The Hanna Neumann conjecture for surface groups

The Hanna Neumann conjecture is a statement about the rank of the intersection of two finitely generated subgroups of a free group. The conjecture was posed by Hanna Neumann in 1957. In 2011, a strengthened version of the conjecture was proved independently by Joel Friedman and by Igor Mineyev. In this paper we show that the strengthened Hanna Neumann conjecture holds not only in free groups but also in non-solvable surface groups. In addition, we show that a retract in a free group and in a surface group is inert. This implies the Dicks–Ventura inertia conjecture for free and surface groups.


Introduction
Let G be a group. We say that G satisfies the Howson property if the intersection of two finitely generated subgroups of G is again finitely generated. This property was introduced by Howson [How54] where he proved that it holds for free groups. In fact, Howson gave an effective bound for the number of generators of the intersection which was improved few years later by H. Neumann [Neu56].
Let d(G) denote the number of generators of a group G. H. Neumann showed that if U and W are non-trivial finitely generated subgroups of a free group, then and she conjectured that, in fact, the factor 2 can be omitted. This conjecture became known as the Hanna Neumann conjecture.
In 1980, W. Neumann improved the result of H. Neumann. For a group G we put d(G) = max{0, d(G) − 1}. W. Neumann showed that if U and W are finitely generated subgroups of a free group F , then and he also conjectured that again the factor 2 can be omitted. This conjecture became known as the strengthened Hanna Neumann conjecture. It was proved independently by Friedman [Fri14] and by Mineyev [Mine12] in 2011. These were also the first proofs of the Hanna Neumann . Then by [Tur96, Theorem 1], φ ∞ (G) is a retract in G, and, thus, by Theorem 1.4, we only have to show that Fix(φ) is inert in φ ∞ (G). By [IT89, Theorem 1], the restriction of φ on φ ∞ (G) is an automorphism. Thus, [DV96, Theorem IV.5.5] gives us the desired result.
Ventura has pointed out to us that the same reduction argument works in the case of a surface group G.
If φ is not an automorphism, then φ(G) has infinite index and, hence, it is free. In particular, φ ∞ (G) is still a retract of G and the argument applies verbatim. The only difference is when 1852 The Hanna Neumann conjecture φ is an automorphism. However, this case was proved already by Wu and Zhang in [WZ14,Corollary 1.5].
Let us briefly describe the structure of the paper. In § 2 we include main definitions and facts that we use in the paper. In § 3 we introduce L 2 -Betti numbers β K[G] k (M ) for K[G]-modules M with K a subfield of C and explain the Atiyah and Lück approximation conjectures. The L 2 -independence and L 2 -Hall properties are discussed in § 4. In § 5 we prove Theorem 1.4. In § 6 we introduce an auxiliary ring L τ [G] which already played an important role in Dicks' simplification of Freidman's proof. We finish the proof of Theorems 1.2 and 1.1 in § 7. In § 8 we reformulate the geometric Hanna Neumann conjecture in terms of an inequality for β . A key step of our proof of Theorem 1.3 is to find a specific submodule of K[G/U ] ⊗ K[G/W ] with trivial β . This is done in § 11. However, previously we present two auxiliary properties. In § 9 we prove a generalization of Howson property for quasi-convex subgroups of hyperbolic groups and for subgroups of limit groups and in § 10 we prove the Wilson-Zalesskii property for quasi-convex subgroups of hyperbolic virtually compact special groups. We finish the proof of Theorem 1.3 in § 12 and we describe also some limitations of our methods in order to extend them to more cases of Conjecture 1.
Remark 1.6. Theorems 1.2 and 1.4 and Corollary 1.5 hold also for fundamental groups of surfaces of non-negative Euler characteristic (i.e. the trivial group, Z/2Z, Z 2 and a, b | a 2 b 2 , the fundamental group of a Klein bottle). However, the results are either trivial, or use simple arguments specific for these cases. On the other hand, it is easy to produce a counter-example of Theorem 1.1 when G is virtually Z 2 .

