Geometrically finite amalgamations of hyperbolic 3-manifold groups are not LERF
aa r X i v : . [ m a t h . G T ] M a y GEOMETRICALLY FINITE AMALGAMATIONS OFHYPERBOLIC -MANIFOLD GROUPS ARE NOT LERF HONGBIN SUN
Dedicated to Professor Boju Jiang on his 80th birthday
Abstract.
We prove that, for any two finite volume hyperbolic 3-manifolds,the amalgamation of their fundamental groups along any nontrivial geomet-rically finite subgroup is not LERF. This generalizes the author’s previouswork on nonLERFness of amalgamations of hyperbolic 3-manifold groups alongabelian subgroups. A consequence of this result is that closed arithmetic hyper-bolic 4-manifolds have nonLERF fundamental groups. Along with the author’sprevious work, we get that, for any arithmetic hyperbolic manifold with di-mension at least 4, with possible exceptions in 7-dimensional manifolds definedby the octonion, its fundamental group is not LERF. Introduction
For a group G and a subgroup H < G , we say that H is separable in G if forany g ∈ G \ H , there exists a finite index subgroup G ′ < G such that H < G ′ and g / ∈ G ′ . G is LERF (locally extended residually finite) or subgroup separable if allfinitely generated subgroups of G are separable.The LERFness of groups is a property closely related with low dimensionaltopology, especially the virtual Haken conjecture on 3-manifolds (which is settled in[Ag3]). The topological significance of LERFness is shown by the following picture:suppose we have a π -injective immersed compact object in a space S M (e.g.a π -injective immersed surface in a 3-manifold), if π ( S ) is a separable subgroupof π ( M ) (which holds if π ( M ) is LERF), then S lifts to an embedded object insome finite cover of M .Among fundamental groups of low dimensional manifolds, it is known that fol-lowing groups are LERF: free groups ([Ha]), surface groups ([Sc]) and geometric3-manifolds groups (the Seifert fibered space case, Sol manifold case and hyperboliccase are settled in [Sc], [Ma] and [Ag3, Wi] respectively). It is also know that fun-damental groups of non-geometric 3-manifolds are not LERF, including: groups ofnontrivial graph manifolds ([NW]), and groups of mixed 3-manifolds ([Sun], andthe first such example is given in [Liu]). So the LERFness of fundamental groupsof compact 1-manifolds, compact 2-manifolds and compact 3-manifolds with emptyor tori boundary is completely determined.For any n ≥ G , there exists a closed smooth n -manifold with fundamental group isomorphic to G . So it is impossible to havea complete criterion of LERFness as dimension ≤ Date : October 15, 2018.2010
Mathematics Subject Classification.
Key words and phrases. locally extended residually finite, graph of groups, hyperbolic 3-manifolds, arithmetic hyperbolic manifolds.The author is partially supported by NSF Grant No. DMS-1510383. to some special class of manifolds. In [Sun], it is shown that, for all arithmetichyperbolic manifolds with dimension at least 4, with possible exceptions in closed4-dimensional manifolds and 7-dimensional manifolds defined by the octonion, theirfundamental groups are not LERF.The result in [Sun] on high dimensional arithmetic hyperbolic manifold groupsis a corollary of 3-dimensional results, including nonLERFness of mixed 3-manifoldgroups, and another result on nonLERFness of Z -amalgamations of finite volumehyperbolic 3-manifold groups ([Sun]). A special family of mixed 3-manifold groupsconsists of Z -amalgamations of hyperbolic 3-manifold groups, so both results areabout abelian amalgamations of finite volume hyperbolic 3-manifold groups.In this paper, we give a more general result on nonLERFness of amalgamationsof finite volume hyperbolic 3-manifold groups. Theorem 1.1.
Let M , M be two finite volume hyperbolic -manifolds, A be anontrivial finitely generated group and i : A → π ( M ) , i : A → π ( M ) be twoinjective homomorphisms with geometrically finite images, then the amalgamation π ( M ) ∗ A π ( M ) is not LERF. Theorem 1.1 implies the most interesting cases of Theorem 1.3 and Theorem 1.4of [Sun]: Z - and Z -amalgamations of finite volume hyperbolic 3-manifold groupsare not LERF.Theorem 1.1 might be a little bit surprising. It is known that all finite volume hy-perbolic 3-manifolds have LERF fundamental groups ([Ag3, Wi]), and geometricallyfinite subgroups are considered to be ”nice” subgroups of hyperbolic 3-manifoldgroups. However, when we take amalgamation of two such LERF groups along anontrivial ”nice” subgroup, we get a nonLERF group. The main reason is thatfinite volume hyperbolic 3-manifold groups have a lot of virtually fibered surfacesubgroups (geometrically infinite subgroups). They are ”not so nice” subgroups ofhyperbolic 3-manifold groups, from geometric group theory point of view. Theo-rem 1.1 may suggest that, although finite volume hyperbolic 3-manifold groups areLERF, they are kind of on the border of LERF groups and nonLERF groups.In [Sun], we used nonLERFness of Z - and Z -amalgamations of finite volumehyperbolic 3-manifold groups to prove that most arithmetic hyperbolic manifoldswith dimension ≥ Theorem 1.2.
For any closed arithmetic hyperbolic -manifold, its fundamentalgroup is not LERF. Along with Theorem 1.1 and 1.2 of [Sun], we get the following corollary.
MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 3
Corollary 1.3.
For any arithmetic hyperbolic manifold with dimension at least and not defined by the octonion (which only show up in dimension ), its funda-mental group is not LERF. This corollary seems also suggest that hyperbolic 3-manifold groups are on theborder of LERF groups and nonLERF groups, since if we increase the dimensionby 1, we goes from LERF groups to nonLERF groups. This is mainly becausethat high dimensional arithmetic hyperbolic manifold groups contain many geo-metrically infinite subgroups which are not as nice as virtually fibered subgroupsin dimension 3. Note that for arithmetic hyperbolic manifolds of simplest type(defined by quadratic forms over totally real number fields), it is known that theirgeometrically finite subgroups are separable ([BHW]).The main idea of the proof of Theorem 1.1 is similar to the proof of Theorem1.3 and 1.4 of [Sun]. To realize the idea as a mathematical proof, we need to takecare of the following two points.The first point is to show that π ( M ) ∗ A π ( M ) has a subgroup with nontrivialinduced graph of group structure that is ”algebraically fibered”. For an amalgama-tion π ( M ) ∗ A π ( M ), by saying it is ”algebraically fibered” , we mean that thereare two fibered cohomology classes in H ( M ; Z ) and H ( M ; Z ) respectively, suchthat their restrictions on H ( A ; Z ) are the same nontrivial cohomology class. Wecare about such algebraically fibered structures since nonseparable subgroups wewill get are constructed by ”pasting” fibered surface subgroups in different vertexpieces together carefully. The ”algebraically fibered” structure makes this pastingconstruction along A applicable.In [Sun], the existence of an ”algebraically fibered” structure on a subgroup of π ( M ) ∗ Z π ( M ) is almost for free, and the existence of an ”algebraically fibered”structure on a subgroup of π ( M ) ∗ Z π ( M ) follows from the work of Przytycki-Wise ([PW]). However, for π ( M ) ∗ A π ( M ) with a general group A , we need toprove the existence of an ”algebraically fibered” structure. Moreover, for a generalgroup A , the corresponding topological space is usually not a genuine fiber bundleover circle, so we need the notion of ”algebraically fibered” structure here.We will use Agol’s construction of virtual fibered structures ([Ag2]) and thevirtual retract property of geometrically finite subgroups ([CDW]) to prove theexistence of an ”algebraically fibered” structure on a subgroup of π ( M ) ∗ A π ( M )with nontrivial induced graph of group structure. The precise statement and itsproof is given in Section 3.The second point is that nonseparable subgroups we will construct are usuallyinfinitely presented, so they may not be carried by π -injective compact objects.The nonseparable subgroups constructed in [Sun] are always realized by π -injectivecompact objects (surfaces or one point unions of surfaces). However, for a generalgeometrically finite subgroup A of a finite volume hyperbolic 3-manifold group π ( M ), and a cohomology class α ∈ H ( M ; Z ), if we consider α as a homomorphism α : π ( M ) → Z , the kernel of its restriction on A might be infinitely generated.Since we will ”paste” fibered surface subgroups in different vertex pieces to getnonseparable subgroups, we may need to ”paste” them along infinitely generatedsubgroups, and get finitely generated infinitely presented nonseparable subgroups. HONGBIN SUN
If we want to realize a finitely generated infinitely presented subgroup by a π -injective object, it must be noncompact. However, since Scott’s topological in-terpretation of separability ([Sc]) do need a compact object in the correspondingcovering space, we will sacrifice the π -injectivity and make sure that the nonsepa-rable subgroup is carried by a compact object.The organization of this paper is as the following. In Section 2, we will reviewsome relevant background on geometric group theory, topology of hyperbolic 3-manifolds, and arithmetic hyperbolic manifolds. In Section 3, we will show thatany nontrivial geometrically finite amalgamation π ( M ) ∗ A π ( M ) of two finitevolume hyperbolic 3-manifold groups has a subgroup that has nontrivial inducedgraph of group structure and is algebraically fibered. In Section 4, we give theproof of Theorem 1.1. In the proof, we first construct a further subgroup of thealgebraically fibered group we got in Section 3, and construct a topological space X with π ( X ) isomorphic to this subgroup. Then we construct a map f : Z → X from a compact 2-dimensional complex Z to X , and show that f ∗ ( π ( Z )) < π ( X )is not separable. In Section 5, we give the proof of Theorem 1.2. In Section 6, weask some further questions related to results in this paper. Acknowledgement.
The author thanks Ian Agol and Daniel Groves for valuableconversations. 2.
Preliminaries
In this section, we review some relevant background on geometric group theory,topology of hyperbolic 3-manifolds, and arithmetic hyperbolic manifolds. Thissection has some overlap with Section 2 of [Sun].2.1.
Subgroup separability.
In this subsection, we review basic concepts andproperties on subgroup separability.
Definition 2.1.
Let G be a group, and H < G be a subgroup, we say that H is separable in G if for any g ∈ G \ H , there exists a finite index subgroup G ′ < G such that H < G ′ and g / ∈ G ′ .It is obvious that finite index subgroups are always separable, so we are mainlyinterested in infinite index subgroups when we talk about separability. Definition 2.2.
A group G is LERF (locally extended residually finite) or subgroupseparable if all finitely generated subgroups of G are separable in G .Here are two basic results on LERFness of groups, and we will use them implicitlyin this paper. • If A and B are two LERF groups, then A ∗ B is also LERF. • If G is a group and G ′ < G is a finite index subgroup, then G is LERF ifand only if G ′ is LERF.A more elementary property on LERFness of groups is that any subgroup ofa LERF group is still LERF, and we state it as in the following lemma. As in[Sun], this property is fundamental for our proof on nonLERFness of groups: toprove a group is not LERF, we only need to find a manageable subgroup (e.g.amalgamation of 3-manifold groups) and show this subgroup is not LERF. MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 5
Lemma 2.3.
Let G be a group and Γ < G be a subgroup. For a further subgroup H < Γ , if H is separable in G , then H is separable in Γ .In particular, if Γ is not LERF, then G is not LERF. Fibered structures of -manifolds and quasi-fibered classes. In thissubsection, we always assume 3-manifolds are compact, connected, oriented, irre-ducible and with empty or tori boundary.By a fibered structure of a 3-manifold, we mean a surface bundle over circlestructure. For a fibered structure of M , there exist a compact oriented surface S and an orientation-preserving surface automorphism f : S → S , such that M ishomeomorphic to the mapping torus S × I/ ( s, ∼ ( f ( s ) ,
1) with respect to thefibered structure. So we get a homology class [
S, ∂S ] ∈ H ( M, ∂M ; Z ) and its dualcohomology class α ∈ H ( M ; Z ). They both correspond to the fibered structure of M , and we also consider α ∈ H ( M ; Z ) as a homomorphism α : π ( M ) → Z .If a 3-manifold M has a fibered structure and b ( M ) >
1, then M has infinitelymany fibered structures. These fibered structures are organized by the Thurstonnorm on H ( M ; R ) (defined in [Th2]). We will not give the definition of Thurstonnorm here, but we need the following facts on Thurston norm (see [Th2]).In general, the Thurston norm is only a semi-norm, and it is a genuine normwhen M is a finite volume hyperbolic 3-manifold. The unit ball of Thurston normis a polyhedron in H ( M ; R ) with finitely many faces. For a top dimensional openface F of the Thurston norm unit ball, let C be the open cone over F . In [Th2],Thurston proved that an integer point α ∈ H ( M ; Z ) ⊂ H ( M ; R ) corresponds toa fibered structure of M if and only if α is contained in an open cone C as aboveand all integer points in C correspond to fibered structures of M . In this case,the open cone C is called a fibered cone . For any real coefficient cohomology class α ∈ C ⊂ H ( M ; R ), it is called a fibered class .All integer fibered classes in H ( M ; Z ) correspond to genuine fibered structuresof M , and their image in P H ( M ; Q ) (the projectivization of H ( M ; Q )) form anopen subset of P H ( M ; Q ). In particular, we have the following lemma on fiberedclasses. Lemma 2.4.
