aa r X i v : . [ m a t h . G T ] J a n A FIBERING THEOREM FOR 3-MANIFOLDS
JORDAN A. SAHATTCHIEVE
Abstract.
We generalize a result of Moon on the fibering of certain 3-manifoldsover the circle. Our main theorems are the following:
Theorem 0.1.
Let M be a compact geometric manifold and let G = π ( M ) .Suppose that U is a finitely generated subgroup of G with | G : U | = ∞ , andsuppose that U contains a non-trivial subnormal subgroup N ⊳ s G . If N isnot infinite cyclic, then M is finitely covered by a bundle over S with fiber acompact surface F such that π ( F ) is commensurable with U . Theorem 0.2.
Let M be a closed 3-manifold. Suppose that G = π ( M ) contains a finitely generated group U of infinite index in G which containsa non-trivial subnormal subgroup N = Z of G , and suppose that N has acomposition series of length n in which at least n − terms are finitely gen-erated. Suppose that N intersects non-trivially the fundamental groups of thesplitting tori given by the Geometrization Theorem and that the intersectionsof N with the fundamental groups of the geometric pieces are non-trivial andnot isomorphic to Z . Then, M has a finite cover which is a bundle over S with fiber a compact surface F such that π ( F ) and U are commensurable. We also obtain the following generalization of a theorem of Griffiths’:
Theorem 0.3.
Let S be a closed surface whose fundamental group containsa finitely generated subgroup U of infinite index, which contains a non-trivialsubnormal subgroup N of π ( S ) . Then, S is the torus or the Klein bottle. Introduction and Statements of Prerequisite Results
The goal of this paper is to communicate a proof of a generalization of the mainresult in [6] in the context of the Geometrization Theorem. The result generalizedherein is itself a generalization of Stallings’ Fibration Theorem.Recall that a fiber bundle over a topological space B is the data ( E, B, η, F ),where E and F are also topological spaces, and η : E → B is a continuous sur-jection, sometimes called the bundle projection , which is possessed of the followingproperties:(1) For every x ∈ E , there exists an open neighborhood U of η ( x ), suchthat η − ( U ) is homeomorphic to U × F under the homeomorphism φ U : η − ( U ) → U × F .(2) The bundle surjection, η , commutes with the projection pr U onto the firstfactor of the trivialization U × F , formulaically η = φ ◦ pr U .(3) Whenever the projection of the point x ∈ E belongs to two different openneighborhoods U i and U j of the type described above, the composition φ U i ◦ φ − U j : U i ∩ U j × F → U i ∩ U j × F is a continuous map, which is theidentity on the first coordinate.The spaces E , B , and F are called the total space, base, and fiber, respectively. In the category of 3-manifolds, all of the maps introduced above are assumed tobe smooth; for more on fiber bundles, see [11]. For the remainder of this paper,we shall be interested exclusively in compact 3-manifolds which are fiber bundleswhose base B is the circle, and we shall refer to them as fibering over S .Consider a 3-manifold M which fibers over S with fiber a compact surface F . Itis obvious that the bundle projection η induces a homomorphism of π ( M ) onto Z whose kernel is precisely π ( F ). One can alternatively think of M as the map-ping torus of F under the automorphism of F given by the holonomy of thebundle. Whatever the case may be, we clearly have the short exact sequence1 −→ π ( F ) −→ π ( M ) −→ Z −→
1. Stallings proved in [10] the celebratedconverse to this:
Theorem 1.1. (Stallings, 1961)
Let M be a compact, irreducible 3-manifold.Suppose that there is a homomorphism π ( M ) → Z whose kernel G is finitelygenerated and not of order 2. Then, M fibers over S with fiber a compact surface F whose fundamental group is isomorphic to G .Proof. See [10]. (cid:3)
The interested reader should also be aware of the following generalizations ofTheorem 1.1:
Theorem 1.2. (Elkalla, 1983)