Preliminaries
Although our main result is about surface groups, many steps of our proof hold in more general contexts of word hyperbolic, limit or virtually special compact groups. In this section, we recall all the relevant definitions and facts about these groups.
Let Y be a geodesic metric space. A subset Z ⊆ Y is called quasi-convex if there exists ≥ 0 such that for any points z 1 , z 2 ∈ Z, any geodesic joining these points is contained in the closed -neighborhood of Z.
A geodesic metric space Y is called (Gromov) hyperbolic if there exists a constant δ ≥ 0 such that for any geodesic triangle Δ in Y , any side of Δ is contained in the closed δ-neighborhood of the union of other sides. A finitely generated group G is said to be hyperbolic if its Cayley graph with respect to some finite generating set is a hyperbolic metric space. Quasi-convex subgroups of G are very important in the study of hyperbolic groups. Such subgroups are themselves hyperbolic and are quasi-isometrically embedded in G (see, for example, [ABCF + 90]). Moreover, for finitely generated subgroups of hyperbolic groups, being quasi-isometrically embedded in G is equivalent to be quasi-convex. The intersection of two quasi-convex subgroups in a hyperbolic group is quasi-convex by a result of Short [Sho91].
For a subgroup H ≤ G, we write H g = gHg −1 . A subgroup H of a group G is called malnormal if for every x ∈ G \ H, H x ∩ H = {1}.
A finitely generated group G is a limit group if, for any finite subset X of G, there exists a homomorphism f : G → F to a free group so that the restriction of f on X is injective. By a result of Wilton [Wil08], a finitely generated subgroup of a limit group is a virtual retract. Therefore, in a limit group all finitely generated subgroups are quasi-isometrically embedded and, in particular, in hyperbolic limit groups finitely generated subgroups are quasi-convex.
A right-angled Artin group (RAAG) is a group which can be given by a finite presentation, where the only defining relators are commutators of the generators. To construct such a group, one usually starts with a finite graph Γ with vertex set V and edge set E. One then defines the corresponding RAAG A(Γ) by the following presentation: We always view A(Γ) as a metric space with respect to the word metric induced by V when considering quasi-convexity of subgroups.
Special cube complexes were introduced in [HW08]. A group is called (compact) special if it is the fundamental group of a non-positively curved (compact) special cube complex. If G is the fundamental group of X, a (compact) special cube complex, thenX, the universal cover of X, is a CAT(0) cubical complex where G acts. By a quasi-convex subgroup of G we mean a subgroup of H with a quasi-convex orbit of vertices inX with respect to the combinatorial metric. A nice group theoretic characterization of these groups is that a group is (compact) special if and only if it is a (quasi-convex) subgroup of a RAAG (see [Hag08,HW08] . Assume G is a virtually compact special hyperbolic group (or, more generally, a GFERF hyperbolic group), and H 1 , . . . , H s are quasi-convex subgroups of G, s ∈ N and g 0 , . . . , g s ∈ G. Then the product g 0 H 1 g 1 . . . g s−1 H s g s is separable in G.

L 2 -Betti numbers, the strong Atiyah conjecture and the Lück approximation
Let G be a discrete group and let l 2 (G) denote the Hilbert space with Hilbert basis the elements of G; thus, l 2 (G) consists of all square summable formal sums g∈G a g g with a g ∈ C and inner product g∈G a g g, The left-and right-multiplication action of G on itself extend to left and right actions of G on l 2 (G). The right action of G on l 2 (G) extends to an action of C[G] on l 2 (G) and so we obtain that the group algebra C[G] acts faithfully as bounded linear operators on l 2 (G). The ring N (G) is the ring of bounded operators on l 2 (G) which commute with the left action of G. We often consider C[G] as a subalgebra of N (G). The ring N (G) satisfies the left and right Ore conditions 1854 The Hanna Neumann conjecture (a result proved by Berberian in [Ber82]) and its classical ring of fractions is denoted by U(G). The ring U(G) can be also described as the ring of densely defined (unbounded) operators which commute with the left action of G.
The computations of L 2 -Betti numbers have been algebraized through the seminal works of Lück [Lüc98a,Lüc98b]. The basic observation is that one can use a dimension function dim U (G) , which is defined for all modules over U(G) and compute the kth L 2 -Betti number of a C[G]-module M using the following formula: We recommend the book [Lüc02] for the definition of dim U (G) and its properties.
The ring U(G) is an example of a * -regular ring. Already in the case G = t ∼ = Z it is quite complicated as a ring (it is isomorphic to L 1 (S 1 )). Therefore, sometimes, it is more convenient to consider a smaller object R C[G] introduced by Linnell and Schick [LS12].
Let K be a subfield of C. We define -modules and use it in order to define the L 2 -Betti numbers (see [Jai19a,Jai19b]). If M is a K[G]-module, then its L 2 -Betti numbers are computed using the formula β is the dimension of K(t)-vector spaces. More generally, the strong Atiyah conjecture (see [Lüc02]) predicts that if G is torsion-free, then all numbers β In this paper, we use the solution of the strong Atiyah conjecture in the case where G is a torsion-free virtually compact special group. Another important conjecture about L 2 -Betti numbers is the Lück approximation conjecture (see [Lüc02]). In this paper, we use the solution of this conjecture in the case of approximation by sofic groups.
In this paper, we consider only the fields K which are subfields of C. Recall that the kth L 2 -Betti number of a group G is defined as b (2)

