplication by a scalar, see [3,Sect. 6.3]). Both maps are functorial in the sense that they commute with the maps induced by the inclusions G i (k) → G(K). Then it is enough to show that the map θ : is an isomorphism. For the multi-index 0 = (0, . . . , 0) we fix a generator [X 0 ] of H n (X 0 ; R). In (3.4) the only contribution to the cohomology of G 0 (k ⊗ Q R) comes from the noncompact factor G 0 (R). Similarly for G(K ⊗ Q R), whose noncompact factors form a product σ G 0 (R) indexed by the set of embeddings σ : K → R fixing k (since G = G 0,K ), that is, by the Galois group Gal(K/k) since K/k is Galois. The Künneth formula then gives an isomorphism H n (S n × · · · × S n ; R) ∼ = σ H n (S n ; R), with the right hand side indexed by Gal(K/k). Let ([S σ ]) σ denote the image of [X 0 ] via (3.5). The

Reflection groups and pseudo-admissibility
In this section we explain how to check if a given hyperbolic reflection group is pseudo-arithmetic. We have applied the method to all examples contained in the tables provided by the program CoxIter developed by Guglielmetti [11]. We have found that all these reflection groups are pseudo-arithmetic for n > 3; see Sect. 4.3 for a summary of the results. 4.1. Ambient groups of Coxeter groups. The references for this section are Vinberg [31,Theorem 5] and Maclachlan [16,Sect. 9]. Let q = −x 2 0 + x 2 1 + · · · + x 2 n denote the standard quadratic form in H n , with associated bilinear form (x, y) → B(x, y) = q(x + y) − q(x) − q(y). Let P ⊂ H n be a Coxeter polytope with r faces whose orthogonal vectors are denoted by e 1 , . . . , e r , and are chosen to have q-norm one. Let Γ ⊂ PO(n, 1) be the Coxeter group associated with P , and let G = (a ij ) ij be its Gram matrix; its entries are a ij = B(e i , e j ). For a subset The trace field of Γ is then the field K = Q({b i 1 ,...,i k }) generated by all the possible "cyclic" products b i 1 ,...,i k , and the ambient group is the group PO f where f is the quadratic form q restricted to the K-vector space V spanned by the v i 1 ,...,i k . Note that the quadratic form f can be determined without explicitly computing the e i ; one proceeds as follows: (1) Extract a minor M of the Gram matrix which has full rank n+1. This minor is then given by M = (B(e m i , e m j )) ij for some sequence m 1 , . . . , m n+1 , and it follows that the vectors e m 1 , . . . e m n+1 are linearly independent over R.
(2) For each j = 1, . . . , n + 1, find a sequence of indices l 1 , . . . , l k with l k = m j such that the product a 1l 1 a l 1 l 2 · · · a l k−1 l k is non-zero (possible since the Coxeter diagram is connected), and set w j = v l 1 ,...,l k . (3) The K-vector space V then admits {w 1 , . . . , w n+1 } as a basis. It follows that f has the matrix representation A = (B(w i , w j )) ij , whose entries can be easily computed from the entries of G.

Quadratic forms over multiquadratic extensions.
For this section we refer to Lam's book [15] and the two papers [5,6] of Elman, Lam and Wadsworth. If k is a field, W (k) will denote the Witt group (or ring) of k (as defined in [15,Chap. II]). Recall that two rank n regular quadratic forms f, f over k are isometric if and only if they represent the same element in W (k), and that 0 ∈ W (k) is represented by any hyperbolic form. Since we will use this group only to compare regular forms of the same rank, we will usually identify quadratic forms with their image in W (k). For a field extension K/k, we let r K k : W (k) → W (K) denote the group homomorphism induced by the inclusion k ⊂ K (sending a quadratic form f to f K , the same form seen over K). If K/k is a quadratic extension with K = k( √ a), a ∈ k, let s : K → k be the k-linear map sending 1 → 0 and √ a → 1. This map induces a group homomorphism s K k : W (K) → W (k) called the transfer map: a quadratic form q ∈ W (K) of rank n is mapped to s K k (q) = s • q, seen as a quadratic form of rank 2n over k. See [15, Sect. VII.3] for details.
Let now K/k be a multiquadratic extension of a global field k. The two results we will need for our criterion are: (1) K/k is an excellent extension, i.e., a quadratic form q is defined over k (by which we mean q ∼ = f K for some quadratic form f over k) if and only if q ∈ im(r K k ). This is a consequence of [5, Theorem 2.13] which states that any extension of a global field k containing a Galois extension of k of even degree is excellent. See also [15,Sect. XII.4].
(2) There is an exact sequence of Witt groups where the product on the right is taken on all fields k ⊂ F ⊂ K with [K : F ] = 2. This is the end of the sequence (1.1) in [6], whose exactness is proven for global fields in Theorem C of the same paper. From these two results we deduce: 4.3. The computations. Let Γ ⊂ PO(n, 1) be a Coxeter group with trace field K and ambient group PO f for some quadratic form f over K. In order to check if Γ is pseudo-admissible, one needs to find a subfield k ⊂ K such that K/k is multiquadratic and a k-form of PO f which is admissible. Using Prop. 4.1, one can already find a candidate for k: the intersection of all codimension 2 subfields F ⊂ K such that s K F (f ) is hyperbolic. If one is then able to find an admissible quadratic form g over k such that f ∼ = g K , one can conclude: Γ is pseudo-arithmetic over K/k with ambient group PO g .
If however the field K is multiquadratic over Q and the form f is itself defined over Q, the computation of such a g becomes superfluous, since any quadratic form over Q of signature (n, 1) is admissible. It turns out that all the examples in the Table 2. Coxeter groups in [28] with trace field of degree 2.
deg Reference [28,  table of Coxeter groups of [11] fulfill this simpler necessary condition for pseudoarithmeticity. The computations were done using the mathematical program Sage [29] and an algorithm of [14] for checking hyperbolicity over number fields. In Tables 1-2, we show a sample of the Coxeter groups we analyzed, namely those from the classification of Roberts [28] having trace field of degree > 1. Each of these Coxeter groups is pseudo-arithmetic over K/Q (where K is the trace field). The indicated quadratic forms were found by brute-force search over quadratic forms of the shape f 0 + ax 2 n , where f 0 = −x 2 0 + x 2 1 + · · · + x 2 n−1 .   Finally, in Figures 2 and 3 we reproduce the diagrams of the Coxeter groups realizing the largest trace fields in even dimensions (n = 4: degree 8) and odd dimensions (n = 5: degree 4).