Suppose that α ∈ H ( M ; Z ) is a fibered class of a -manifold M ,then for any β ∈ H ( M ; Z ) , there exists N ∈ Z + , such that for any n > N , nα ± β ∈ H ( M ; Z ) are both fibered classes. The following definition of quasi-fibered class is given in [FV]:
Definition 2.5.
For a cohomology class α ∈ H ( M ; Z ) − { } , α is a quasi-fiberedclass if α lies on the closure of a fibered cone C .In the proof of the virtual fibering conjecture, the last step is Agol’s criteria forvirtual fiberings ([Ag2]). In [Ag2] (see an alternative proof in [FK]), Agol showedthat if the fundamental group of a compact irreducible 3-manifold M with emptyor tori boundary satisfies the RFRS property (residually finite rational solvable),then M is virtually fibered. We will not give the definition of RFRS here. Alongwith Wise and his collaborators’ works on geometric group theory (e.g. [Wi]),Agol showed that all finite volume hyperbolic 3-manifolds have virtually RFRSfundamental group ([Ag3]), and solved the virtual fibering conjecture.In [Ag2], Agol actually proved that, if π ( M ) is RFRS, then for any nontrivialnon-fibered cohomology class α ∈ H ( M ; Z ), there exists a finite cover f : M ′ → M HONGBIN SUN such that f ∗ ( α ) lies on the boundary of a fibered cone of H ( M ′ ; R ), i.e. f ∗ ( α )is a quasi-fibered class. In [FV], the following proposition on quasi-fibered classesis proved (Corollary 5.2 of [FV]). Its proof is a direct application of the result in[Ag2], and we modify the statement in [FV] a little bit. Theorem 2.6.
Let M be a -manifold with virtually RFRS fundamental group,then there exists a finite regular cover p : M ′ → M , such that for any nontrivialclass α ∈ H ( M ; Z ) , p ∗ ( α ) ∈ H ( M ′ ; Z ) is a quasi-fibered class. So the process of pulling back a cohomology class to get a quasi-fibered classnot only work for each cohomology class individually, but also work for all of themsimultaneously.2.3.
Infinite volume hyperbolic -manifolds. For infinite volume hyperbolic3-manifolds with finitely generated fundamental groups, there is a rich theory onsuch manifolds. In this paper, we are mainly interested in such manifolds that coverfinite volume hyperbolic 3-manifolds.For a finite volume hyperbolic 3-manifold M , we have the following dichotomy fora finitely generated infinite index subgroup A < π ( M ) (the proof of this dichotomyis a combination of results in [Th1], [Ca] and [Ag1, CG]):(1) A is a geometrically finite subgroup of π ( M ). Equivalently, A is (relatively)quasiconvex in the (relative) hyperbolic group π ( M ), from geometric grouptheory point of view.(2) A is a geometrically infinite subgroup of π ( M ). It is equivalent to that A is a virtually fibered surface subgroup of M .Here we do not give the definition of geometrically finite and geometrically infinitesubgroups, the readers only need to know that if A is not a virtually fibered surfacesubgroup, then it is a geometrically finite subgroup.If A < π ( M ) is a nontrivial finitely generated infinite index subgroup, the fol-lowing lemma implies b ( A ) ≥
1. Actually, it holds for any nontrivial infinitelycovolume discrete torsion-free subgroup of Isom + H , and it is well-known for ex-perts on hyperbolic 3-manifolds. Lemma 2.7. If A is a nontrivial finitely generated subgroup of Isom + H that actsfreely and properly discontinuously on H with infinite covolume, then the first bettinumber b ( A ) ≥ .Proof. Since A acts freely and properly discontinuously on H , Y = H /A is aninfinitely volume hyperbolic 3-manifold with finitely generated fundamental group.By the tameness of hyperbolic 3-manifolds ([Ag1],[CG]), Y is homeomorphic tothe interior of a compact 3-manifold X .Since Y = H /A has infinite volume, X can not be a closed 3-manifold. Since Y is a quotient of H , it is irreducible, and so does X . So no component of ∂X is a sphere. Otherwise X can only be a 3-ball, which contradicts with that A is anontrivial group.So X is a compact 3-manifold with boundary and each component of ∂X haspositive genus. Then a canonical application of the duality theorem on 3-manifolds(half lives, half dies) implies b ( A ) = b ( X ) ≥ b ( ∂X ) ≥ . (cid:3) MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 7
Although Lemma 2.7 is well-known, it is fundamental for this paper. It is aclassical result that amalgamations along trivial subgroups (free products) of LERFgroups are still LERF, but Theorem 1.1 implies nontrivial geometrically finite amal-gamations of finite volume hyperbolic 3-manifold groups (which are LERF groups)are not LERF. The difference is mainly rooted in Lemma 2.7, since we will use b ( A ) ≥ α ∈ H ( M ; Z ), its restrictionon A might be a nontrivial homomorphism to Z .2.4. Virtual retractions of hyperbolic -manifold groups.Definition 2.8. For a group G and a subgroup A < G , we say that A is a virtualretraction of G if there exists a finite index subgroup G ′ < G and a homomorphism φ : G ′ → A , such that A < G ′ and φ | A = id A .In [CDW], it is shown that (relatively) quasiconvex subgroups of virtually com-pact special (relative) hyperbolic groups are virtual retractions. The celebratedvirtually compact special theorem of Wise and Agol ([Wi] for cusped manifoldsand [Ag3] for closed manifolds) implies that groups of finite volume hyperbolic3-manifolds are virtually compact special. These two results together imply thefollowing theorem. Theorem 2.9.
Let M be a finite volume hyperbolic -manifold, A < π ( M ) bea geometrically finite subgroup (i.e. A is not a virtually fibered surface subgroup),then A is a virtual retraction of π ( M ) . Arithmetic hyperbolic manifolds.
In this subsection, we briefly review thedefinition of arithmetic hyperbolic manifolds of simplest type. They are defined byquadratic forms over number fields. Since all arithmetic hyperbolic 4-manifolds arein the simplest type, this definition is sufficient for this paper. Most material inthis subsection can be found in Chapter 6 of [VS].Recall that the hyperboloid model of H n is defined as the following. Equip R n +1 with a bilinear form B : R n +1 × R n +1 → R defined by B (cid:0) ( x , · · · , x n , x n +1 ) , ( y , · · · , y n , y n +1 ) (cid:1) = x y + · · · + x n y n − x n +1 y n +1 . Then the hyperbolic space H n can be identified with I n = { ~x = ( x , · · · , x n , x n +1 ) | B ( ~x, ~x ) = − , x n +1 > } . The hyperbolic metric is induced by the restriction of B ( · , · ) on the tangent spaceof I n .The isometry group of H n consists of all linear transformations of R n +1 thatpreserve B ( · , · ) and fix I n . Let J = diag(1 , · · · , , −
1) be the ( n + 1) × ( n + 1)matrix defining the bilinear form B ( · , · ), then the isometry group of H n isIsom( H ) ∼ = P O ( n, R ) = { X ∈ GL ( n + 1 , R ) | X t JX = J } / ( X ∼ − X ) . The orientation preserving isometry group of H n isIsom + ( H n ) ∼ = SO ( n, R ) , which is the component of SO ( n, R ) = { X ∈ SL ( n + 1 , R ) | X t JX = J } that contains the identity matrix. HONGBIN SUN
Let K ⊂ R be a totally real number field, and σ = id, σ , · · · , σ k be all theembeddings of K into R . Let f ( x ) = n +1 X i,j =1 a ij x i x j , a ij = a ji ∈ K be a nondegenerate quadratic form defined over K with negative inertia index 1(as a quadratic form over R ). If for any l >
1, the quadratic form f σ l ( x ) = n +1 X i,j =1 σ l ( a ij ) x i x j is positive definite, then we can use K and f to define an arithmetic hyperbolicgroup.Let A be the ( n + 1) × ( n + 1) matrix defining the quadratic form f . Since thenegative inertia index of A is 1, the special orthogonal group of f : SO ( f ; R ) = { X ∈ SL ( n + 1 , R ) | X t AX = A } is conjugate to SO ( n, R ) by a matrix P (satisfying P t AP = J ). SO ( f ; R ) hastwo components, and let SO ( f ; R ) be the component that contains the identitymatrix.Let O K be the ring of algebraic integers in the field K . Then we can form theset of algebraic integer points SO ( f ; O K ) = { X ∈ SL ( n + 1 , O K ) | X t AX = A } ⊂ SO ( f ; R ) . The theory of arithmetic groups implies that SO ( f ; O K ) = SO ( f ; O K ) ∩ SO ( f ; R )conjugates to a lattice of Isom + ( H n ) (by the matrix P ), i.e. the correspondingquotient space of H n has finite volume. For simplicity, we still use SO ( f ; O K ) todenote its P -conjugation in SO ( n, R ) ∼ = Isom + ( H n ).Here SO ( f ; O K ) ⊂ Isom + ( H n ) is called the arithmetic group defined by thenumber field K and quadratic form f , and H n /SO ( f ; O K ) is a finite volume hy-perbolic arithmetic n -orbifold. A hyperbolic n -manifold (orbifold) M is called anarithmetic hyperbolic n -manifold (orbifold) of simplest type if M is commensurablewith H n /SO ( f ; O K ) for some K and f .For this paper, the most important property of arithmetic hyperbolic manifoldsof simplest type is that they contain many finite volume hyperbolic 3-manifolds astotally geodesic submanifolds. This can be seen by diagonalizing the matrix A andtaking indefinite 4 × Algebraically fibered structures on π ( M ) ∗ A π ( M )In this section, we construct ”algebraically fibered” structures on certain sub-groups of geometrically finite amalgamations of finite volume hyperbolic 3-manifoldgroups. The construction of an algebraically fibered structure on certain subgroupis the first step for building an ideal model of geometrically finite amalgamationsfor which we can construct nonseparable subgroups. Definition 3.1.
For a group G , by an algebraically fibered structure on G , we meana nontrivial homomorphism G → Z with finitely generated kernel. MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 9
The main result in this section is the following theorem, and we will prove it insubsection 3.1. In subsection 3.2, we will prove a related result on virtually fiberedboundary slopes of cusped hyperbolic 3-manifolds, which is quite interesting byitself.
Theorem 3.2.
Let M , M be two finite volume hyperbolic -manifolds, A be anontrivial finitely generated group, i : A → π ( M ) and i : A → π ( M ) be twoinjective homomorphisms with geometrically finite images. Then there exist finitecovers N , N of M , M respectively, such that the following holds. (1) i − ( π ( N )) and i − ( π ( N )) are the same subgroup A ′ ≤ A (and we iden-tify A ′ with their images in π ( N ) and π ( N ) ). (2) There exist fibered classes α ∈ H ( N ; Z ) and α ∈ H ( N ; Z ) , such that α | A ′ = α | A ′ as homomorphisms from A ′ to Z , and the restricted homo-morphisms are surjective.Moreover, the group π ( M ) ∗ A π ( M ) contains a subgroup isomorphic to π ( N ) ∗ A ′ π ( N ) . The subgroup π ( N ) ∗ A ′ π ( N ) is still a nontrivial geometrically finiteamalgamation of finite volume hyperbolic -manifold groups, and it has an alge-braically fibered structure. Construct algebraically fibered structures on subgroups.
In this sub-section, we give the proof of Theorem 3.2. Before proving Theorem 3.2, we startwith the following proposition.
Proposition 3.3.