Y. Antolín and A. Jaikin-Zapirain
where K is arbitrary subfield of C. In the case when G is a virtually limit group, we have a good control of its L 2 -Betti numbers.
Proposition 3.3. Let G be a virtually limit group and K a subfield of C. Then In particular, Proof. See, for example, [BK17, Corollary C].
If U is a subgroup of a group G, then R K[G] is a flat right R K[U ] -module and for every left This implies the following result.

The L 2 -Hall property for surface groups
Let U be a subgroup of G. The embedding of U into G induces the corestriction map We say that the group G is L 2 -Hall, if for every finitely generated subgroup U of G, there exists a subgroup H of G of finite index containing U such that U is L 2 -independent in H.
The L 2 -independence can also be characterized in terms of R K [G] .
Lemma 4.1. Let G be a group and K a subfield of C. Then a subgroup U of G is L 2 -independent if and only if where cor : -module and we are done.
is a semi-simple algebra, Lemma 4.1 implies that in order to show that U is L 2 -independent in G, one has to prove that ker(cor) = {0}. In the case of virtually limit groups, we can give also the following description.
Proposition 4.2. Let G be a virtually limit group and let H 1 ≤ H 2 be two finitely generated subgroups of G. Let K be a subfield of C. Consider the exact sequence Then H 1 is L 2 -independent in H 2 if and only if β

The Hanna Neumann conjecture
Proof. We have the following exact sequence of Tor-functors: By Proposition 3.3, Tor In addition, the Shapiro lemma provides canonical isomorphisms   Let P be a pro-p group. We denote by d(P ) the minimal cardinality of a topological generating set of P . If P is finitely generated and L is an assuming that all dim Fp H k (P i , L) are finite. The limit always exists and it does not depend on the chain (see [Jai19b,Proposition 11.2]). An (infinite) Demushkin pro-p group is a Poincaré duality pro-p group of cohomological dimension 2. For the purposes of this paper, it is enough to know that the fundamental group of a closed surface is residually finite 2-group and its pro-2 completion is Demushkin. First, let us present the following result whose proof is essentially contained in the proof of [JS19, Proposition 7.2].

Proposition 4.5. Let P be an infinite Demushkin pro-p group and let H be a proper closed subgroup of P such that the map
In the proof of [JS19, Proposition 7.2], it is shown that L is an one-relator F p [[P ]]-module. Thus, we can produce an exact sequence On the other hand, using the exact sequence we obtain that In the penultimate equality we have used that H is free and P is Demushkin.
The previous proposition leads to a criterion for L 2 -independence of a subgroup of a free or a surface group.
Proposition 4.6. Let G be a finitely generated free group or a surface group and U a retract Proof. Without loss of generality, we assume that U is non-trivial and proper. Thus, G is infinite and U is a free group. First consider the case where G is a surface group.
As U is a retract, Let P be the pro-2 completion of G. As we have mentioned, G is a Demushkin pro-2 group. Let P 1 > P 2 > P 3 > · · · be a chain of open normal subgroups of P with trivial intersection. We put G i = G ∩ P i . Let H be the closure of U in P . As U is a retract of G, H is a free pro-2 group, and, thus, it is a proper subgroup of P .
The condition H 1 (U ; Consider two exact sequences Tensoring the first sequence with F 2 over Z, we obtain another exact sequence of F 2 [G]-modules, Therefore, the sequence ]-modules. In particular, The Hanna Neumann conjecture The long exact sequences of Tor-functors implies that Observe that and, because we assume that U is not trivial, Putting all limits together, we obtain that The remaining case is the case where G is a finitely generated free group. The proof works verbatim just bearing in mind that in Proposition 4.5 one has to change P to be a free pro-p group in the hypothesis, and in the conclusion β In addition, in the proof of Proposition 4.6, one has that the groups G i are free and, hence, dim Q Tor Proof of Theorem 4.4. Let G be a finitely generated free group or a surface group and U a finitely generated subgroup of G. There exists a subgroup S of finite index in G, containing U and such that U is a retract of S (see [Hal49,Sco78]). Now, we can apply Proposition 4.6.

