Geometric construction of homology classes in Riemannian manifolds covered by products of the hyperbolic plane

We study the homology of Riemannian manifolds of finite volume that are covered by a product $(\mathbb{H}^2)^r = \mathbb{H}^2 \times \ldots \times \mathbb{H}^2$ of the real hyperbolic plane. Using a variation of a method developed by Avramidi and Nyguen-Phan, we show that any such manifold $M$ possesses, up to finite coverings, an arbitrarily large number of compact oriented flat totally geodesic $r$-dimensional submanifolds whose fundamental classes are linearly independent in the real homology group $H_r(M;\mathbb{R})$.


Introduction
Let M be a Riemannian manifold of finite volume that is covered by a product (H 2 ) r = H 2 × . . . × H 2 . If r = 1, then M is a hyperbolic surface and its homology is well understood. Otherwise, M can be a complicated object. For example, let d > 0 be a square-free integer and consider the real quadratic number field F = Q( √ d) with its two distinct real embeddings σ 1 , σ 2 : F → R. Let O F be the ring of integers of F . Then the group SL 2 (O F ) acts properly discontinuously on the product H 2 × H 2 by γ · (z 1 , z 2 ) := (σ 1 (γ) · z 1 , σ 2 (γ) · z 2 ), where σ i (γ) · z i is the action of SL 2 (R) on H 2 by fractional linear transformations. For any torsion-free subgroup of finite index Γ ⊂ SL 2 (O F ), the quotient Γ\(H 2 × H 2 ) is a Riemannian manifold of finite volume that is covered by H 2 × H 2 . It is called a Hilbert modular surface and is an irreducible locally symmetric space of higher rank.
The homology of such a locally symmetric space is in general hard to compute, and even if one can do so, the geometric meaning of the homology classes is often lost during the computation. We choose a more geometric approach going back to Millson [12], in which one studies homology classes that are the fundamental classes of totally geodesic submanifolds.
Promising candidates for such submanifolds are the compact flat totally geodesic submanifolds of dimension equal to the rank of the locally symmetric space. It is known that these submanifolds exist in any nonpositively curved locally symmetric space of finite volume, and Pettet and Souto proved in [16,Theorem 1.2] that they are non-peripheral, which means they cannot be homotoped outside of every compact subset of the locally symmetric space.
This suggests that these submanifolds might contribute to the homology of the locally symmetric space. Avramidi and Nyguen-Phan [1] have investigated this question for the locally symmetric space M = SL n (Z)\ SL n (R)/ SO(n) and proved that, in fact, up to finite coverings, the compact flat totally geodesic submanifolds of dimension equal to the rank of M contribute to the homology group of M with coefficients in a field.
We prove the following theorem, which shows that this is also true for all Riemannian manifolds of finite volume that are covered by products of H 2 : In particular, it follows that the r-th Betti number of a Riemannian manifold of finite volume that is covered by (H 2 ) r can be made arbitrarily large by going to a finite covering space of the manifold.
Our proof proceeds as follows: Using induction on dim(M ), one finds irreducible Riemannian manifolds M 1 , . . . , M k for some k ∈ N and a finite covering M 1 × . . . × M k → M . An application of the Künneth theorem for homology now shows that it suffices to prove Theorem 1.1 for irreducible manifolds. So assume that M is irreducible. If r = 1, then M is a hyperbolic surface and there exists a finite covering surface M of M whose genus is at least n. The surface M has n distinct simple closed geodesics whose real homology classes are linearly independent in H 1 (M ; R). This proves the claim for r = 1. If r > 1, then Margulis' arithmeticity theorem implies that M is arithmetic, by which we mean that it is finitely covered by a quotient of (H 2 ) r by an arithmetically defined lattice in SL 2 (R) r .
The goal of this article is to explain our proof of Theorem 1.1 for arithmetic manifolds. It is structured as follows: In Section 2, we fix our notation for algebraic groups, discuss arithmetically defined lattices, and state Margulis' arithmeticity theorem. In Section 3, we describe the arithmetically defined lattices in SL 2 (R) r using quaternion algebras. In Section 5, we study flats in symmetric spaces, and in Section 6 we discuss geometric cycles. Finally, in Section 7, we describe our construction of the covering M → M and the submanifolds F 1 , . . . , F n ⊂ M for an arithmetic manifold M , which is based on the techniques developed by Avramidi and Nyguen-Phan in [1].
The material covered in this article evolved from the author's doctoral thesis [22], in which the reader can find more details on our construction and the proof of Theorem 1.1.