Let M be a finite volume hyperbolic -manifold, and A < π ( M ) be a nontrivial geometrically finite subgroup. Then for any nontrivial homomor-phism γ : A → Z , there exist a finite cover M ′ of M and a quasi-fibered class β ∈ H ( M ′ ; Z ) , such that the following hold.For the subgroup A ′ = A ∩ π ( M ′ ) , β | A ′ = γ | A ′ holds as homomorphisms from A ′ to Z .Proof. Since
A < π ( M ) is a geometrically finite subgroup, by Theorem 2.9, thereexists a finite cover M ′′ of M , such that A < π ( M ′′ ) and there is a retract homo-morphism φ : π ( M ′′ ) → A .By taking the composition, we get a nontrivial homomorphism δ = γ ◦ φ : π ( M ′′ ) → Z , which gives a cohomology class δ ∈ H ( M ′′ ; Z ).Then by Theorem 2.6, there exists a finite cover p : M ′ → M ′′ , such that β = p ∗ ( δ ) ∈ H ( M ′ ; Z ) is a quasi-fibered class.For A ′ = A ∩ π ( M ′ ), it is easy to see that β | A ′ = p ∗ ( δ ) | A ′ = δ ◦ p ∗ | A ′ = γ ◦ φ ◦ p ∗ | A ′ = γ ◦ φ | A ′ = γ | A ′ . (cid:3) Now we are ready to prove Theorem 3.2. Actually, what we will prove is strongerthan the statement of Theorem 3.2. We can start with any nontrivial homomor-phism γ : A → Z , then the restriction of the algebraically fibered structure on A ′ can be arbitrarily close to the restriction of γ on A ′ (as homomorphisms to Z ). Proof.
By abusing notation, we still use A to denote its images in π ( M ) and π ( M ). Lemma 2.7 implies that there exists a nontrivial homomorphism γ : A → Z . By Proposition 3.3, there exist a finite cover M ′ → M and a quasi-fibered class β ′ ∈ H ( M ′ ; Z ), such that for A ′′ = A ∩ π ( M ′ ), β ′ | A ′′ = γ | A ′′ holds. Since β ′ is a quasi-fibered class of M ′ , there exists a fibered class β ∈ H ( M ′ ; Z )which is (arbitrarily) close to β ′ in P H ( M ′ ; Q ) (under the projectivization). More-over, we can assume that β | A ′′ : A ′′ → Z is a nontrivial homomorphism.Since π ( M ) is LERF, A ′′ < π ( M ) is separable. So there exists a finite cover M ′ of M such that A ∩ π ( M ′ ) = A ′′ . Now we apply Proposition 3.3 again to A ′′ < π ( M ′ ) and β | A ′′ : A ′′ → Z . Then there exist a finite cover N ′ of M ′ and aquasi-fibered class β ∈ H ( N ′ ; Z ), such that for A ′ = A ′′ ∩ π ( N ′ ), β | A ′ = β | A ′ holds. Moreover, since A ′ < A ′′ is a finite index subgroup, β | A ′ is a nontrivialhomomorphism from A ′ to Z .Since π ( M ′ ) is LERF, A ′ < π ( M ′ ) is separable. So there exists a finite cover N ′ of M ′ such that A ′′ ∩ π ( N ′ ) = A ′ . Moreover, since A ′ < π ( N ′ ) is a geometricallyfinite subgroup, it is a virtual retraction of a finite index subgroup of π ( N ′ ).By abusing notation, we denote the corresponding finite cover (such that A ′ is aretraction of π ( N ′ )) by p : N ′ → M ′ .As a summary, we are in the following situation now. We have finite covers N ′ and N ′ of M and M respectively, with A ′ = A ∩ π ( N ′ ) = A ∩ π ( N ′ ) and A ′ is aretraction of π ( N ′ ) via a retract homomorphism φ : π ( N ′ ) → A ′ . Moreover, wehave a fibered class p ∗ ( β ) ∈ H ( N ′ ; Z ) and a quasi-fibered class β ∈ H ( N ′ ; Z ),such that p ∗ ( β ) | A ′ = β | A ′ are nontrivial homomorphisms to Z . Now we need tomodify β to a fibered class.Since β ∈ H ( N ′ ; Z ) is a quasi-fibered class, there exists δ ∈ H ( N ′ ; Z ) and N ∈ Z + , such that for any n > N , α = nβ + δ ∈ H ( N ′ ; Z ) is a fibered class.For the restricted homomorphism δ | A ′ : A ′ → Z , δ = δ | A ′ ◦ φ : π ( N ′ ) → Z isa cohomology class in H ( N ′ ; Z ). Since p ∗ ( β ) is a fibered class of N ′ , by Lemma2.4, for n large enough, α = np ∗ ( β ) + δ is also a fibered class of N ′ , and itsrestriction on A ′ is a nontrivial homomorphism to Z .So we have α | A ′ = np ∗ ( β ) | A ′ + δ | A ′ = nβ | A ′ + δ | A ′ ◦ φ | A ′ = nβ | A ′ + δ | A ′ = α | A ′ . If α | A ′ = α | A ′ are surjective homomorphisms to Z , then we just take N = N ′ and N = N ′ .If α | A ′ = α | A ′ is not surjective, since it is nontrivial, the image is d Z < Z for some d ≥
2. Then we take cyclic covers q : N → N ′ and q : N → N ′ corresponding to kernels of π ( N ′ ) α −→ Z → Z /d Z and π ( N ′ ) α −→ Z → Z /d Z respectively. Then we have A ′ < π ( N ) and A ′ < π ( N ), while q ∗ ( α ) and q ∗ ( α ) are surjective homomorphisms from π ( N ) and π ( N ) to d Z respectively.So d q ∗ ( α ) ∈ H ( N ; Z ) and d q ∗ ( α ) ∈ H ( N ; Z ) are both primitive cohomologyclasses, and they have the same restriction on A ′ . By abusing notation, we use α and α to denote d q ∗ ( α ) and d q ∗ ( α ). Then N and N are desired finite coversof M and M , with primitive fibered classes α ∈ H ( N ; Z ) and α ∈ H ( N ; Z ),such that α | A ′ = α | A ′ are surjective homomorphisms to Z .Since π ( N ) ∩ A = π ( N ) ∩ A = A ′ , π ( N ) ∗ A ′ π ( N ) is a nontrivial geo-metrically finite amalgamation of π ( N ) and π ( N ). There is an obvious homo-morphism from π ( N ) ∗ A ′ π ( N ) to π ( M ) ∗ A π ( M ), and it is injective by thecanonical form of elements in an amalgamation product.Since α : π ( N ) → Z and α : π ( N ) → Z agree with each other on A ′ = π ( N ) ∩ π ( N ) < π ( N ) ∗ A ′ π ( N ), they induce a homomorphism α : π ( N ) ∗ A ′ π ( N ) → Z . MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 11
Let Σ and Σ be connected fibered surfaces of N and N corresponding toprimitive fibered classes α and α respectively, and let K be the kernel of thesurjective homomorphism α | A ′ = α | A ′ : A ′ → Z . Since α | A ′ = α | A ′ : A ′ → Z is surjective, it is easy to see that the kernel of α : π ( N ) ∗ A ′ π ( N ) → Z is π (Σ ) ∗ K π (Σ ).Since Σ and Σ are compact surfaces, and the kernel π (Σ ) ∗ K π (Σ ) isgenerated by π (Σ ) and π (Σ ), π (Σ ) ∗ K π (Σ ) is finitely generated. So π ( N ) ∗ A ′ π ( N ) is algebraically fibered. (cid:3) Remark 3.4.
For a general geometrically finite subgroup A ′ of a finite volumehyperbolic 3-manifold group, the kernel K may not be finitely generated. So π (Σ ) ∗ K π (Σ ) may not be finitely presented.3.2. Virtually fibered boundary slopes on cusps.
In this subsection, we studyvirtually fibered boundary slopes on boundary components of cusped hyperbolic 3-manifolds. It is not directly related with the proof of Theorem 1.1, but it naturallyshows up in the study of matching fibered structures of 3-manifolds along Z sub-groups. Although the Z -amalgamation case is dealt in [Sun] by using the result in[PW], the following results are quite interesting by themselves. Definition 3.5.
Let M be a cusped hyperbolic 3-manifold, T be a boundary com-ponent of M , and a be a slope on T . We say that a is a virtually fibered boundaryslope if there exist a finite cover M ′ of M , an elevation T ′ of T in M ′ , and a fiberedstructure of M ′ , such that the corresponding fibered surface in M ′ intersects with T ′ along (parallal copies of) an elevation of a in T ′ .For the A ∼ = Z case of Theorem 3.2, we basically just find virtually fiberedboundary slopes on M and M such that they match with each other under thepasting. Moreover, the proof of Theorem 3.2 implies that, for any torus boundarycomponent T of a cusped hyperbolic 3-manifold, the set of virtually fibered bound-ary slopes form an open dense subset of P H ( T ; Q ). Moreover, in the followingproposition, we prove there are only finitely many slopes on T that may not bevirtually fibered boundary slopes. Proposition 3.6.
Let M be a cusped finite volume hyperbolic -manifold, and T be a boundary component of M , then all but finitely many slopes on T are virtuallyfibered boundary slopes.Proof. Since π ( T ) < π ( M ) is a geometrically finite subgroup, by Theorem 2.9,there is a finite cover M ′′ of M such that π ( T ) < π ( M ′′ ) and there is a retracthomomorphism π ( M ′′ ) → π ( T ). In particular, the torus T lifts to M ′′ and thehomomorphism H ( T ; Z ) → H ( M ′′ ; Z ) induced by inclusion is injective.By Theorem 2.6, there is a finite cover p : M ′ → M ′′ , such that for any nontrivial α ∈ H ( M ′′ ; Z ), p ∗ ( α ) ∈ H ( M ′ ; Z ) is a quasi-fibered class. Take any elevation T ′ of T in M ′ and we will prove that all but finitely many slopes on T ′ are boundaryslopes of fibered surfaces of M ′ .Let i be the inclusion map i : T ′ → M ′ . To show that a slope a on T ′ is theboundary slope of a fibered surface, we need only to show that there is a fiberedclass α ∈ H ( M ′ ; Z ), such that i ∗ ( α ) ∈ H ( T ′ ; Z ) is a nonzero multiple of the dualof a . Actually, since i ∗ : H ( M ′ ; R ) → H ( T ′ ; R ) is represented by an integer entrymatrix, it suffices to show that there is a fibered class α ∈ H ( M ′ ; R ), such that i ∗ ( α ) ∈ H ( T ′ ; R ) lies in the line containing the dual of a . By the fact that H ( T ; R ) → H ( M ′′ ; R ) is injective, i ∗ : H ( T ′ ; R ) → H ( M ′ ; R )is also injective. So the dual homomorphisms H ( M ′′ ; R ) → H ( T ; R ) and i ∗ : H ( M ′ ; R ) → H ( T ′ ; R ) are surjective.For any slope a on T ′ , let the line in H ( T ′ ; R ) containing the dual of a bedenoted by l a . Since i ∗ : H ( M ′ ; R ) → H ( T ′ ; R ) is surjective, ( i ∗ ) − ( l a ) is acodimension-1 hyperplane in H ( M ′ ; R ) going through the origin. Since coveringmaps always induce injective homomorphisms on real coefficient cohomology, we canidentify H ( M ′′ ; R ) as a subspace of H ( M ′ ; R ). Since the covering map induces anisomorphism between H ( T ; R ) and H ( T ′ ; R ) and H ( M ′′ ; R ) → H ( T ; R ) is sur-jective, we can see that ( i ∗ ) − ( l a ) ∩ H ( M ′′ ; R ) is also a codimension-1 hyperplanein H ( M ′′ ; R ), and different slopes on T ′ correspond to different codimension-1hyperplanes in H ( M ′′ ; R ).There are only finitely many codimension-1 hyperplanes in H ( M ′′ ; R ) that donot intersect with top dimensional (open) faces of the Thurston norm unit ballof M ′′ , and these hyperplanes give us finitely many possible exceptional slopes.For any slope a such that the corresponding codimension-1 hyperplane ( i ∗ ) − ( l a ) ∩ H ( M ′′ ; R ) intersects with a top dimensional open face F of the Thurston norm unitball of M ′′ , it clearly does not contain the whole face F by dimensional reason. Since F lies in the closure of an (open) fibered face F ′ of M ′ , we have F ′ ∩ ( i ∗ ) − ( l a ) = ∅ .So there is a fibered class α ∈ H ( M ′ ; R ), such that i ∗ ( α ) ∈ l a , thus the slope a in T ′ is the boundary slope of a fibered surface in M ′ . (cid:3) Proposition 3.6 implies that there are only finitely many slopes on T that may notbe virtually fibered boundary slopes. However, we do not have any single exampleof slope that is known not to be a virtually fibered boudnary slope. So maybe it isnot too optimistic to ask whether all slopes are virtually fibered boundary slopes.We do not have an answer for a general hyperbolic 3-manifold. However, sincearithmetic hyperbolic manifolds have a lot of symmetries in their finite covers, thesesymmetries imply that those finitely many possible exceptional slopes are actuallyvirtually fibered boundary slopes. So we have an affirmative answer for cuspedarithmetic hyperbolic 3-manifolds. Proposition 3.7.