The proof of Theorem 1.4
In this section, we prove Theorem 1.4. A similar argument is used later in our proof of Theorem 1.2. A key observation is the following proposition. 1859 Y. Antolín and A. Jaikin-Zapirain Proposition 5.1. Let G be a surface group or a free group and H a subgroup of G. Let K be a subfield of C.
is of global dimension 2 and for such rings a submodule of a module of projective dimension 1 is also of projective dimension 1.
In order to show part (2) we have to prove that Tor . As M is of projective dimension 1 and Tor Proposition 5.2. Let G be a free group or a surface group and U an Proof. Without loss of generality we may assume that G = U . Hence, G is infinite and U is free.
is also of projective dimension 1. Therefore, by Proposition 5.1(2), β As As U t ∩ H are free groups, M/L is of projective dimension 1 as a K[H]-module, and, thus, Thus, from the exact sequence we obtain that Tor Proof of Theorem 1.4. Let U be a retract of G and H a subgroup of G. By Proposition 4.6, U is L 2 -independent in G. Thus, the theorem follows from Proposition 5.2.
Dicks and Ventura conjectured that every compressed subgroup of a free group is also inert. We finish this section with the following natural question.
Question 2. Is any compressed subgroup of a free group also L 2 -independent? 1860 The Hanna Neumann conjecture

The structure of acceptable L τ [G]-modules
Let L be a field and let τ : G → Aut(L) be a homomorphism. We denote by L τ [G] the twisted group ring: its underlying additive group coincides with the ordinary group ring L[G], but the multiplication is defined as follows: The main advantage of working with L τ [G]-modules instead of L[G]-modules is stated in the following lemma.
Lemma 6.1 [Jai17, Claim 6.3]. Let G be a group and L a field. Let τ : G → Aut(L) and H = ker τ . Assume that H is of finite index in G. Then: (2) up to isomorphism, L is the unique irreducible L τ [G]-module on which H acts trivially.
Our next task is to prove a version of the strong Atiyah conjecture for L τ [G]-modules where G is a torsion-free virtually compact special group. We use the fact that a torsion-free virtually compact special group G has the factorization property. This means that any map from G to a finite group factors through a torsion-free elementary amenable group. This was proved by Schreve (see Corollary 2.6, Lemma 2.2 and the proof of Theorem 1.1 in [Sch14]). Proof. Recall that G is residually finite. Using the factorization property, we can construct a chain G ≥ H > T 1 > T 2 > · · · of normal subgroups of G with trivial intersection such that for each i, A i = G/T i is torsion-free elementary amenable.
As τ sends the elements of T i to the trivial automorphism of L, abusing slightly the notation we can construct L Let M be a finitely presented L τ [G]-module and let Then from (1) we obtain that In particular, again taking (1) into account, we conclude that The groups B i are torsion-free elementary amenable groups, and, thus, they satisfy the strong Atiyah conjecture [Lin93]. Hence, the rings R L[B i ] are division rings. Therefore, by Proposition 3.2, there exists i such that In this paper, acceptable L τ [G]-modules appear using the construction presented in the following lemma.

This leads to the following exact sequence of L[H]-modules:
Using the long exact sequence for Tor, we obtain that

The proofs of Theorems 1.2 and 1.1
In this section, we finish the proof of Theorem 1.2 and deduce from it Theorem 1.1.
If U or W is of infinite index, then U ∩ xW x −1 is free. Thus, we obtain

A module theoretic reformulation of the geometric Hanna Neumann conjecture for limit groups
Let G be a group and let K be a field. Let

Y. Antolín and A. Jaikin-Zapirain
For every k ≥ 1, we put This definition is different from that used in [Jai17]. In light of [Bro82, Proposition III.2.2] and the Lück approximation (Proposition 3.2) one sees that two definitions are closely related. However, we do not claim that these two definitions always define the same invariant. We are very grateful to Mark Shusterman who suggested this new definition to us. In the following proposition, we collect the main properties of β Proposition 8.1. Let G be a group and let K be a subfield of C. (

of finite index in G. Then for any left K[G]-module N and any
Proof.
(1) This follows directly from the definitions.
( (3) From the long exact sequence for the Tor functor, corresponding to the exact sequence