Algebraic groups and Margulis' arithmeticity theorem
We consider algebraic groups as special cases of group schemes and identify them with their functors of points (see [21,13]). By this, we mean the following: Let R be a commutative ring. A group scheme over R is a functor G : Alg R → Grp from the category of commutative R-algebras to the category of groups that is representable by a finitely generated R-algebra. We denote this algebra by O(G). One can think of G as a group functor defined by polynomial equations with coefficients in R. In fact, by choosing a surjection π : R[X 1 , . . . , X n ] → O(G), we obtain for each commutative R-algebra A a natural inclusion which identifies the group G(A) with the vanishing set of the ideal ker(π) ⊂ R[X 1 , . . . , X n ] in A n . For a topological R-algebra A, we put on G(A) the unique weakest topology for which the above inclusion (and thus any such inclusion) is continuous with respect to the product topology on A n .
The extension of scalars of a group scheme G over R to some ring extension S/R is the group scheme G S : Alg S → Grp, A → G(res S (A)). Here, res S (A) denotes A as an S-algebra. If the extension map σ : R → S is not clear from the context, then we write res σ (A) instead of res S (A) and G σ instead of G S .
An algebraic group over a field K, or in short a K-group, is simply a group scheme G over K. Its group G(K) of points with values in an algebraic closure K of K is then an affine variety in the space K n equipped with the Zariski topology and the group operations are polynomial maps. The algebraic group G is said to be connected or finite if this affine variety is connected or finite, respectively. Let L/K be a finite separable field extension. The restriction of scalars of an algebraic group G over L to K is the functor Res L/K H : . This is an algebraic group over K (see [13, p. 57]) and the natural isomorphism Every algebraic group G over a number field F has an integral form. This is a group scheme G 0 over the ring of integers O F of F together with an This notion is independent of the choice of the integral form, because the groups of O F -points of any two integral forms of G are commensurable with each other (see [17,Proposition 4.1]).
By an important theorem of Borel and Harish-Chandra [2], any arithmetic subgroup Γ ⊂ H(Q) of a semisimple algebraic group H over Q is a lattice in H(R). The following definition describes all lattices in the group of real points of an algebraic group over R that are constructed in this way: is compact, and an arithmetic subgroup Γ ⊂ H(Q) such that ∆ is commensurable with Φ(H(Γ)).
Non-arithmetically defined lattices are known to exist in some real algebraic groups of rank one. For example, in SL 2 (R) this follows from the existence of uncountably many non-commensurable hyperbolic surfaces (see [9, p. 63]).
Margulis' celebrated arithmeticity theorem states that in a real algebraic group of higher rank, or similarly in a real Lie group of higher rank, all irreducible lattices are arithmetically defined (see [11,Theorem IX.1.11]): Theorem 2.2 (Margulis' arithmeticity theorem). Let G be a connected semisimple R-group without R-anisotropic almost R-simple factors and with rank R (G) > 1. Then every irreducible lattice in G(R) 0 is arithmetically defined.