Let M be a cusped arithmetic hyperbolic -manifold, and T bea boundary component of M , then all slopes on T are virtually fibered boundaryslopes.Proof. By Proposition 3.6, there are finitely many slopes a , a , · · · , a n on T , suchthat all other slopes on T are virtually fibered boundary slopes. We fix a slope a = a , and use the arithmetic property of M to show that a is actually a virtuallyfibered boundary slope.Recall that, for any cusped arithmetic hyperbolic 3-manifold M , π ( M ) is com-mensurable with P SL ( O d ) for some square free d ∈ Z + (Theorem 8.2.3 of [MR]).Here O d is the ring of algebraic integers of field Q ( √− d ) . We can identify π ( M ) as a subgroup of P SL ( C ) ∼ = Isom + ( H ). Up to conju-gation, we haveComm( π ( M )) = Comm( P SL ( O d )) = P GL ( Q ( √− d )) . For any Γ < Isom + ( H ), its commensurator is defined byComm(Γ) = { g ∈ Isom + ( H ) | g Γ g − ∩ Γ is a finite index subgroup of both Γ and g Γ g − } . MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 13
We assume that the cusp T corresponds to a parabolic fixed point ∞ ∈ C ∪{∞} = ∂ H (under the upper half-space model). Then π ( T ) is a subgroup of the parabolicstabilizer of ∞ in P GL ( Q ( √− d )): { (cid:18) r (cid:19) | r ∈ Q ( √− d ) } . For any γ = (cid:18) r (cid:19) ∈ π ( T ) with r = a + b √− d and a, b ∈ Q , we use arg( a + b √− d ) as a coordinate of γ .Suppose that the (possibly exceptional) slope a corresponds to a parabolic ele-ment h = (cid:18) p + q √− d (cid:19) . Since there are only finitely many possibly exceptionalslopes on T , there exists ǫ > T with coordinate in(arg( p + q √− d ) , arg( p + q √− d ) + ǫ )are virtually fibered boundary slopes.Then for any s, r ∈ Q , with arg( s + r √− d ) ∈ (0 , ǫ ), we consider g = (cid:18) s + r √− d
00 ( s + r √− d ) − (cid:19) ∈ P GL ( Q ( √− d )) = Comm( π ( M )) . So π ( M ) ∩ gπ ( M ) g − is a finite index subgroup of both π ( M ) and gπ ( M ) g − .Let M ′ be the finite cover of M corresponding to π ( M ) ∩ gπ ( M ) g − , then M ′ covers M in two different ways.We consider M and M ′ as quotients of H by M = H /π ( M ) and M ′ = H /π ( M ) ∩ gπ ( M ) g − . There are two covering maps p, p ′ : M ′ = H /π ( M ) ∩ gπ ( M ) g − → M = H /π ( M )defined by p ( x ) = x and p ′ ( x ) = g − x . Then the induced homomorphisms onfundamental groups p ∗ , p ′∗ : π ( M ′ ) = π ( M ) ∩ gπ ( M ) g − → π ( M ) are given by p ∗ ( h ) = h and p ′∗ ( h ) = g − hg respectively.Let T ′ be the boundary torus of M ′ corresponding to ∞ ∈ ∂ H . For the slope a on T , it corresponds to h = (cid:18) p + q √− d (cid:19) ∈ π ( T ) < π ( M ). Then under thecovering map p ′ : M ′ → M , the elevation slope a ′ of a on T ′ corresponds to gh n g − = (cid:18) n ( s + r √− d ) ( p + q √− d )0 1 (cid:19) ∈ π ( T ′ ) < π ( M ′ )for some n ∈ Z + . Under the covering map p : M ′ → M , the projection of theslope a ′ on T still corresponds to matrix (cid:18) n ( s + r √− d ) ( p + q √− d )0 1 (cid:19) , whichhas coordinatearg (cid:0) n ( s + r √− d ) ( p + q √− d ) (cid:1) ∈ (arg( p + q √− d ) , arg( p + q √− d ) + ǫ ) , since arg( s + r √− d ) ∈ (0 , ǫ ).Since we have assumed that all slopes on T with coordinate in (arg( p + q √− d ) , arg( p + q √− d ) + ǫ ) are virtually fibered boundary slopes, a ′ is a virtu-ally fibered boundary slope on T ′ ⊂ ∂M ′ (via the covering map p ). Since thecovering map p ′ : M ′ → M maps a ′ to a , a is a virtually fibered boundary slope on T ⊂ ∂M . (cid:3) Construction of nonseparable subgroups
In this section, for a nontrivial geometrically finite amalgamation π ( M ) ∗ A π ( M ) with an algebraically fibered structure, we construct a nonseparable sub-group of it (we still use the notation π ( M ) ∗ A π ( M ), instead of π ( N ) ∗ A ′ π ( N )as in Theorem 3.2).The first step is to find a further subgroup of π ( M ) ∗ A π ( M ) (denoted by π ( N ) ∗ A ′ ,A ′′ π ( N )). This subgroup has an induced graph of group structure withtwo vertices and two edges, and also has an induced algebraically fibered structure.The remaining part of the construction is more topological. We first constructa space X that has a graph of space structure with π ( X ) ∼ = π ( N ) ∗ A ′ ,A ′′ π ( N ).Then we use the cycle in the dual graph of X to construct a compact 2-complex Z and a map f : Z → X , by pasting fibered surfaces in vertex pieces of X togethercarefully. Then we show that f ∗ ( π ( Z )) < π ( X ) is not separable, by assuming itis separable and using Scott’s topological interpretation of separability ([Sc]) to geta contradiction.4.1. A further subgroup of algebraically fibered π ( M ) ∗ A π ( M ) . In thissubsection, for a nontrivial geometrically finite amalgamation π ( M ) ∗ A π ( M )that has an algebraically fibered structure, we construct a subgroup such that itfits into the ideal model for constructing nonseparable subgroups in [Sun]. Proposition 4.1.
Suppose that π ( M ) ∗ A π ( M ) is a nontrivial geometricallyfinite amalgamation of two finite volume hyperbolic -manifold groups, which hasan algebraically fibered structure and satisfies conditions in Theorem 3.2. Then ithas a subgroup π ( N ) ∗ A ′ ,A ′′ π ( N ) such that the following conditions hold. (1) N , N are finite covers of M , M respectively. (2) There are two nontrivial groups A ′ , A ′′ and four injective homomorphisms i ′ : A ′ → π ( N ) , i ′ : A ′ → π ( N ) , i ′′ : A ′′ → π ( N ) , i ′′ : A ′′ → π ( N ) , such that their images in π ( N ) and π ( N ) are geometrically finite anddisjoint from each other except at the identity. Then π ( N ) ∗ A ′ ,A ′′ π ( N ) is isomorphic to the group with a graph of group structure induced by thesefour injective homomorphisms. (3) There are two elements g ∈ π ( M ) − π ( N ) and g ∈ π ( M ) − π ( N ) ,such that A ′ = A ∩ π ( N ) and A ′′ = g Ag − ∩ π ( N ) hold in π ( M ) ;while A ′ = A ∩ π ( N ) and A ′′ = g Ag − ∩ π ( N ) hold in π ( M ) . Herewe identify A ′ and A ′′ with their images in π ( N ) and π ( N ) . (4) There are fibered classes β ∈ H ( N ; Z ) and β ∈ H ( N ; Z ) such that β | A ′ = β | A ′ and β | A ′′ = β | A ′′ , and they are all surjective homomor-phisms to Z . (5) The homomorphism H ( A ′ ∗ A ′′ ; Z ) → H ( N ; Z ) induced by the inclusionis injective, and the image is a retraction of H ( N ; Z ) . The proof is similar to the proof of Lemma 4.5 of [Sun].
Proof.
Since
A < π ( M ) is a geometrically finite subgroup, its limit set is a properclosed subset of S ∞ = ∂ H . Take a loxodromic element g ∈ π ( M ) − { e } , suchthat both of its limit points do not lie in the limit set of A . For a large enough n ∈ Z + , the obvious homomorphism from A ∗ g n Ag − n to π ( M ) is injective, andthe image is geometrically finite. For simplicity, we denote the image by A ∗ g Ag − . MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 15
Similarly, there is also an element g ∈ π ( M ) − { e } such that the subgroupof π ( M ) generated by A and g Ag − is isomorphic to A ∗ g Ag − , and it isgeometrically finite.Now we apply LERFness of finite volume hyperbolic 3-manifold groups to A ∗ g Ag − < π ( M ) and the virtual retract property of A ∗ g Ag − < π ( M ). Thenthere exist a finite cover p : N → M , such that g π ( N ), A ∗ g Ag − < π ( N ),and A ∗ g Ag − is a retraction of π ( N ). By the same construction, we get a finitecover p : N → M with the same property. Actually, since we do not needcondition (5) for N , a simpler construction works.Then we can get the desired group π ( N ) ∗ A ′ ,A ′′ π ( N ) by identifying A <π ( N ) with A < π ( N ), and identifying g Ag − < π ( N ) with g Ag − < π ( N ).So actually A ′ ∼ = A ′′ ∼ = A .Then it is easy to see that conditions (1), (2), (3) in the proposition hold.We take β = p ∗ ( α ) ∈ H ( N ; Z ) and β = p ∗ ( α ) ∈ H ( N ; Z ). Since A ′ and A ′′ are just conjugations of A , condition (4) holds.The fact that A ∗ g Ag − = A ′ ∗ A ′′ is a retraction of π ( N ) implies that H ( A ′ ∗ A ′′ ; Z ) is a retraction of H ( N ; Z ). So the inclusion induces an injectivehomomorphism H ( A ′ ∗ A ′′ ; Z ) → H ( N ; Z ), with the image being a retraction of H ( N ; Z ). So condition (5) holds.It remains to show that π ( N ) ∗ A ′ ,A ′′ π ( N ) is isomorphic to a subgroup of π ( M ) ∗ A π ( M ), i.e. the obvious homomorphism is injective.By van Kampen theorem, π ( N ) ∗ A ′ ,A ′′ π ( N ) is isomorphic to h π ( N ) , π ( N ) , t | i ′ ( a ′ ) = i ′ ( a ′ ) , i ′′ ( a ′′ ) = ti ′′ ( a ′′ ) t − for any a ′ ∈ A ′ , a ′′ ∈ A ′′ i . Recall that A ′ ∼ = A ′′ ∼ = A , with i ′′ ( a ) = g i ′ ( a ) g − and i ′′ ( a ) = g i ′ ( a ) g − for g ∈ π ( M ) − π ( N ) and g ∈ π ( M ) − π ( N ).Then each element σ in π ( N ) ∗ A ′ ,A ′′ π ( N ) can be written in the form of σ = t k h t k h · · · t k n h n t k n +1 . Here k , k n +1 ∈ Z , k , · · · , k n ∈ Z − { } , and each h i is a nontrivial product ofelements in π ( N ) and π ( N ). Moreover, we also have the following conditionsfor h i : • If h i is a product of more than one terms, then each term does not lie in i ′ ( A ′ ) < π ( N ) or i ′ ( A ′ ) < π ( N ). • If h i is just (the product of) one element in π ( N ) or π ( N ), then – if h i ∈ i ′′ ( A ′′ ) < π ( N ), then either k i ≥ k i +1 ≤ – if h i ∈ i ′′ ( A ′′ ) < π ( N ), then either k i ≤ k i +1 ≥ j : π ( N ) ∗ A ′ ,A ′′ π ( N ) → π ( M ) ∗ A π ( M )is defined by j ( n ) = n if n ∈ π ( N ) , j ( n ) = n if n ∈ π ( N ) , j ( t ) = g g − . By the above form of elements in π ( N ) ∗ A ′ ,A ′′ π ( N ), the choice of g and g ,and the canonical form of elements in an amalgamation, it is routine to check that j is injective. So π ( N ) ∗ A ′ ,A ′′ π ( N ) is isomorphic to a subgroup of π ( M ) ∗ A π ( M ). (cid:3) Realize π ( N ) ∗ A ′ ,A ′′ π ( N ) as the fundamental group of a space. Allproofs in the previous part of this paper are in algebraic fashion, since we onlyworked on the group level and did not realize π ( M ) ∗ A π ( M ) and π ( N ) ∗ A ′ ,A ′′ π ( N ) as fundamental groups of topological spaces. In the following part of thispaper, when we construct nonseparable subgroups, we need to realize π ( N ) ∗ A ′ ,A ′′ π ( N ) as the fundamental group of a topological space. In this subsection, weconstruct this topological space, and develop some definition for the convenience offurther constructions.For finite volume hyperbolic 3-manifolds N , N as in Proposition 4.1, we takecovering spaces p ′ : ˜ N ′ → N and p ′ : ˜ N ′ → N corresponding to A ′ < π ( N ) and A ′ < π ( N ) respectively. Although having isomorphic fundamental groups, ˜ N ′ and˜ N ′ may not be homeomorphic to each other. However, since hyperbolic manifoldsare all Eilenberg-Maclane spaces ( K ( π, f ′ : ˜ N ′ → ˜ N ′ . Similarly, for covering spaces p ′′ : ˜ N ′′ → N and p ′′ : ˜ N ′′ → N corresponding to A ′′ < π ( N ) and A ′′ < π ( N ) respectively, we also have ahomotopy equivalence f ′′ : ˜ N ′′ → ˜ N ′′ . For most part of this paper, the readers canjust think f ′ and f ′′ as homeomorphisms.Now we construct a space X from N ⊔ N ⊔ ˜ N ′ × I ⊔ ˜ N ′′ × I by the followingpasting maps: p ′ : ˜ N ′ × { } → N , p ′ ◦ f ′ : ˜ N ′ × { } → N ,p ′′ : ˜ N ′′ × { } → N , p ′′ ◦ f ′′ : ˜ N ′′ × { } → N . Here we apply p ′ , p ′′ , f ′ , f ′′ to slices of ˜ N ′ × I and ˜ N ′′ × I by restricting to the firstcoordinate.By van Kampen theorem, π ( X ) is isomorphic to π ( N ) ∗ A ′ ,A ′′ π ( N ).We also need to define a notion of immersed objects in X that give desirednonseparable subgroups. This will play the role of properly immersed surfaces inmixed 3-manifolds, and ”properly immersed singular surfaces” in N ∪ c ∪ c N asin [Sun]. The main difference is that the immersed object we will construct is not π -injective. Definition 4.2.