2) Observe that R K[H] ⊗ K[H] K[G] is isomorphic to R K[G] as a right K[G]-module and dim
Observe that This finishes the proof of part (3). 1866 The Hanna Neumann conjecture In the following proposition, we give an algebraic reinterpretation of the sum which appears in Conjecture 1.
Proposition 8.2. Let G be a limit group and let K a subfield of C. Let U and W be two finitely generated subgroups of G. Then Proof. First let us show that (4) Indeed, Proposition 3.4 implies that β (K). Now, from Proposition 3.3, it follows that β Corollary 8.3. Conjecture 1 for a limit group G is equivalent to the following statement: for any finitely generated subgroups U and W of G,

The strengthened Howson property for hyperbolic limit groups
The Howson property for limit groups was proved by Dahmani [Dah03]. In the case of hyperbolic limit group, we can prove the strengthened Howson property (see the statement of Theorem 9.1). In fact, the strengthened Howson property holds for the family of stable subgroups of a given group.
Let f : R ≥1 × R ≥0 → R ≥0 be a function. Let H ≤ G be finitely generated groups, and fix some word metrics.
A quasi-geodesic γ in G is f -stable if for any (λ, )-quasi-geodesic η with endpoints on γ, we have η is contained in the f (λ, )-neighborhood of γ. The subgroup H is f -stable in G if the inclusion of H is a quasi-isometrically embedding (with respect to the word metrics) and the image of any geodesic in H is an f -stable quasi-geodesic in G. A subgroup H is stable if it is f -stable for some f as previously.
Examples of stable subgroups include quasi-convex subgroups of hyperbolic groups, subgroups quasi-isometrically embedded in the cone-off graph of relatively hyperbolic groups and convex cocompact subgroups of mapping class groups or Out(F n ). Note that any stable subgroup must be word hyperbolic and that being a stable subgroup is a property preserved under conjugation. See [AMST19] and references therein for details. 1867 Y. Antolín and A. Jaikin-Zapirain Theorem 9.1. Let U and W be two stable subgroups of a finitely generated group G. Then for almost all x ∈ U \G/W , the subgroup U ∩ xW x −1 is finite. In particular, if U is torsion-free, the sum x∈U \G/W d(U ∩ xW x −1 ) is finite.
Proof. The theorem follows from [AMST19, Lemma 4.2] which states that, under the hypothesis of our theorem, there is a constant D ≥ 0 such that whenever |U g 1 ∩ W g 2 | = ∞ for some g 1 , g 2 ∈ G then the cosets g 1 U and g 2 W have intersecting D-neighborhoods.
Suppose that U ∩ W g is infinite. Then gW intersects the D-neighborhood of U . By multiplying g by an element of U on the left, we can assume that gW is a distance at most D from the identity. Thus, by multiplying g by an element of W on the right, we can assume that the length of g is less than D. Therefore, for all UxW ∈ U \G/W , having no representative in the ball of radius D and center the identity, the subgroup U ∩ xW x −1 is finite.
By [AMST19, Lemma 3.1], the intersection of stable subgroups is stable and, hence, finitely generated. Therefore, the 'in particular' claim follows.
As hyperbolic limit groups are torsion-free and every finitely generated subgroup is quasiconvex (and, hence, stable), we obtain the following corollary.
Corollary 9.2. Let G be a hyperbolic limit group and let U and W be two finitely generated subgroups of G.
The strengthened Howson property is not true for limit groups that are not hyperbolic. A simple example can be constructed on abelian groups, because all conjugates of a subgroup are equal, regardless of conjugating by representatives of different cosets. This is essentially the only reason for which the strengthened Howson property fails for limit groups. For limit groups, or more generally relatively hyperbolic groups, one has a similar statement to Theorem 9.1 if one restricts to non-parabolic intersections.
Let G be a group and H = {H λ } λ∈Λ a collection of subgroups. Let H be the disjoint union λ∈Λ H λ . A group G is hyperbolic relative to a family of subgroups H if it admits a finite relative presentation with linear relative isoperimetric inequality. The group G has a finite relative presentation with respect to H if G is generated by a finite set X together with the collection of subgroups in H and it is subject to a finite number of relations involving elements of X and elements of H, formally G = ( X | * ( * λ∈Λ H λ ))/ R , with X and R finite. Here R denotes the normal closure of R in X | * ( * λ∈Λ H λ ). Let R be all the words over H that represent trivial elements. The relative presentation has linear isoperimetric inequality, if there is a constant C such that for every w ∈ (X ∪ H) * representing 1 in G, then w is equal in X | * ( * λ∈Λ H λ ) to a product of conjugates of elements of R ∪ R using at most C (w) + C conjugates of R. Here (w) denotes the length of w. An important property that will be used is that the Cayley graph of G with respect to X ∪ H, denoted Γ(G, X ∪ H), is hyperbolic.