Unit groups in quaternion algebras
The arithmetically defined lattices in SL 2 (R) r can be described using quaternion algebras. Let K be a field of characteristic zero. A quaternion algebra over K is an algebra D over K for which there exists a vector space basis {1, i, j, k} and a, b ∈ K × such that (3.1) We then call {1, i, j, k} a quaternionic basis for D. The reduced norm of an The reduced norm of x ∈ D is preserved by every automorphism of D and is therefore independent of the choice of the quaternionic basis. For example, in the case of the matrix algebra D = M 2 (K), we have N (x) = det(x). An element x ∈ D is invertible if and only if N (x) = 0. We write D 1 for the group of units of reduced norm one in D.
The general linear group associated to a quaternion algebra D over K is the algebraic group GL D : Alg K → Grp, A → (A ⊗ K D) × . We extend the reduced norm to tensor products A ⊗ K D for K-algebras A using (3.2) and define the special linear group SL D : Alg K → Grp, A → (A ⊗ K D) 1 .
We now study quaternion algebras over number fields. Let F be a number field. The analog for a quaternion algebra D over F of the ring of integers of F is an order. This is a subring Λ ⊂ D that is a finitely generated O F -submodule of D and spans D over F . For example, the subring M 2 (O F ) ⊂ M 2 (F ) is an order. A quaternion algebra has many orders, but the groups of units of any two of its orders are commensurable with each other (see [8,Lemma 4.6.9]).
For an order Λ ⊂ D, we define its general linear group GL Λ : This is a group scheme over O F and an integral form of the algebraic group GL D . If Λ is the O F -span of a quaternionic basis for D, then we again extend the reduced norm using (3.2) and define the special linear group SL Λ : 1 , which is an integral form of SL D . It follows that for any order Λ ⊂ D, the group of units of reduced norm one Λ 1 := Λ ∩ D is an arithmetic subgroup of SL D (F ).
For any real embedding σ : F → R, the algebra D ⊗ F res σ (R) is either isomorphic to M 2 (R) or is a division algebra. In the first case, we say D is split at σ and otherwise ramified at σ. This now leads us to the following: Definition 3.1. A subgroup ∆ ⊂ SL 2 (R) r is said to be derived from a quaternion algebra if there exists a totally real number field F , a quaternion algebra D over F that is split at exactly r distinct real embeddings σ 1 , . . . , σ r : F → R, an isomorphism τ i : D ⊗ F res σ i (R)  − → M 2 (R) in Definition 3.1 are not unique, but any two choices for τ i differ only by conjugation with a matrix in GL 2 (R) by the Skolem-Noether theorem. So up to commensurability and conjugation in GL 2 (R) r , the subgroup of SL 2 (R) r derived from a quaternion algebra depends only on the isomorphism class of the quaternion algebra.
A subgroup derived from a quaternion algebra is readily seen to be an arithmetically defined lattice in SL 2 (R) r . In fact, the converse direction also holds (see [22,Theorem 5.44]): We write D = (a, b) F for the quaternion algebra determined by (3.1). The field E := F ( √ a) is a splitting field for D. By this, we mean that it is a field extension E/F so that D ⊗ F E ∼ = M 2 (E). A quadratic extension E/F is a splitting field for D if and only if for all of the places v of F where D is ramified, the local completion E v /F v is a quadratic extension (see [10,Theorem 7.3.3 and its proof]). Here, E v is the completion of E at some place w of E lying above v. Note that for any two places w and w of E lying above v, the corresponding completions of E are F v -isomorphic to each other by [15,Proposition II.9.1], which justifies the notation E v . Moreover, for any Next, we show that the constants a, b ∈ F × defining the isomorphism class of the quaternion algebra (a, b) F can always be chosen in a certain way. We will need this in our computations later in Section 7.

Lemma 3.4.
Let F be a number field. Then for every quaternion algebra D over F , there exist a, b ∈ O F in the ring of integers of F such that D is isomorphic to (a, b) F and such that for any real embedding σ : F → R at which D is split, we have σ(a) > 0.
Proof. We use the Grunwald-Wang theorem [19, p. 29] to construct a splitting field E/F for D as follows: By this theorem, there exists a quadratic extension E/F whose local completions E v /F v at the places v of F where D is ramified are quadratic extensions, and such that at all real places v of F at which D is split, they are trivial extensions. Then E/F is a splitting field for D by the criterion stated above. Since

Adeles and congruence subgroups
We will use adeles and congruence subgroups to construct subgroups of finite index in arithmetic groups. Let F be a number field with ring of integers O F . We denote by A f,F the ring of finite adeles of F and by O f,F the ring of integral finite adeles of F (see [17, pp. 10-13]). We consider F as a subring of A f,F by the diagonal embedding F → A f,F , and similarly We have (see [22,Proposition 4.39]): For a subgroup of the integral points Γ ⊂ G(O F ), the group Γ(a) coincides with ker(Γ → G(O F /aO F )) and has finite index in Γ. We call Γ(a) the principal congruence subgroup of Γ of level a. More generally, any subgroup of Γ that contains a principal congruence subgroup has finite index in Γ and is called a congruence subgroup of Γ.
It is possible that there exist subgroups of finite index in G(O F ) that are not congruence subgroups. Examples of such subgroups in SL 2 (Z) were already known to Fricke and Klein in the 19th century (see [18, p. 299]). The group scheme G is said to have the congruence subgroup property if every subgroup of finite index in G(O F ) is a congruence subgroup. Chevalley [4] proved in 1951 that GL 1 has this property: (Chevalley). Let F be a number field. Then for every subgroup