For the space X as above, a generalized immersed surface in X isa pair ( Z, f ) where Z is a connected compact 2-complex and f : Z → X is a mapsuch that the following conditions hold.(1) Z is constructed by pasting finitely many intervals ⊔ I j to finitely manycompact oriented surfaces ⊔ S i , by identifying all end points of intervalswith distinct points in surfaces.(2) The f -image of each S i either entirely lies in N or entirely lies in N .Moreover, f | S i : S i → N k is a π -injective immersion for the corresponding k ∈ { , } .(3) The f -image of each I j either entirely lies in ˜ N ′ × I or entirely lies in ˜ N ′ × I .Moreover, let p k : ˜ N ′ k × I → I be the projection to the second factor, then p k ◦ f | I j : I j → I is the identity map on interval, for the corresponding k ∈ { , } .For a generalized immersed surface ( Z, f ) in X , the induced homomorphism f ∗ : π ( Z ) → π ( X ) may not be injective. MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 17
Construct a generalized immersed surface that carries a nonsepa-rable subgroup.
In this subsection, for the space X as above with fundamentalgroup π ( N ) ∗ A ′ ,A ′′ π ( N ), we construct a generalized immersed surface ( Z, f ) in X , such that f ∗ ( π ( Z )) is the candidate of a nonseparable subgroup of π ( X ).The construction of this nonseparable subgroup is similar to the constructionin [Sun], and the proof of nonseparability essentially follows the idea in [Liu] and[RW]. The only difference is that we do not require the nonseparable subgroup iscarried by a π -injective map, and the following statement on the construction of( Z, f ) is more complicated.
Proposition 4.3.
Let π ( N ) ∗ A ′ ,A ′′ π ( N ) be a group that has a graph of groupstructure, an algebraically fibered structure, and satisfies all conditions in Propo-sition 4.1. Let X be the topological space with fundamental group isomorphic to π ( N ) ∗ A ′ ,A ′′ π ( N ) constructed in subsection 4.2, then there exists a generalizedimmersed surface ( Z, f ) such that the following conditions hold. (1) Z is a quotient space of n compact connected oriented surfaces { S ,i } i =1 , { S ,j } nj =1 and n intervals { I k } nk =1 (with n ≥ ), by identifyingall end points of intervals with points in surfaces. Moreover, all surfaces { S ,j } nj =1 are homeomorphic to each other. (2) For each interval I k ⊂ Z , one of its end point lies in some S ,i and the otherend point lies in some S ,j . There are n + 1 intervals connecting S , tosurfaces in { S ,j } nj =1 , and n − intervals connecting S , to surfaces in { S ,j } nj =1 . Moreover, all these end points have different images in X . (3) There are two intervals (say I and I ) connecting S , to S , . (4) For each S ,i and S ,j , the restriction of f on this surface is an embeddinginto N and N respectively, and the image is a fibered surface. Moreover,the images of { S ,j } nj =1 are n parallel copies of an oriented fibered surfacein N . (5) Let γ , , γ , ∈ H ( N ; Z ) be fibered classes of N corresponding to fiberedsurfaces S , , S , respectively, and γ ∈ H ( N ; Z ) be the fibered class of N corresponding to one of the parallel fibered surfaces S ,j . Then we have γ , | A ′ = ( n + 1) γ | A ′ , γ , | A ′′ = nγ | A ′′ ,γ , | A ′ = ( n − γ | A ′ , γ , | A ′′ = nγ | A ′′ , as homomorphisms to Z . Moreover, γ | A ′ and γ | A ′′ are surjective homo-morphisms. (6) There exist closed embedded oriented circles c ′ in ˜ N ′ and c ′′ in ˜ N ′′ such thatthe following hold. The algebraic intersection numbers of p ′ ( c ′ ) with S , and S , in N are n + 1 and n − respectively, the algebraic intersectionnumbers of p ′′ ( c ′′ ) with S , and S , in N are both n , and the algebraicintersection numbers of p ′ ◦ f ′ ( c ′ ) and p ′′ ◦ f ′′ ( c ′′ ) with each S ,j in N areexactly . (7) The following statement holds for all triples ( p ′ ( c ′ ) , S , , n + 1) , ( p ′ ( c ′ ) , S , , n − , ( p ′′ ( c ′′ ) , S , , n ) , ( p ′′ ( c ′′ ) , S , , n ) , ( p ′ ◦ f ′ ( c ′ ) , S ,j , , ( p ′′ ◦ f ′′ ( c ′′ ) , S ,j , . We only state it for the triple ( p ′ ( c ′ ) , S , , n + 1) , and the statements forother triples are similar. There exist n + 1 points a , a , · · · , a n +1 in c ′ ∩ ( p ′ ) − ( S , ) such thatthe following hold. (a) The points a , a , · · · , a n +1 follow the orientation of c ′ , and the localalgebraic intersection number of p ′ ( c ′ ) and S , at each a i is . (b) We take the oriented subarc of c ′ from a to a i , then slightly moveit along the positive direction of c ′ and get an oriented subarc ρ i ⊂ c ′ whose end points are away from ( p ′ ) − ( S , ) . Then the algebraicintersection number of p ′ ( ρ i ) and S , is i − . (c) p ′ ( a ) , p ′ ( a ) , · · · , p ′ ( a n +1 ) are n + 1 points on S , that are identi-fied with end points of intervals in Z . Moreover, f maps these n + 1 intervals to ˜ N ′ × [0 , . (8) There are exactly n + 1 intervals in Z connecting S , to { S ,j } that aremapped to ˜ N ′ × [0 , , they give a one-to-one correspondence between n + 1 points in c ′ × { } ( a , a , · · · , a n +1 as above) and n + 1 points in c ′ × { } such that this correspondence preserves the cyclic order on the orientedcircle c ′ . Moreover, the f -image of these intervals lie in c ′ × [0 , , they aredisjoint from each other, and their projections to c ′ are embedded (possiblydegenerate) subarcs of c ′ . The same statement holds for n − edges in Z connecting S , to { S ,j } that are mapped to ˜ N ′ × [0 , , n edges in Z connecting S , to { S ,j } that are mapped to ˜ N ′′ × [0 , , and n edges in Z connecting S , to { S ,j } that are mapped to ˜ N ′′ × [0 , . The readers should compare Proposition 4.3 with Proposition 4.8 of [Sun]. Al-though the statement of Proposition 4.3 is more complicated than Proposition 4.8of [Sun], it just follows the same idea. Since we have a more complicated space X in the current situation, the statement gets more complicated. Proof.