A subgroup U G is relatively quasi-convex if U is a quasi-convex set in Γ(G, X ∪ H). Being relatively quasi-convex is independent of the generating set X.
Conjugates of elements of H are called parabolic. Non-parabolic infinite-order elements are called loxodromic and, indeed, they act as a loxodromic isometry of Γ(G, X ∪ H). Note that parabolic subgroups are bounded subsets of Γ(G, X ∪ H) and, therefore, they are relatively quasi-convex.
Connecting with the previous notion of stability, if a subgroup of a relatively hyperbolic group is quasi-convex and has no non-trivial parabolic elements, then it is stable. 1868 The Hanna Neumann conjecture Lemma 9.3. Let U and W be two relatively quasi-convex subgroups of a finitely generated, relatively hyperbolic group G. Then for almost all x ∈ U \G/W , the subgroup U ∩ xW x −1 does not contain a loxodromic element.
Proof. The key arguments of this proof are contained in [HW09,Lemma 8.4] whose proof we follow closely. We assume that X is a finite generating set of G.
Suppose that U ∩ gW g −1 does contain a loxodromic element f . As f is a loxodromic isometry of Γ = Γ (G, X ∪ H), the subgroup f has two different accumulation points {f ∞ , f −∞ } ∈ ∂Γ, the Gromov boundary of Γ. As f ∈ gW g −1 , we have that f g ∈ gW and note that f g has also {f ∞ , f −∞ } as accumulation points because it is at finite Hausdorff X-distance from f . Thus, the accumulation points of [HW09,Lemma 8.3] there are bi-infinite geodesics γ U and γ W in Γ from f −∞ to f ∞ and g −1 f −∞ to g −1 f ∞ , respectively, and they are at a finite Hausdorff X-distance from f and g −1 fg , respectively. Finally, the vertices of γ U lie in the σ-neighborhood of U and the vertices of γ W lie on the σ-neighborhood of W , where σ is the quasi-convexity constant of U and W .
Note that gγ W has the same end points at infinite as γ U . Now, by [HW09, Lemma 8.2], there is a constant L, only depending on Γ, such that the vertices of the geodesics γ U and gγ W are at most L Hausdorff X-distance of each other. Thus, f and f g are at Hausdorff X-distance at most L + 2σ. This implies that U and gW have intersecting D = L + 2σ neighborhoods. By multiplying g by an element of U on the left, we can assume that gW is a distance at most D from the identity. Thus, by multiplying g by an element of W on the right, we can assume that the length of g is less than D. Therefore, for all UxW ∈ U \G/W , having no representative in the X-ball of radius D and center the identity, the subgroup U ∩ xW x −1 does not contain loxodromic elements.
Theorem 9.4. Let G be a limit group and let U and W be two finitely generated subgroups of G. Then for almost all x ∈ U \G/W , the subgroup U ∩ xW x −1 is abelian. In particular, the sum x∈U \G/W χ(U ∩ xW x −1 ) is finite.
Proof. The case where G is hyperbolic follows from Theorem 9.2.
If G is a non-hyperbolic limit group, then G is finitely generated and hyperbolic relative to the family H of maximal abelian non-cyclic subgroups (see [Dah03,Theorem 4.5]).
Recall that by [Dah03,Proposition 4.6], finitely generated subgroups of limit groups are relatively quasi-convex. In particular, U and W are relatively quasi-convex. By Lemma 9.3, for almost all x ∈ U \G/W the subgroup U ∩ xW x −1 does not contain a loxodromic element. As limit groups are torsion-free, this implies that for almost all x ∈ U \G/W , the subgroup U ∩ xW x −1 is contained in a parabolic subgroup and, hence, it is abelian. Moreover, because limit groups have the Howson property [Dah03,Theorem 4.7], each U ∩ xW x −1 is finitely generated and, hence, a limit group. Hence, χ(U ∩ xW x −1 ) is well-defined and zero when U ∩ xW x −1 is abelian.
Theorem 9.1 implies also that a quasi-convex subgroup of a hyperbolic virtually compact special group is virtually malnormal.
Corollary 9.5. Let G be a hyperbolic virtually compact special group and H a quasi-convex subgroup of G. Then H is virtually malnormal and a virtual retract.
Proof. By Theorem 2.1, H is a virtual retract.
As G is virtually a subgroup of a RAAG, G is virtually torsion-free and residually finite. By passing to a finite index subgroup of G, we can assume that H is torsion-free. By Theorem 9.1, there is only a finite number of double cosets HxH such that H x ∩ H = {1}. By Theorem 2.2, each of these double cosets is separable. As a finite collection of disjoint separable sets is separable, there exists a normal subgroup N of G of finite index that separates these double cosets. Hence, H is malnormal in HN .