Polar regular elements and flats
We will use polar regular elements, as introduced by Mostow in [14, p. 12], to algebraically describe the maximal flat subspaces of a symmetric space.
Let G be a connected linear semisimple Lie group. By [7, p. 431], each g ∈ G has a unique decomposition with g u , g h , g e ∈ G such that g u ,g h , and g e correspond in one (and thus any) embedding G → GL n (R) to a unipotent, a hyperbolic, and an elliptic matrix, respectively, and such that they all commute with each other. We call a matrix in GL n (R) semisimple if it is diagonalizable over C. A semisimple matrix is called hyperbolic if all its eigenvalues are real and positive, and it is called elliptic if all its eigenvalues have absolute norm one. We call (5.1) the real Jordan decomposition of g.
where C G (g h ) denotes the centralizer of g h in G.
Let X G be the symmetric space associated to G. A flat in X G is a connected totally geodesic submanifold of X G whose curvature tensor vanishes. A flat is called maximal if it is of maximal dimension among all flats in X G . We write G A := {g ∈ G : g · A = A} for the stabilizer group of flat A ⊂ X G .
By [14,Lemma 5.2], we have the following relationship between polar regular elements in G and maximal flats in X G : Proposition 5.2. Let G be a connected linear semisimple Lie group and let g ∈ G be polar regular. Then there exists a unique maximal flat A ⊂ X G in the symmetric space associated to G such that g · A = A. Moreover, the centralizer C G (g) is a subgroup of G A and acts transitively on A.
The maximal flats in (H 2 ) r are the products of geodesic lines in H 2 . An element g = (g 1 , . . . , g r ) ∈ SL 2 (R) r is polar regular if and only if each g i has two distinct real eigenvalues.
Let Γ ⊂ G be a lattice. We say that a flat A ⊂ X G is Γ-compact if the quotient Γ A \A is compact. Then the image of A in Γ\X G is also compact. By [14,Lemma 8.3'], the set of Γ-compact maximal flats is dense in the space of all maximal flats: Theorem 5.4 (Density of Γ-compact flats). Let G be a connected linear semisimple Lie group. Let Γ ⊂ G be a lattice and let A ⊂ X G be a maximal flat. Then for every open neighborhood of the identity U ⊂ G, there exists some u ∈ U such that u · A is a Γ-compact maximal flat in X G that is stabilized by a polar regular element of Γ.

Geometric cycles
Let G be a linear semisimple Lie group and let X G be the symmetric space associated to G. For a closed subgroup H ⊂ G, we can always find a maximal compact subgroup K ⊂ G such that K H := K ∩ H is a maximal compact subgroup of H. We write X H := H/K H . Then X H is diffeomorphic to a Euclidean space and the inclusion H → G induces a closed embedding j H : X H → X G whose image is a totally geodesic submanifold of X G (see [20, p. 213]).
Consider now a torsion-free lattice Γ ⊂ G and the corresponding locally symmetric space Γ\X G . The map j H descends to an immersion into the locally symmetric space Γ\X G , but this map will in general not be an embedding. In the arithmetic setting, we can always obtain an embedding by passing to a subgroup of finite index (see [ In the situation of the above theorem, we call the image of j H|Γ in Γ \X G a geometric cycle. An effective strategy to show that the fundamental class of a geometric cycle is nontrivial in the homology of Γ \X G is to find another geometric cycle in Γ \X G such that their intersection product is nontrivial.

Construction of flat submanifolds
We now explain our construction of compact flat totally geodesic submanifolds with linearly independent homology classes in an arithmetic Riemannian manifold covered by (H 2 ) r . This will finish the proof of Theorem 1.1. As we have seen, it suffices to consider for this task a quotient M = ∆\(H 2 ) r , where ∆ ⊂ SL 2 (R) r is a subgroup derived from a quaternion algebra.
Throughout this section, we fix the following assumptions: F is a totally real number field and we denote by {σ 1 , . . . , σ d } the set of all distinct real embeddings F → R. Further, D = (a, b) F is a quaternion algebra over F that is split at the first r embeddings and ramified at the remaining ones. Λ ⊂ D is an order and Γ ⊂ Λ 1 is a torsion-free subgroup of finite index. For We assume that F ⊂ R and σ 1 is the identity embedding. Further, we assume that a, b ∈ O F and σ 1 (a), . . . , σ r (a) > 0. This is justified by Lemma 3.4. We also assume that Λ is the O F -span of a quaternionic basis {1, i, j, k} for D and that with respect to this basis, τ 1 is given by This is justified by Remark 3.2. In particular, we have τ 1 (D) ⊂ M 2 (F ( √ a)). In this setting, we have an action of D 1 on (H 2 ) r by g · (z 1 , . . . , z r ) := (τ 1 (g) · z 1 , . . . , τ r (g) · z r ), where τ i (g) · z i is the action of SL 2 (R) on H 2 by fractional linear transformations. We extend this action to an action of D × on (H 2 ) r through the maps τ i as above by defining the action of GL 2 (R) on H 2 as follows:

Building a configuration of flats
The following lemma shows that two generic geodesic lines in H 2 can be slightly perturbed without changing the way they intersect. We denote by ∂ ∞ H 2 the boundary at infinity of the hyperbolic plane (see [5, p. 27]).
is an open neighborhood of the identity in SL 2 (R). By the construction of U , the endpoints of two geodesic lines u · L 1 and v · L 2 with u, v ∈ U in ∂ ∞ H 2 are linked if and only if whose of L 1 and L 2 are linked. So the statement of the lemma follows.
each have two distinct real eigenvalues. (4) Each B j is stabilized by an element β j ∈ D × such that τ 1 (β j ), . . . , τ r (β j ) each have two distinct real eigenvalues and Proof. We start with the first condition. For this, we choose geodesic lines only if i ≤ j (See Figure 2 for an example with n = 3). Set A i := L i × · · · × L i and B j := M j × · · · × M j . The first condition is now satisfied. By Lemma 7.2, there exists an open neighborhood of the identity U ⊂ SL 2 (R) so that we may perturb each L i and M j by isometries in U without invalidating the first condition. The product U r is an open neighborhood of the identity in SL 2 (R) r . So for each i ∈ {1, . . . , n}, there exists by Theorem 5.4 an element u i ∈ U r such that u i · A i is a Γ-compact flat that is stabilized by an element α i ∈ D 1 for which τ 1 (α i ), . . . , τ r (α i ) each have two distinct real eigenvalues. We now replace each A i by u i · A i and the first three conditions are satisfied.
For the last condition, we choose an element x 0 ∈ D × with (x 0 ) 2 = a and x 0 / ∈ F . Then for each k ∈ {1, . . . , r}, we have (τ k (x 0 )) 2 = σ k (a)I 2 and τ k (x 0 ) / ∈ R · I 2 , where I 2 ∈ M 2 (R) is the identity matrix. Recall that σ k (a) > 0 by assumption. So the minimal polynomial of τ k (x 0 ) over R is . Hence, each τ k (x 0 ) has two distinct real eigenvalues and τ 1 (x 0 ) is diagonalizable over F ( √ a). So by Proposition 7.1, there exists a unique maximal flat B 0 ⊂ (H 2 ) r that is stabilized by x 0 . The group SL 2 (R) r acts transitively on the set of all maximal flats in (H 2 ) r , and so for each j ∈ {1, . . . , n}, we can find some T j ∈ SL 2 (R) r with B j = T j · B 0 .
By the real approximation theorem [13,Theorem 25.70], the image of D 1 in SL 2 (R) r under the maps τ 1 , . . . , τ r is dense. Since the subsets U T j ⊂ SL 2 (R) r are open, there exist x j ∈ D 1 and v j ∈ U with (τ 1 (x j ), . . . , τ r ( We now replace each B j by v j · B j and then all four conditions are satisfied.