By Proposition 4.1 (4), there are fibered classes β ∈ H ( N ; Z ) and β ∈ H ( N ; Z ) such that β | A ′ = β | A ′ , β | A ′′ = β | A ′′ hold, and they are both surjective homomorphisms to Z .By Proposition 4.1 (5), the homomorphism H ( A ′ ∗ A ′′ ; Z ) → H ( N ; Z ) inducedby inclusion is injective, with a retract homomorphism φ : H ( N ; Z ) → H ( A ′ ∗ A ′′ ; Z ). So for the homomorphism δ : A ′ ∗ A ′′ → Z (equivalently H ( A ′ ∗ A ′′ ; Z ) → Z )defined by δ | A ′ = β | A ′ , δ | A ′′ = 0, the composition γ = δ ◦ φ : H ( N ; Z ) → Z givesa cohomology class γ ∈ H ( N ; Z ).Since β is a fibered class of N , by Lemma 2.4, for large enough n ∈ Z + , γ , = nβ + γ, γ , = nβ − γ ∈ H ( N ; Z )are both fibered classes of N . Then we take γ = β as the desired fibered class of N . Since γ | A ′ = β | A ′ and γ | A ′′ = 0, it is easy to check that condition (5) holds.For example, we have γ , | A ′ = nβ | A ′ + γ | A ′ = ( n + 1) β | A ′ = ( n + 1) β | A ′ = ( n + 1) γ | A ′ . Since γ = β , γ is a primitive class in H ( N ; Z ), and its restrictions on A ′ and A ′′ are surjective homomorphisms to Z . Since γ , | A ′ has image ( n + 1) Z and γ , | A ′′ has image n Z , γ , is a primitive class in H ( N ; Z ). The same argumentimplies γ , is also a primitive class.Then we take connected oriented fibered surfaces S , and S , in N corre-sponding to fibered classes γ , and γ , respectively, take 2 n parallel copies of the MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 19 connected oriented fibered surface in N corresponding to γ , and denote them by { S ,j } nj =1 . The construction of these fibered surfaces satisfies condition (4).Since γ | A ′ : A ′ → Z is surjective and π ( ˜ N ′ ) ∼ = A ′ , there exists a closed em-bedded oriented circle c ′ in ˜ N ′ , such that γ ([ c ′ ]) = 1. Then we have γ , ([ c ′ ]) =( n + 1) γ ([ c ′ ]) = n + 1 and γ , ([ c ′ ]) = ( n − γ ([ c ′ ]) = n −
1. Similarly, thereexists a closed embedded oriented circle c ′′ in ˜ N ′′ , such that γ ([ c ′′ ]) = 1, and γ , ([ c ′′ ]) = γ , ([ c ′′ ]) = n . These equalities imply that condition (6) holds. We canalso homotopy c ′ and c ′′ such that their images in N and N have general positionwith S , , S , and S ,j .Now we consider condition (7). We only work on the triple ( p ′ ( c ′ ) , S , , n + 1),and the same argument works for all other triples. Actually, the requirementsin condition (7) instruct us how to choose the points a , a , · · · , a n +1 . Since wesupposed that p ′ ( c ′ ) intersects with S , transversely, each point in p ( c ′ ) ∩ S , has local algebraic intersection number ±
1, and their sum is γ , ([ c ′ ]) = n + 1. Westart with a point a ∈ c ′ with local algebraic intersection number 1, then follow theorientation of c ′ and sum algebraic intersection numbers of points in c ′ ∩ ( p ′ ) − ( S , )we have visited. Let a i be the first point we get total algebraic intersection number i , then it is obvious that we have local algebraic intersection number 1 at each a i ,and desired properties in condition (7) (a) and (7) (b) hold.Condition (7) (a) and (7) (b) provides us 2 n = ( n + 1) + ( n −
1) points in c ′ ∩ ( p ′ ) − ( S , ∪ S , ), 2 n = n + n points in c ′′ ∩ ( p ′′ ) − ( S , ∪ S , ), 2 n = 1 × n pointsin c ′ ∩ ( p ′ ◦ f ′ ) − ( ∪ nj =1 S ,j ) and 2 n = 1 × n points in c ′′ ∩ ( p ′′ ◦ f ′′ ) − ( ∪ nj =1 S ,j ).These points will be pasted with end points of intervals in Z , and we will see thatthese numbers of points fulfill the requirements in condition (2).Then we pair the 2 n points in c ′ ∩ ( p ′ ) − ( S , ∪ S , ) with the 2 n points in c ′ ∩ ( p ′ ◦ f ′ ) − ( ∪ nj =1 S ,j ), and pair the 2 n points in c ′′ ∩ ( p ′ ) − ( S , ∪ S , ) with the2 n points in c ′′ ∩ ( p ′′ ◦ f ′′ ) − ( ∪ nj =1 S ,j ), such that these pairings preserve cyclic orderson c ′ and c ′′ . To fulfill condition (3), we need to pair one point in c ′ ∩ ( p ′ ) − ( S , )with the point in c ′ ∩ ( p ′ ◦ f ′ ) − ( S , ), and one point in c ′′ ∩ ( p ′′ ) − ( S , ) with thepoint in c ′′ ∩ ( p ′′ ◦ f ′′ ) − ( S , ).Now we construct the map on intervals { I k } nk =1 . For each pair of points in c ′ and c ′′ that are identified by the above pairing, we construct a map on an interval toconnect these two points. For example, if these two points are q ∈ c ′ ∩ ( p ′ ) − ( S , )and q ∈ c ′ ∩ ( p ′ ◦ f ′ ) − ( S , ), then we use an embedded subarc δ of c ′ from q to q to connect these two points. Then we get a map from [0 ,
1] to ˜ N ′ × [0 ,
1] defined by t → ( δ ( t ) , t ) connecting ( q ,
0) and ( q , c ′ × [0 , n maps from [0 ,
1] to ˜ N ′ × I , and 2 n mapsfrom [0 ,
1] to ˜ N ′′ × I . We paste their endpoints with the corresponding points in( S , ⊔ S , ) ⊔ ( ⊔ nj =1 S ,j ), and get the desired 2-complex Z . The map f is alreadygiven during our construction. So we get a generalized immersed surface ( Z, f ) thatsatisfies all desired conditions. (cid:3)
Construction of the covering space of X corresponding to f ∗ ( π ( Z )) . In this subsection, we figure out the topology of the covering space ˆ X of X corre-sponding to f ∗ ( π ( Z )) < π ( X ), and show that f : Z → X lifts to an embedding ˆ f : Z ֒ → ˆ X . In the next subsection, with the knowledge that Z lifts to be embeddedin ˆ X , we suppose f ∗ ( π ( Z )) < π ( X ) is separable, then apply Scott’s topologicalinterpretation of separability ([Sc]) and get a contradiction. The correspondingcovering space in [Sun] is easy to figure out, since the edge spaces in [Sun] arevery simple. Our current case is more complicated, so we give a more detailedconstruction of the covering space.Let ˆ N , and ˆ N , be the infinite cyclic cover of N corresponding to (the kernelof) γ , and γ , respectively, and let ˆ N be the infinite cyclic cover of N corre-sponding to γ . Let ˆ N ′ be the infinite cyclic cover of ˜ N ′ corresponding to γ , | A ′ ,and ˆ N ′′ be the infinite cyclic cover of ˜ N ′′ corresponding to γ , | A ′′ . Similarly, letˆ N ′ be the infinite cyclic cover of ˜ N ′ corresponding to γ | A ′ , and ˆ N ′′ be the infinitecyclic cover of ˜ N ′′ corresponding to γ | A ′′ .Note that ˆ N ′ is homeomorphic to the infinite cyclic cover of ˜ N ′ corresponding to γ , | A ′ , since γ , | A ′ and γ , | A ′ are both nonzero multiples of γ | A ′ , and they havethe same kernel. Moreover, ˆ N ′ is homotopic equivalent to ˆ N ′ , since A ′ = π ( ˜ N ′ )is isomorphic to A ′ = π ( ˜ N ′ ) and γ , | A ′ has the same kernel as γ | A ′ . The samestatement also holds for ˆ N ′′ .Let N ∗ = N ∪ ˜ N ′ × [0 , ∪ ˜ N ′′ × [0 ,
1] be the subspace of X correspondingto one vertex and two edges of the dual graph of X . Technically we should take˜ N ′ × [0 , − ǫ ] and ˜ N ′′ × [0 , − ǫ ], or saying that X is a quotient space of N ∗ , butwe abuse notation here. Note that N ⊂ N ∗ is a deformation retract of N ∗ .Let ˆ N ∗ , and ˆ N ∗ , be the infinite cyclic cover of N ∗ corresponding to γ , : π ( N ) → Z and γ , : π ( N ) → Z respectively. Since π ( ˜ N ′ ) = A ′ and π ( ˜ N ′′ ) = A ′′ , γ , | A ′ is an ( n + 1)-multiple of a primitive element in H ( A ′ ; Z ) and γ , | A ′′ isan n -multiple of a primitive element in H ( A ′′ ; Z ), ˆ N ∗ , is the union of ˆ N , , n + 1copies of ˆ N ′ × [0 ,
1] and n copies of ˆ N ′′ × [0 , N ∗ , is the union of ˆ N , , n − N ′ × [0 ,
1] and n copies of ˆ N ′′ × [0 , Z has a graph of space structure. Its dual graph G has 2 + 2 n vertices,and they correspond to { S , , S , } ∪ { S ,j } nj =1 . There are 4 n edges in G and eachof them connects one vertex in { S , , S , } to one vertex in { S ,j } nj =1 , and theycorrespond to intervals in Z . Each edge also has a marking in { , } , correspondingto whether f maps this edge to ˜ N ′ × [0 ,
1] or ˜ N ′′ × [0 ,
1] respectively.Let S ∗ , be the union of S , and all its adjacent edges in Z , and S ∗ , be the unionof S , and all its adjacent edges in Z . Then f | S ∗ , : S ∗ , → N ∗ and f | S ∗ , : S ∗ , → N ∗ have liftings ˆ f | S ∗ , : S ∗ , → ˆ N ∗ , and ˆ f | S ∗ , : S ∗ , → ˆ N ∗ , respectively. For S ∗ , ,it is clear that ˆ f | S ∗ , maps edges in S ∗ , with marking 1 to copies of ˆ N ′ × [0 , N ′′ × [0 , f | S ∗ , .Moreover, condition (7) in Proposition 4.3 implies that the n + 1 edges in S ∗ , with marking 1 are mapped to n + 1 different copies of ˆ N ′ × [0 ,
1] in ˆ N ∗ , (see theproof of Proposition 4.10 of [Sun]). The same statement also holds for the n edgesin S ∗ , with marking 2, the n − S ∗ , with marking 1 and the n edges in S ∗ , with marking 2. MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 21
Then we take 2 n copies of ˆ N and denote them by { ˆ N ,j } nj =1 , each endowed witha lifting of S ,j ⊂ N to S ,j ⊂ ˆ N ,j , and two marked points in S ,j correspondingto end points of intervals in Z that are pasted to S ,j .For each S ,j ⊂ Z , there are two intervals in Z connected with it. ˆ f | S ∗ , andˆ f | S ∗ , map these two intervals to some copy of ˆ N ′ × [0 ,
1] and ˆ N ′′ × [0 ,
1] in ˆ N ∗ , or ˆ N ∗ , . Then we can paste the ends of ˆ N ′ × [0 ,
1] and ˆ N ′′ × [0 ,
1] with secondcoordinate 1 to ˆ N ,j by maps ˆ N ′ × { } → ˆ N ′ → ˆ N ,j and ˆ N ′′ × { } → ˆ N ′′ → ˆ N ,j that send the end points of intervals in ˆ N ′ × { } and ˆ N ′′ × { } to the correspondingtwo marked points in S ,j ⊂ ˆ N ,j . In the composition ˆ N ′ × { } → ˆ N ′ → ˆ N ,j , thefirst map is a homotopy equivalence and the second one is a covering map. It is thelifting of p ′ ◦ f ′ : ˜ N ′ × { } → ˜ N ′ → N to the corresponding infinite cyclic covers.After pasting 2 n copies of ˆ N ′ × { } and 2 n copies of ˆ N ′′ × { } in ˆ N ∗ , and ˆ N ∗ , with 2 n copies of ˆ N by the above process, we get a space Y with a graph of spacestructure such that its dual graph is isomorphic to the dual graph of Z . Moreover,we have an embedding ˆ f : Z ֒ → Y that induces an isomorphism on the dual graphs.Then we prove that the space Y constructed above is the covering space of X corresponding to the subgroup f ∗ ( π ( Z )) < π ( X ). Lemma 4.4.
Let ˆ X be the covering space of X corresponding to the subgroup f ∗ ( π ( Z )) < π ( X ) , then ˆ X is homeomorphic to the space Y constructed above.Moreover, f : Z → X lifts to an embedding ˆ f : Z ֒ → ˆ X .Proof. We first construct a covering map π : Y → X , such that the embeddingˆ f : Z ֒ → Y is a lifting of f : Z → X , then show that ˆ f : Z → Y is a π -surjectivemap. The existence of a lifting ˆ f : Z → Y implies f ∗ ( π ( Z )) < π ( Y ), and π -surjectivity implies f ∗ ( π ( Z )) = π ( Y ). So Y is homeomorphic to the coveringspace of X corresponding to f ∗ ( π ( Z )), and ˆ f : Z ֒ → Y is an embedded lifting of f : Z → X .The covering map π : Y → X is almost obvious. On ˆ N ∗ , ⊂ Y and ˆ N ∗ , ⊂ Y ,we take their covering maps to N ∗ in the construction of ˆ N ∗ , and ˆ N ∗ , . Oneach ˆ N ,j ⊂ Y , we take its covering map to N . Then we get a well-defined map π : Y → X . The only thing we need to check is that π is a local homeomorphismnear each ˆ N ,j , since all the other points in Y lie in the interior of ˆ N ∗ , ⊂ Y orˆ N ∗ , ⊂ Y . This local homeomorphism property is easy to check and we leave it tothe readers.The definition of ˆ f : Z → Y immediately implies that π ◦ ˆ f = f , so ˆ f : Z ֒ → Y is an embedded lifting of f : Z → X .Now we show that ˆ f : Z → Y is π -surjective. At first, by the construction of Y ,the restriction of ˆ f on each vertex space of Z to the corresponding vertex piece of Y induces an isomorphism on their fundamental groups. Since the fundamental groupof each edge space of Y is a subgroup of fundamental groups of its two adjacentvertex spaces, the van-Kampen theorem implies that π ( Y ) is generated by thefundamental group of its vertex spaces, and the fundamental group of the dualgraph of Y (by choosing a ”section” s : G → Y from the dual graph G to Y ). Sinceˆ f : Z → Y induces an isomorphism on the dual graph, we can assume the section s : G → Y factors through ˆ f : Z → Y . Then the groups of vertex spaces of Y and the group of the dual graph of Y (via a section) are both contained in ˆ f ∗ ( π ( Z )).So ˆ f : Z → Y is π -surjective. (cid:3) Nonseparability of f ∗ ( π ( Z )) < π ( X ) . In this subsection, we prove thatthe subgroup f ∗ ( π ( Z )) < π ( X ) constructed in Proposition 4.3 is not separable. Proposition 4.5.