The Wilson-Zalesskii property in virtually compact special hyperbolic groups
Let G be a residually finite group and let U and W two subgroups of G. We say that U and W satisfy the Wilson-Zalesskii property if Here U denotes the closure of U in the profinite completion G of G. When G is virtually free, the Wilson-Zalesskii property for G was proved by Wilson and Zalesskii in [WZ98, Proposition 2.4] for every pair of finitely generated subgroups (see also [RZ96,Lemma 3.6] for the case when U and W are cyclic). In this section, we show that a pair of quasi-convex subgroups of a virtually compact special hyperbolic group satisfies the Wilson-Zalesskii property. Our argument essentially follows the original argument of Wilson and Zalesskii. It uses a beautiful idea of double trick that goes back to the work of Long and Niblo [LN91]. Let us start with the following useful lemma.
Lemma 10.1. Let G be a residually finite group and let U and W be two finitely generated subgroups of G. Let H be a subgroup of G of finite index. Assume that Proof. Note that one always has that U ∩ W ⊆ U ∩ W . Let v ∈ U ∩ W . Then we can write v = u 1 u 2 = w 1 w 2 , where u 1 ∈ U , u 2 ∈ U ∩ H, w 1 ∈ W and w 2 ∈ W ∩ H. Thus, Therefore, there are u 3 ∈ U ∩ H and w 3 ∈ W ∩ H such that k = u 3 w −1 3 . Hence, Lemma 10.2. Let G be a hyperbolic virtually compact special group, U a malnormal retract of G and W a quasi-convex subgroup of G.
Proof. As U is a retract in G, U is quasi-isometrically embedded in G and, hence, quasi-convex. By [Git96, Lemma 5.2], G * U G is hyperbolic and by [HW15, Theorem A], it is virtually compact special.
For the sake of the proof, let G denote a copy of G and U , W and K the corresponding copies of U , W and K in G . Let B = G * U =U G and A = W, W B. From Bass-Serre theory (or normal forms on amalgamated free products), it follows easily that the natural map W * K=K W → W, W G * U =U G is injective and, hence, an isomorphism. Fix finite generating sets Y and Y of G and G , respectively, and let X = Y ∪ Y a finite generating set of B. In [Git96] terminology, a path p on the Cayley graph of Γ(B, X) is in normal 1870 The Hanna Neumann conjecture form if p is the concatenation of subpaths p ≡ p 1 p 2 . . . p n , such that the label of each p i is the label of a geodesic word either on Γ(G, Y ) or on Γ(G , Y ), no two labels of consecutive subpaths p i and p i+1 lie in the same set Y or Y and, finally, no label represents an element of U = U except maybe the label of p 1 . Now, [Git96,Lemma 4.1] in view of [Git96, Lemma 5.2] claims that there is a constant C such that any geodesic path in Γ(B, X) is in the C-neighborhood of any path in normal form with the same endpoints. From this, it easily follows that A is quasi-convex in B. Indeed, let q be any geodesic path in Γ(B, X) with endpoints in A. By the equivariance of the action, we can assume that q goes from 1 to a ∈ A. As A ∼ = W * K W , there is a path p in normal form p ≡ p 1 . . . p n from 1 to a where the labels of each p i represents an element of W or W . By the mentioned result of Gitik, q is in the C-neighborhood of p. Let σ denote the quasi-convexity constant of W and W as subspaces of Γ(G, Y ) and Γ(G , Y ), respectively. We claim that each p i is in the σ-neighborhood of A. Indeed, let a 0 = 1 and a i ∈ A be the element represented by the label of p 1 p 2 . . . p i−1 . Suppose that the label of p i is a word in Y . The case where the label of p i is a word in Y is analogous. Then a −1 i p i is a geodesic path in Γ(G, Y ) with endpoints in W , and therefore lies in the σ-neighborhood of W . Thus, p i lies in the σ-neighborhood of a i W ⊆ A as claimed. Therefore, p is in the σ-neighborhood of A. Let G = {g : g ∈ G} be a group isomorphic to G such that the map g → g is an isomorphism between G and G . Put F = G * U G . In the following, we identify U and U in F . Let P be a subgroup of F generated by W and W . As G ∩ G = U , we obtain that W ∩ W = W ∩ W ∩ U = K, and so P ∼ = W * K W .
Observe that G is a retract of F . Hence, the closure G of G in F is isomorphic to G. Thus, the closures of U , W and K in F are isomorphic to the closures of U , W and K in G, respectively. In particular, F is isomorphic to the profinite amalgamated product G * U G , and so G ∩ G = U .
As W is quasi-convex in G, it is virtually a retract (Theorem 2.1). Hence, W = W . Thus, the closures of K in W and G are isomorphic. Therefore, P is isomorphic to the profinite amalgamated product W * K W .
By Lemma 10.2, P is quasi-convex in F and F is virtually compact special. Therefore, by Theorem 2.1, P is virtually retract in F , and so the closure P of P in F is isomorphic its profinite completion W * K W . Hence, W ∩ W = K. On the other hand, We conclude that U ∩ W = K.
Corollary 10.4. Let G be a hyperbolic limit group and let U and W two finitely generated subgroups of G. Then for every normal subgroup T of G of finite index, there exists a finite-index Proof. Assume that for every normal subgroup H of G of finite index there exists x H ∈ (UH ∩ W H) \ (U ∩ W )T . Let G > H 1 > H 2 > · · · be a chain of normal subgroups of G of finite index that form a base of neighbors of 1 in the profinite topology of G. Without loss of generality, we may assume that there exists Clearly, x ∈ U ∩ W . By Theorem 10.3, U and W satisfy the Wilson-Zalesskii property. Hence, x ∈ U ∩ W . Therefore, there exists n such that if i ≥ n, x H i ∈ (U ∩ W )T . We have arrived at a contradiction.