Controlling the intersections in the quotient
Our next goal is to find a finite covering space of Γ\(H 2 ) r in which the image of the flats A i and B j from Proposition 7.3 are embedded submanifolds and to control the intersections of these submanifolds. In order to ease notation, we fix throughout this subsection two maximal flats A, B ⊂ (H 2 ) r that are either disjoint or intersect transversally in a single point and we fix α ∈ D 1 and β ∈ D × that stabilizes A and B, respectively. We assume that τ 1 (α), . . . , τ r (α) and τ 1 (β), . . . , τ r (β) each have two distinct real eigenvalues and that τ 1 (β) is diagonalizable over F ( √ a). We also assume that A is Γ-compact. Proposition 7.4. There exists a subgroup of finite index Γ cent ⊂ Γ such that every element of Γ cent which stabilizes A commutes with α, and every element of Γ cent which stabilizes B commutes with β.
Proof. By Proposition 7.1, the centralizer We now use the rings A f,F and O f,F from Section 4. For each i ∈ {1, . . . , m}, is of finite index in Γ, and the proof is complete.
We can now show that the images of A and B in some finite covering space of Γ\(H 2 ) r are embedded submanifolds: Proof. The algebraic group SL D is connected and semisimple. Because of α ∈ SL D (F ), there exists by [13, pp. 33-35] a unique smallest F -subgroup C SL D (α) of SL D such that for all fields K with F ⊂ K, we have Since τ 1 (α) is a diagonalizable matrix, we see that C SL D (α) becomes isomorphic to GL 1 over an algebraic closure of F , so C SL D (α) is connected and reductive.
Let G := Res F/Q (SL D ) and H := Res F/Q C SL D (α) . Then G is a connected semisimple Q-group and H is a connected reductive Q-subgroup of G. For each i ∈ {r+1, . . . , d}, we choose an isomorphism ρ i : where H is the real Hamilton quaternion algebra. The isomorphisms τ 1 , . . . , τ r and ρ r+1 , . . . , ρ d induce an isomorphism maps j H (X H ) onto the flat A. Note that H(Q) = C D 1 (α). So by Propositions 7.1 and 7.4, we have Γ A = Γ ∩ H(Q) for every subgroup of finite index Γ ⊂ Γ cent . Hence, by Theorem 6.1, there exists a subgroup of finite index Γ 0 ⊂ Γ cent such that for all subgroups of finite index Γ ⊂ Γ 0 , the map Γ A \A → Γ \(H 2 ) r is a closed embedding whose image is an orientable flat totally geodesic submanifold.
Similarly, we obtain a subgroup of finite index Γ 1 ⊂ Γ cent such that for every subgroup of finite index Γ ⊂ Γ 1 , the map Γ B \B → Γ \(H 2 ) r is a closed embedding whose image is an orientable flat totally geodesic submanifold. We set Γ emb := Γ 0 ∩ Γ 1 and the proof is complete.

Lemma 7.6. Let Γ ⊂ Γ emb be a subgroup of finite index. Then Γ A is a disjoint union of copies of A, that is, for any
Proof. Let γ ∈ Γ and assume that γA ∩ A = ∅. Then there exist x 1 , x 2 ∈ A with x 2 = γx 1 , and so we have Γ x 1 = Γ x 2 . Since the map Γ A \A → Γ \(H 2 ) r is injective by Proposition 7.5, it follows that Γ A x 1 = Γ A x 2 . So there exists some δ ∈ Γ A with x 1 = δx 2 . Hence, we have δγx 1 = x 1 , and because Γ is torsion-free, this implies that γ = δ −1 . So we have γ ∈ Γ A , or, in other words, γA = A. The statement for Γ B can be proven analogously. Our next task is to find a finite covering space of the locally symmetric space Γ\(H 2 ) r in which we can control the intersection of the images of A and B. We start with some technical results:

subgroup of finite index. Then for any
Proof. If there exists some δ ∈ Γ B with γ 1 A = δγ 2 A, then we also have Proof. Let π : (H 2 ) r → Γ \(H 2 ) r be the projection map. We write F := π(A) and G := π(B). Since F and G are closed totally geodesic submanifolds of Γ \(H 2 ) r and F is compact, it follows that F ∩ G is a compact manifold. In particular, F ∩ G has only finitely many path-connected components. Thus, it suffices to show that for all γ 0 , γ 1 ∈ Γ for which there is a continuous path in F ∩ G connecting a point in π(γ 0 A ∩ B) to a point in π(γ 1 A ∩ B), we have Let c : [0, 1] → A ∩ B be such a path and choose preimages and is a covering map. By the lifting property of p B and the fact that is a disjoint union of copies of A by Lemma 7.6, and so c(0) = x 0 ∈ γ 0 A implies that the image of c must be fully contained in γ 0 A. In particular, On the other hand, using π( c(1)) = c(1) = π(x 1 ) and the injectivity of j B , we see that Because of x 1 ∈ γ 1 A, this shows that c(1) ∈ δγ 1 A for some δ ∈ Γ B . In conclusion, we have c(1) ∈ γ 0 A ∩ δγ 1 A, and so Lemma 7.6 implies that δγ 1 A = γ 0 A. Hence, by Lemma 7.8, we have Corollary 7.10. Let Γ ⊂ Γ emb be a subgroup of finite index. Then there exist γ 1 , . . . , γ m ∈ Γ such that the intersection of the images of A and B in Γ \(H 2 ) r is the image of the projection map Proof. By Proposition 7.9, there exist γ 1 , . . . , γ m ∈ Γ with . . , m}. So by Lemma 7.8, there exists some δ ∈ Γ B with γA = δγ i A. Hence, there is some y ∈ A with x = δγ i y. From γ i y = δ −1 x ∈ B and Γ x = Γ δγ i y = Γ γ i y, we deduce that The next two lemmas will be used in the proof of Proposition 7.13.