For the space X constructed in subsection 4.2 and the generalizedimmersed surface ( Z, f ) constructed in Proposition 4.3, f ∗ ( π ( Z )) is a nonseparablesubgroup of π ( X ) = π ( N ) ∗ A ′ ,A ′′ π ( N ) . The proof is similar to the proof of Proposition 4.10 of [Sun], and the ideagoes back to [Liu] and [RW]. The only difference is that, in the current situation, f : Z → X is not a π -injective map. However, the subgroup f ∗ ( π ( Z )) < π ( X ) isstill manageable, since we showed that Z lifts to be embedded in ˆ X (Lemma 4.4). Proof.
We suppose that f ∗ ( π ( Z )) < π ( X ) is separable, and get a contradiction.Recall that we have constructed the covering space ˆ X of X corresponding to f ∗ ( π ( Z )). Lemma 4.4 implies that f : Z → X lifts to an embedding ˆ f : Z ֒ → ˆ X .Since Z is a compact space, by Scott’s topological interpretation of separability([Sc]), there exists a finite cover ¯ X → X such that ˆ X → X factors through ¯ X , and f : Z → X lifts to an embedding ¯ f : Z ֒ → ¯ X .We first prove the following lemma. Lemma 4.6.
There exists a nontrivial cohomology class ζ ∈ H ( ¯ X ; Z ) , such that ζ | ¯ f ∗ ( π ( Z )) = 0 , as a homomorphism from π ( ¯ X ) to Z .Proof. Each vertex piece of ¯ X is a finite volume hyperbolic 3-manifold (finite coverof N or N ), and its intersection with Z is a (possibly disconnected) orientedsurface. So we get a cohomology class in each vertex piece, and we will showthat these cohomology classes in vertex pieces can be ”pasted together” to get thedesired cohomology class on ¯ X .Each edge space of ¯ X is a finite cover of ˜ N ′ × [0 ,
1] or ˜ N ′′ × [0 , N ′ × [0 , N ′ × [0 , X , and we denote them by ¯ N and ¯ N . Then they are finitecovers of N and N respectively.Since Z intersects with all vertex pieces of ˆ X , Z also intersects with all vertexpieces of ¯ X . So Z ∩ ¯ N = S , or S , , and we assume that Z ∩ ¯ N = S , . Then¯ N → N is a d -sheet cyclic cover along S , for some d ∈ Z + . It is easy tocheck that, in the dual graph of ¯ X , there are ( d, n + 1) edges with marking 1 thatgo through the vertex corresponding to ¯ N . Here ( d, n + 1) denotes the greatestcommon divisor of d and n + 1. Moreover, condition (7) of Proposition 4.3 impliesthat each edge space as above contains exactly n +1( d,n +1) many ¯ f -images of intervalsin Z . The proof of this claim is same with the argument in Proposition 4.10 of[Sun]. It is an elementary application of covering space theory, so we do not givethe proof here.So there are exactly n +1( d,n +1) many intervals in Z that are mapped to ¯ N ′ × [0 , N contains exactly n +1( d,n +1) many ¯ f -images of S ,j in Z . Moreover, ¯ N → N is a d -sheet cyclic cover implies ¯ N ′ → N ′ is a d ( d,n +1) -sheetcyclic cover on the edge space. Since γ | A ′ : A ′ → Z is surjective, ¯ N → N is a d ( d,n +1) -sheet cyclic cover corresponding to γ ∈ H ( N ; Z ). MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 23
Let ¯ p : ¯ N → N and ¯ p : ¯ N → N be the restrictions of the covering map¯ X → X . Then S , ⊂ ¯ N is dual to the cohomology class1 d (¯ p ) ∗ ( γ , ) ∈ H ( ¯ N ; Z ) , and the union of n +1( d,n +1) many S ,j in ¯ N is dual to the cohomology class n + 1( d, n + 1) · d ( d,n +1) (¯ p ) ∗ ( γ ) = n + 1 d (¯ p ) ∗ ( γ ) ∈ H ( ¯ N ; Z ) . Since γ , | A ′ = ( n + 1) γ | A ′ , the restriction of these two cohomology classes givethe same homomorphism from π ( ¯ N ′ ) < A ′ to Z . So the cohomology classes on ¯ N and ¯ N agree with each other on ¯ N ′ × [0 , X defined byoriented surfaces in Z agree with each other on edge spaces. The Mayer-Vietorissequence implies these cohomology classes on vertex space can be pasted to a co-homology class on ¯ X . However, such a class is not unique, and may not vanish on¯ f ∗ ( π ( Z )).Now we construct a homotopic nontrivial map π : ¯ X → S such that the com-position π ◦ ¯ f : Z → ¯ X → S is a constant map. Then we get the desired nontrivialcohomology class ζ ∈ H ( ¯ X ; Z ) by pulling-back a generator of H ( S ; Z ) via π ∗ ,since ζ | ¯ f ∗ ( π ( Z )) = 0 holds.On each vertex space ¯ X v of ¯ X , since Z ∩ ¯ X v is a fibered surface of ¯ X v , it induces amap π | ¯ X v : ¯ X v → S such that π | ¯ X v ( Z ∩ ¯ X v ) = 1 = e i ∈ S . Now we take an edgespace ¯ X e of ¯ X , and assume ¯ X e = ¯ N ′ × [0 , N ′ × [0 ,
1] agree with eachother, the induced maps ¯ N ′ × { } → S and ¯ N ′ × { } → S are homotopy to eachother.So there is a map π ′′ : ¯ N ′ × [0 , → S extending the maps already defined on¯ N ′ × { , } induced from vertex spaces. The problem is that π ′′ may not map edgesof Z in ¯ N ′ × [0 ,
1] to 1 ∈ S , even up to homotopy. Take two points p ∈ c ′ × { } and p ∈ c ′ × { } , such that there is an interval I ⊂ Z such that ¯ f (0) = p and¯ f (1) = p for end points 0 , ∈ I . Then σ = π ′′ ◦ ¯ f | I : I → S maps both 0 , ∈ I to 1 ∈ S . Let σ ′ : I → S be the inverse path of σ in S , then we define a map π ′ : ¯ N ′ × [0 , → S by π ′ ( n, t ) = (cid:26) π ′′ ( n, t ) if t ∈ [0 , σ ′ ( t − · π ′′ ( n,
1) if t ∈ [1 , . Here the operation in σ ′ ( t − · π ′′ ( n,
1) is the multiplication of S . The map π ′ hasthe same definition as π ′′ on the two boundaries of ¯ N ′ × [0 , f ( I ) ∪ ( { p } × [1 , S relative to the boundary.For any other interval I ′ ⊂ Z with ¯ f ( I ′ ) ⊂ ¯ N ′ × [0 , f ( I ′ ) ∩ ( ¯ N ′ × { } ) = { p ′ } . Condition (8) of Proposition 4.3 implies pairings preserve cyclic orders andintervals in c ′ × [0 ,
1] are disjoint from each other, so the null-homotopy condition isalso true for ¯ f ( I ′ ) ∪ ( { p ′ } × [1 , ⊂ ¯ N ′ × [0 , π ′ relativeto ¯ N ′ × { , } and resize ¯ N ′ × [0 ,
2] to ¯ N ′ × [0 ,
1] to get the desired π | ¯ X e : ¯ X e =¯ N ′ × [0 , → S , such that it maps ¯ f ( Z ) ∩ ¯ X e to 1 ∈ S .So we get a map π : ¯ X → S such that π ◦ ¯ f : Z → S is a constant map.Since the restriction of π on each vertex space corresponds to a nontrivial firstcohomology class, π is homotopically nontrivial. So the proof is done. (cid:3) Now we continue with the proof of Proposition 4.5.Let k , · · · , k s be positive integers such that the restriction of ζ on each vertex oredge space of ¯ X is a k i -multiple of a primitive cohomology class. Let K be the leastcommon multiple of k , · · · , k s . Then we take the cyclic cover of ¯ X correspondingto the kernel of π ( ¯ X ) ζ −→ Z → Z /K Z , and get a finite cover q : ˘ X → ¯ X such that Z lifts to be embedded into ˘ X . Then K q ∗ ( ζ ) is a primitive cohomology class in H ( ˘ X ; Z ), and its restriction on each edge and vertex space of ˘ X is also primitive.Let p : ˘ X → X be the induced finite cover from ˘ X to X .Let ˘ N and ˘ N be the elevations of N and N in ˘ X such that S , is containedin ˘ N and S , is contained in ˘ N . Since I and I are edges in Z connecting S , and S , with f ( I ) ⊂ ˜ N ′ × [0 ,
1] and f ( I ) ⊂ ˜ N ′′ × [0 , N ′ × [0 ,
1] and ˘ N ′′ × [0 ,
1] in ˘ X connecting ˘ N and ˘ N . Then ˘ N ′ and ˘ N ′′ are finitecovers of ˜ N ′ and ˜ N ′′ respectively.Since the restriction of ζ on ¯ N ∪ ¯ N ′ × [0 , ∪ ¯ N ′′ × [0 , ⊂ ¯ X is dual to thefibered surface S , ⊂ ¯ N , , and S , is also a fibered surface in N , the restrictedmap p | : ˘ N ∪ ˘ N ′ × [0 , ∪ ˘ N ′′ × [0 , → N ∪ ˜ N ′ × [0 , ∪ ˜ N ′′ × [0 , γ , ∈ H ( N ; Z ) and its restrictions on A ′ and A ′′ respectively.Let p | ˘ N : ˘ N → N be the restriction of p on ˘ N , and suppose this is a degree- D cover. Then the surface S , in ˘ N is dual to D ( p | ˘ N ) ∗ ( γ , ), which is also equal to K q ∗ ( ζ ) | π ( ˘ N ) . Since γ , | A ′ is an ( n + 1)-multiple of a primitive cohomology classin H ( N ′ ; Z ), γ , | A ′′ is an n -multiple of a primitive class in H ( N ′′ ; Z ), and therestriction of K q ∗ ( ζ ) | π ( ˘ N ) = D ( p | ˘ N ) ∗ ( γ , ) on π ( ˘ N ′ ) and π ( ˘ N ′′ ) are primitiveclasses, we have that D is a multiple of n ( n + 1) and the following equation hold:( n + 1) · deg( ˘ N ′ → N ′ ) = deg( ˘ N → N ) = n · deg( ˘ N ′′ → N ′′ ) . Similarly, by applying the same argument to ˘ N → N , we getdeg( ˘ N ′ → N ′ ) = deg( ˘ N → N ) = deg( ˘ N ′′ → N ′′ ) . So we get a contradiction, and f ∗ ( π ( Z )) < π ( X ) is not separable. (cid:3) Proof of Theorem 1.1.
In this subsection, we give the proof of Thoerem 1.1.The proof is just a combination of Proposition 3.2, Proposition 4.1, Proposition 4.3and Proposition 4.5.
Proof.
We start with a nontrivial geometrically finite amalgamation π ( M ) ∗ A π ( M ) of finite volume hyperbolic 3-manifolds M and M . Then Proposition3.2 produces a subgroup of π ( M ) ∗ A π ( M ) with a two-vertex one-edge dualgraph and an algebraically fibered structure. For simplicity, we still denote it by π ( M ) ∗ A π ( M ).Then Proposition 4.1 gives us a further subgroup π ( N ) ∗ A ′ ,A ′′ π ( N ) of π ( M ) ∗ A π ( M ), which has a two-vertex two-edge graph of group structure, andan induced algebraically fibered structure. The subgroup π ( N ) ∗ A ′ ,A ′′ π ( N ) alsosatisfies other conditions in Proposition 4.1. MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 25
In subsection 4.2, we constructed a space X such that π ( X ) ∼ = π ( N ) ∗ A ′ ,A ′′ π ( N ). In Proposition 4.3, we constructed a generalized immersed surface ( Z, f )in X . Then Proposition 4.5 implies that f ∗ ( π ( Z )) is a nonseparable subgroup of π ( X ) ∼ = π ( N ) ∗ A ′ ,A ′′ π ( N ).Lemma 2.3 implies that f ∗ ( π ( Z )) is also a nonseparable subgroup of π ( M ) ∗ A π ( M ), thus π ( M ) ∗ A π ( M ) is not LERF. (cid:3) NonLERFness of closed arithmetic hyperbolic -manifold groups In this section, we prove that Theorem 1.1 implies Theorem 1.2. Actually, The-orem 1.1 also directly implies Corollary 1.3, by applying a similar argument.The proof of Theorem 1.2 follows the same idea in [Sun], except that we have astronger nonLERFness result (Theorem 1.1) in this paper.
Proof.