Constructions of submodules with trivial β K[G] 1
In this section, we assume that G is an L 2 -Hall hyperbolic limit group. For example, by Theorem 4.4, G can be the fundamental group of a closed orientable surface. Let K be a subfield of C. Let W be a finitely generated subgroup of G. Then, because G is L 2 -Hall, there exists a normal subgroup H of G of finite index such that W is L 2 -independent in W H. Let Proof. By Corollary 9.2, there are only finitely many double cosets UxW such that U ∩ xW x −1 is non-trivial. By a result of Minasyan (Theorem 2.2), each of these double cosets is separable. It is easy to see that a finite family of disjoint separable sets is separable. Therefore, there exists a normal subgroup H 0 of G of finite index that separates these cosets.
Let UsW be a double coset with U ∩ sW s −1 non-trivial. Recall that By Corollary 8.3, we are done.

The geometric Hanna Neumann conjecture beyond the surface groups
As we have shown in order to settle the case of hyperbolic limit groups of Conjecture 1, it is enough to prove the L 2 -Hall property for these groups. We strongly believe that L 2 -Hall property holds for arbitrary limit groups.
In the case of limit groups, the generalized Howson property holds if one replaces d by χ (Theorem 9.4) because the reduced Euler characteristic for finitely generated abelian groups is zero. However, if G is a limit group, we do not know whether the double cosets with respect to two finitely generated subgroups are separated (in the hyperbolic case this follows from [Mina06]) and we do not know whether the Wilson-Zalesskii property holds for every pair of finitely generated subgroups of a limit group 1 .
As quasi-convex subgroups of hyperbolic virtually compact special groups satisfy the Howson property we may ask whether they satisfy also the conclusion of the geometric Hanna Neumann conjecture. This is not true. Fix a natural number d. In a 2-generated free group F , we can find a finitely generated malnormal subgroup H with d(H) = d. Then, by [Git96,Corollary 5.3] and [HW15, Corollary B], G = F * H F is hyperbolic and virtually special compact. The two copies of F are quasi-convex in G, and their intersection has rank d.
In addition, we want to mention that a hyperbolic virtually special compact group is not always L 2 -Hall with respect to a quasi-convex subgroup. For example, take a free non-abelian