Lemma 7.11. For the centralizers of α and β in
Proof. We only show C D (α) ∩ C D (β) = F . This suffices, because taking the tensor product with A f,F commutes with taking the centralizer. So suppose to the contrary that there exists x ∈ D with x / ∈ F , αx = xα, and βx = xβ. Note that α and β do not commute with each other, because otherwise we would have A = B by Proposition 7.1, in contradiction to our assumptions on A and B. This implies α, β / ∈ F , and so we see that {1, x, α, β} is a linearly independent subset with four distinct elements of the linear subspace C D (x) ⊂ D. Since dim F (D) = 4, it follows that C D (x) = D, and so we have x ∈ Z(D) = F . But this contradicts the assumption x / ∈ F .
To simplify the notation, we will from now write C G (g) := {h ∈ G : gh = hg} for a group G whenever multiplication with g is defined, even if g ∈ G.
Recall from Proposition 7.1 that C D × (α) and C D × (β) act by orientationpreserving isometries on the flats A and B, respectively. The next two propositions will, combined with Corollary 7.10, allow us to control the intersection of the images of A and B in a finite covering space of Γ\(H 2 ) r . Proposition 7.13. For every γ ∈ Λ 1 that is in the closure of C Λ 1 (β)C Λ 1 (α) in SL Λ (O f,F ), there exist x ∈ C D × (β) and y ∈ C D × (α) such that γ = xy and such that x acts by orientation-preserving isometries on (H 2 ) r .

Proof.
Step 1: , so their product is also closed and contains the set C Λ 1 (β)C Λ 1 (α), hence also the closure point γ of this set.
Step 2: Next, we find some c ∈ A f,F with cx ∈ D × . To achieve this, we observe that x is a solution in D ⊗ F A f,F of the homogeneous system of linear equations x α = (γαγ −1 )x , x β = βx . The coefficients of this system are in F . Let B be an F -basis for the space of solutions of this system in D. Then the solution space in D ⊗ F A f,F is the A f,F -span of B. In particular, B = ∅. Moreover, the function x → N (x) on the solution space in D can be expressed in coordinates with respect to B by some multivariate polynomial P ∈ F [X 1 , . . . , X m ]. Because of N (x ) = 0, we have P = 0. So since F is an infinite field, there exists a solution with nonzero reduced norm in D, that is, there exists an element x ∈ D × satisfying xα = (γαγ −1 )x, xβ = βx.
The element x −1 x ∈ D ⊗ F A f,F commutes with β. It also commutes with α because from the above two linear systems of equations, we deduce that So by Lemma 7.11, we have x = cx ∈ D × for some c ∈ A f,F as required.
Let y := c −1 y . Then y = c −1 (x ) −1 γ = x −1 γ ∈ D × , and so we have γ = xy with x ∈ C D × (β) and y ∈ C D × (α). It remains to show that x acts by orientation-preserving isometries on (H 2 ) r . We do this in the next two steps.
Step 3: Lemma 7.12 and µ 2 (O f,F ) acts transitively on the fibers of the multiplication map by this lemma. Hence, for the images under this map, we obtain Consequently, we have Λ 1 (a) ⊂ U and so (7.2) implies that On the other hand, γ is in the closure of C Λ 1 (β)C Λ 1 (α) and so we have (7.3). This proves the claim and thus finishes this step.
Step 4: Finally, we show that x = cx acts by orientation-preserving isometries on (H 2 ) r . Assume to the contrary that this is not the case. Then there must exist some i ∈ {1, . . . , r} with det( We can extend σ i to a real embedding σ i : E → R as in the proof of Lemma 3.4, because by assumption we have σ i (a) > 0.
Recall that τ 1 (D × ) ⊂ M 2 (E). So τ 1 induces an F -algebra homomorphism D → M 2 (E) and hence also an F -homomorphism GL D → Res E/F GL 2 . By [17, p. 15], we have (Res E/F GL 2 )(A f,F ) ∼ = GL 2 (A f,E ), and so we get a continuous group homomorphism that extends τ 1 on D × . The matrix Φ(β) = τ 1 (β) ∈ GL 2 (E) is by assumption diagonalizable over E with two distinct eigenvalues. So there exists a one-dimensional subspace L ⊂ E 2 which is invariant under τ 1 (β). The corresponding eigenspace in (A f,E ) 2 of τ 1 (β) is A f,E · L, and so every matrix in M 2 (A f,E ) that commutes with τ 1 (β) stabilizes A f,E · L. Let now v ∈ L be a nonzero vector and let ∈ {1, 2} be such that the -th coordinate of v is v = 0. Consider the map Note that s is multiplicative and so its image is contained in Moreover, s is continuous because Φ is continuous and A f,E is a topological E-algebra. We have s(C D × (β)) ⊂ (E × ) 2 , and so by writing x = c −1 cx , we see that is contained in E × and has only negative images under σ i because of σ i (c 2 ) < 0. Note that ⊂ V + , and so we obtain Since s is continuous, the preimage This contradicts the result from the previous step. So x must act by orientation-preserving isometries on (H 2 ) r and the proof is complete. Proposition 7.14. There exists a subgroup of finite index Γ prod ⊂ Γ emb such that every γ ∈ Γ prod with γA ∩ B = ∅ can be written as γ = xy with x ∈ C D × (β) and y ∈ C D × (α) so that x acts by orientation-preserving isometries on (H 2 ) r .
We can now show the following result about the intersection of the images of A and B in a finite covering space of the locally symmetric space Γ\(H 2 ) r : Proof. We denote as above by F and G the images of A and B in Γ \(H 2 ) r , respectively. By Corollary 7.10, there exist γ 1 , . . . , γ m ∈ Γ such that the projection map induces a surjection Because of Γ ⊂ Γ prod , we can apply Proposition 7.14 and write each γ i as γ i = x i y i for some x i ∈ C D × (β) and y i ∈ C D × (α) such that x i acts by orientation-preserving isometries on (H 2 ) r . Now choose orientations A + on A and B + on B, and let F and G carry the induced orientations as described in Remark 7.7. By Proposition 7.1, we have x i · B + = B + and y i · A + = A + , and so we obtain . This shows that F and G intersect if and only if A and B intersect. Furthermore, we see that in this case the intersection of F and G is transverse and the intersection number is the same in each point of intersection, because for each i ∈ {1, . . . , m}, the action of x i maps the intersection A + ∩ B + to the intersection γ i A + ∩ B + while also preserving the orientation of the ambient space (H 2 ) r .