Let M be a closed arithmetic hyperbolic 4-manifold, then [VS] implies that M is an arithmetic hyperbolic manifold of simplest type. So there exist a totallyreal number field K and a nondegenerate quadratic form f : K → K defined over K , such that the negative inertial index of f is 1, and f σ is positive definite forany non-identity embedding σ : K → R . Moreover, π ( M ) is commensurable with SO ( f ; O K ). So to prove π ( M ) is not LERF, we need only to prove SO ( f ; O K )is not LERF.We diagonalize the quadratic form f such that the symmetric matrix defining f is A = diag( k , k , k , k , k ) with k , k , k , k > k < f has two quadratic subforms defined by diag( k , k , k , k )and diag( k , k , k , k ) respectively. These two subforms satisfy the conditions fordefining arithmetic groups in Isom + ( H ), and we denote them by f and f respec-tively. SO ( f ; O K ) and SO ( f ; O K ) are both subgroups of SO ( f ; O K ) < Isom + ( H ).Each of them fix a 3-dimensional totally geodesic hyperplane in H , and we denotethem by P and P respectively. Then P and P intersect with each other perpen-dicularly along a 2-dimensional totally geodesic plane P . So M i = P i /SO ( f i ; O K )is a hyperbolic 3-orbifold for each i = 1 ,
2. Moreover, it is easy to see that SO ( f ; O K ) ∩ SO ( f ; O K ) = SO ( f ; O K ), with f be defined by diag( k , k , k ).Then SO ( f ; O K ) is the subgroup of SO ( f ; O K ) that fixes P . It is easy to seethat P/SO ( f ; O K ) is a totally geodesic 2-orbifold Σ in M ∩ M .We first take a torsion-free finite index subgroup Λ < π (Σ), and consider it assubgroups of π ( M ) and π ( M ). By applying LERFness of hyperbolic 3-manifold(orbifold) groups, we get torsion free finite index subgroups Λ i < SO ( f i ; O K ) for i = 1 ,
2, with Λ ∩ SO ( f ; O K ) = Λ ∩ SO ( f ; O K ) = Λ, and S = P/ Λ has largeenough product neighborhood in N = P / Λ and N = P / Λ . Then there isan obvious map from N ∪ S N to H /SO ( f ; O K ), and the induced map on thefundamental group Λ ∗ Λ Λ → SO ( f ; O K ) is injectiveSo SO ( f ; O K ) contains a subgroup isomorphic to Λ ∗ Λ Λ . Λ ∗ Λ Λ is thefundamental group of N ∪ S N , which is the quotient space of two closed hyperbolic3-manifolds N , N by pasting along a closed totally geodesic subsurface S . Anytotally geodesic subsurface in a hyperbolic 3-manifold gives a geometrically finitesubgroup in the 3-manifold group. By Theorem 1.1, Λ ∗ Λ Λ is not LERF, soLemma 2.3 implies SO ( f ; O K ) and π ( M ) are not LERF. (cid:3) As in [Sun], we have the following corollary of Corollary 1.3, which is aboutnonLERFness of non-arithmetic hyperbolic manifold groups. This corollary coversall known examples of non-arithmetic hyperbolic manifolds with dimension ≥ Corollary 5.1.
For the following non-arithmetic hyperbolic manifolds: • M m is a nonarithmetic hyperbolic m -manifold given by constructions in [GPS] or [BT] , with m ≥ , • M m is a nonarithmetic hyperbolic m -manifold given by the reflection groupof some finite volume polyhedron in H m , with m ≥ ,the fundamental group π ( M ) is not LERF. Further questions
In the proof of Theorem 1.1, the nonseparable subgroups we constructed areusually infinitely presented. Moreover, the nonseparable subgroups we constructedfor closed arithmetic hyperbolic 4-manifold groups are always infinitely presented.So we ask the following two questions on the existence of finitely presented nonsep-arable subgroups.
Question 6.1.
For a general group A , do geometrically finite amalgamations π ( M ) ∗ A π ( M ) of finite volume hyperbolic 3-manifold groups contain finitelypresented nonseparable subgroups? Question 6.2.
Do closed arithmetic hyperbolic 4-manifold groups contain finitelypresented nonseparable subgroups?Theorem 1.1 implies that nontrivial geometrically finite amalgamations of finitevolume hyperbolic 3-manifold groups are not LERF. Then it is natural to ask aboutamalgamations of finite volume hyperbolic 3-manifold groups along a group that isgeometrically infinite in at least one of the vertex groups.
Question 6.3.
Let M , M be two finite volume hyperbolic 3-manifolds, A be anontrivial group, i : A → π ( M ) and i : A → π ( M ) be two injective grouphomomorphisms. If i ( A ) < π ( M ) is a geometrically infinite subgroup, then inwhat circumstance, is π ( M ) ∗ A π ( M ) LERF?Since all geometrically infinite subgroups of a finite volume hyperbolic 3-manifoldgroup are virtual fibered subgroups, A is always a surface group or a free group. Soin this case, the abstract group structure of the edge group is not very complicated.Actually, all results in Section 4 still work if i ( A ) < π ( M ) is geometricallyfinite and i ( A ) < π ( M ) is geometrically infinite (we need to modify the proof ofProposition 4.1 a little bit). So there are examples of nonLERF amalgamations oftwo finite volume hyperbolic 3-manifold groups along a subgroup that is geometri-cally finite in one group and geometrically infinite in the other group. However, inthis case, it seems not easy to get the algebraically fibered structure as in Section3, so the general case is difficult to deal with. For a geometrically infinite amal-gamation, condition (5) in Proposition 4.1 never holds, so our current technique isnot applicable in this case.In [Sun], for 3-manifolds with empty or tori boundary, a topological criterion onLERFness of groups of such 3-manifold is proved (in terms of geometric structureson 3-manifolds). Results of [Sun] and this paper together imply that, for almost MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 27 all arithmetic hyperbolic manifolds with dimension ≥ Question 6.4.
For compact 3-manifolds with higher genus boundary, is there atopological criterion on LERFness of their fundamental groups?
Question 6.5.
Whether all finite volume hyperbolic manifolds with dimension atleast 4 have nonLERF fundamental groups?For Question 6.4, the author propose the following possible criterion. For a com-pact 3-manifold with higher genus boundary, we first do sphere and disc decompo-sitions, then do the torus decomposition without doing the annulus decomposition.Under the torus decomposition, if there are two adjacent pieces that are Seifertfibered spaces or finite volume hyperbolic 3-manifolds (geometric 3-manifold withtori boundary), then the result in [Sun] implies this manifold has nonLERF funda-mental group. If the above picture does not show up, then the author expects thefundamental group to be LERF.Daniel Groves informed the author that, under the torus decomposition of M ,if all pieces are hyperbolic 3-manifolds with higher genus boundary, then π ( M )is LERF. This follows from [BW] and [PW]. In [BW], it is proved that, for a 3-manifold M as above, all finitely generated subgroups of π ( M ) are quasiconvex,and [PW] shows that such quasiconvex subgroups are separable.For Question 6.5, the main difficulty is that we do not know whether the fun-damental group of a general finite volume hyperbolic manifold with dimension ≥ ≥ T of a cuspedhyperbolic 3-manifold, all but finitely many boundary slopes on T are virtuallyfibered boundary slopes. So it is natural to ask the following question. Question 6.6.
For a cusped hyperbolic 3-manifold M and a boundary component T ⊂ ∂M , are all slopes on T virtually fibered boundary slopes?Proposition 3.7 implies the answer for cusped arithmetic hyperbolic 3-manifoldsis positive. However, it seems not easy to prove it for a general cusped hyperbolic3-manifold.If we want to use the proof of Proposition 3.6, then we need to know more aboutthe shape of the Thurston norm unit ball when taking a finite cover. If in somefinite cover M ′ of M , all codimension-1 hyperplanes in H ( M ′ ; R ) intersects withsome open face of the Thurston norm unit ball, then all slopes on T are virtuallyfibered boundary slopes. However, it seems we have very few knowledge on theshape of Thurston norm unit ball when taking a finite cover such that the firstbetti number increases. If one want to develop a version of Agol’s construction of virtual fibered structurein [Ag2] relative to a fixed slope on T , then there is some difficulty on the behaviorof norm-minimizing surfaces when doing Dehn filling along the fixed slope. References [Ag1] I. Agol,
Tameness of hyperbolic -manifolds , preprint 2004, https://arxiv.org/abs/math/0405568 [Ag2] I. Agol, Criteria for virtual fibering , J. Topol. 1 (2008), no. 2, 269 - 284.[Ag3] I. Agol,
The virtual Haken conjecture , with an appendix by I. Agol, D. Groves, J. Manning,Documenta Math. 18 (2013), 1045 - 1087.[BT] M. Belolipetsky, S. Thomson,
Systoles of hyperbolic manifolds , Algebr. Geom. Topol. 11(2011), no. 3, 1455 - 1469.[BHW] N. Bergeron, F. Haglund, D. Wise,
Hyperplane sections in arithmetic hyperbolic mani-folds , J. Lond. Math. Soc. (2) 83 (2011), no. 2, 431 - 448.[BW] H. Bigdely, D. Wise.
Quasiconvexity and relatively hyperbolic groups that split , MichiganMath. J. 62 (2013), no. 2, 387 - 406.[CG] D. Calegari, D. Gabai,
Shrinkwrapping and the taming of hyperbolic -manifolds , J. Amer.Math. Soc. 19 (2006), no. 2, 385 - 446.[Ca] R. Canary, A covering theorem for hyperbolic -manifolds and its applications , Topology35 (1996), no. 3, 751 - 778.[CDW] E. Chesebro, J. DeBlois, H. Wilton, Some virtually special hyperbolic -manifold groups ,Comment. Math. Helv. 87 (2012), no. 3, 727 - 787.[FK] S. Friedl, T. Kitayama, The virtual fibering theorem for -manifolds , Enseign. Math. 60(2014), no. 1 - 2, 79 - 107.[FV] S. Friedl, S. Vidussi, The Thurston norm and twisted Alexander polynomials , J. ReineAngew. Math. 707 (2015), 87 - 102.[GPS] M. Gromov, I. Piatetski-Shapiro,
Nonarithmetic groups in Lobachevsky spaces ,Inst. Hautes ´Etudes Sci. Publ. Math. No. 66 (1988), 93 - 103.[Ha] M. Hall,
Coset representations in free groups , Trans. Amer. Math. Soc. 67 (1949), 421 -432.[Liu] Y. Liu,
A characterization of virtually embedded subsurfaces in -manifolds , Trans. Amer.Math. Soc. 369 (2017), no. 2, 1237 - 1264.[MR] C. Maclachlan, A. Reid, The arithmetic of hyperbolic -manifolds , Graduate Texts inMathematics, 219. Springer-Verlag, New York, 2003.[Ma] A. Mal’cev, On homomorphisms onto finite groups , Ivanov. Gos. Ped. Inst. Ucen. Zap. 18(1958), 49 - 60.[NW] G. Niblo, D. Wise,
Subgroup separability, knot groups and graph manifolds , Proc. Amer.Math. Soc. 129 (2001), no. 3, 685 - 693.[PW] P. Przytycki, D. Wise,
Mixed -manifolds are virtually special , preprint 2012, http://arxiv.org/abs/1205.6742 .[RW] H. Rubinstein, S. Wang, π -injective surfaces in graph manifolds , Comment. Math. Helv.73 (1998), no. 4, 499 - 515.[Sc] P. Scott, Subgroups of surface groups are almost geometric , J. London Math. Soc. (2) 17(1978), no. 3, 555 - 565.[Sun] H. Sun,
NonLERFness of arithmetic hyperbolic manifold groups , preprint 2016, https://arxiv.org/abs/1608.04816 .[Th1] W. Thurston,
The geometry and topology of three-manifolds , Princeton lecture notes,1979, available at .[Th2] W. Thurston,
A norm for the homology of -manifolds , Mem. Amer. Math. Soc. 59 (1986),no. 339, i - vi and 99 - 130.[VS] E. Vinberg, O. Shvartsman, Discrete groups of motions of spaces of constant curvature ,Geometry, II, 139 - 248, Encyclopaedia Math. Sci., 29, Springer, Berlin, 1993.[Wi] D. Wise,
The structure of groups with a quasiconvex hierarchy , preprint, . MALGAMATIONS OF HYPERBOLIC 3-MANIFOLD GROUPS ARE NOT LERF 29
Department of Mathematics, University of California at Berkeley, Berkeley, CA94720, USA
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