Finishing the proof of the main theorem
We now use Proposition 7.15 from the previous subsection to show that the flats that we have constructed in Proposition 7.3 give us a family of linearly independent real homology classes. For this, we will use de Rham cohomology (see [3]).
Let M be a smooth oriented n-manifold. We write H * dR (M ) for the de Rham cohomology groups of a smooth oriented manifold M and H * dR,c (M ) for the de Rham cohomology groups of M with compact support. (7.7) In the above proposition, we equip S 1 ∩ S 2 with the natural orientation induced by the orientations on M , S 1 , and S 2 . If S 1 and S 2 are of complementary dimensions in M , then this orientation is simply given by the intersection numbers of S 1 and S 2 .
We now use this to finish the proof of our main result. Recall that we call a Riemannian manifold M covered by (H 2 ) r arithmetic if it is finitely covered by a quotient of (H 2 ) r by an arithmetically defined lattice in SL 2 (R) r . The following theorem completes our proof of Theorem 1.1: Proof. As explained before, we can and will assume that M = Γ\(H 2 ) r with Γ as in the beginning of this section. By Proposition 7.3, there exist maximal flats A 1 , . . . , A n and B 1 , . . . , B n in (H 2 ) r such that for all i, j ∈ {1, . . . , n}, Proposition 7.15 can be applied to the flats A = A i and B = B j . So there exists a subgroup of finite index Γ ⊂ Γ such that the images of A 1 , . . . , A n and B 1 , . . . , B n in M := Γ \(H 2 ) r are closed orientable flat totally geodesic r-dimensional submanifolds. We denote them by F 1 , . . . , F n and G 1 , . . . , G n and choose orientations on them as in Remark 7.7. By Proposition 7.16, we have Furthermore, by what we know about the intersections of F i and G j from Proposition 7.15, this sum is nonzero if and only if F i ∩ G j = ∅, which is the case if and only if i ≤ j. It follows that the matrix is upper triangular with nonzero entries on the diagonal, hence is in GL n (R). Since the map H r dR,c (M ) × H r dR (M ) → R, (ω, τ ) → M ω ∧ τ is bilinear, it follows that the cohomology classes η F 1 , . . . , η Fn ∈ H r dR,c (M ) are linearly independent. They are mapped by the Poincaré duality isomorphism to the images of the fundamental classes [F 1 ], . . . , [F n ] in H r (M ; R), and so these homology classes are also linearly independent.