Let M be a P -irreducible, compact and connected3-manifold. If G = π ( M ) contains a non-trivial subnormal subgroup N such that N is contained in an indecomposable and finitely generated subgroup U of infiniteindex in G , and if G is U -residually finite, then either (i) the Poincare associateof M is finitely covered by a manifold, which is a fiber bundle over S with fiber acompact surface F , such that there is a subgroup V of finite index in both π ( F ) and U , or (ii) N is isomorphic to Z .Proof. See [2]. (cid:3)
Recall that a 3-manifold is called P -irreducible if it is irreducible , i.e. everyembedded sphere in M bound a ball in M , and if additionally there are no embedded2-sided projective planes in M . A subgroup N of a group G is called subnormal, ifthere is a composition series N = N n ⊳ N n − ⊳ N n − ⊳ ... ⊳ N = G . Throughout thispaper we shall use the notation N ⊳ s G to stand for the relationship of subnormalityof N in G .The term indecomposable here refers to indecomposability with respect to the(non-abelian) free product of groups, and U -residually finite means that for every g ∈ G − U one can find a quotient of G in which g is mapped to non-identity elementunder the natural epimorphism. Theorem 1.3. (Moon, 2005)
Let M be a closed 3-manifold, which is either atorus sum X S T X , or X S T , where each X i is either a Seifert fibered space ora hyperbolic manifold. If G = π ( M ) contains a finitely generated subgroup U ofinfinite index which contains a non-trivial normal subgroup N of G , which intersectsnon-trivially the fundamental group of the splitting torus, and such that N ∩ π ( X i ) is not isomorphic to Z , then M has a finite cover which is a bundle over S withfiber a compact surface F , and π ( F ) is commensurable with U .Proof. See [6]. (cid:3)
FIBERING THEOREM FOR 3-MANIFOLDS 3
It is this latter result that this paper proposes to generalize. The reader mayfind it helpful to become acquainted with Moon’s paper [6] - many of the proofs ofthis paper are borrowed from [6] with minor changes to obtain the necessary scopeof generality. 2.
Compact Seifert Fibered Spaces
Our work depends on the following results of Griffiths:
Theorem 2.1. (Griffiths, 1962)
Let S be a surface whose fundamental group G contains a finitely generated subgroup U of infinite index, which contains a non-trivial normal subgroup N of G . Then, S is homeomorphic to either the torus orthe Klein bottle.Proof. See Theorem 6.1 in [4]. (cid:3)
Theorem 2.2. (Griffiths, 1967)
Let G be the fundamental group of an ordinaryFuchsian space (such as a compact orientable surface with or without boundary).Suppose that G is not abelian and not isomorphic to Z ∗ Z = < a, b : a = 1 , b =1 > or < a , b , a , b : a = 1 , b = 1 , a = 1 , b = 1 , a b a b = 1 > , and supposethat U is a finitely generated subgroup of G . If U contains a non-trivial subnormalsubgroup of G , then U is of finite index in G .Proof. See Theorem 14.7 in [3]. (cid:3)
Theorem 2.3. (Elkalla, 1983)
Let G be a group and let U be a finitely generatedsubgroup of infinite index. If U contains a non-trivial subnormal subgroup of G ,then G is indecomposable with respect to the free product of groups.Proof. See Theorem 1.5 in [2]. (cid:3)
Lemma 2.4.
Let X be a 2-dimensional, non-spherical, good orbifold whose funda-mental group is infinite. Then, π orb ( X ) has no non-trivial finite normal subgroups.Proof. Straight from definitions, X has a cover which is a closed surface S . Now, S is covered by either R . (or S .) In the former case, S will carry a constant metricof either flat or negative curvature and thus will be a quotient of either E or H by a discrete subgroup of their respective isometry group, whereas in the latter S will be a quotient of S , with its natural metric of constant positive curvature, by adiscrete subgroup of its isometry group. Thus, we have the tower of orbifold covers Y −→ S −→ X , where Y = E or H . Now, suppose that N ⊳ π orb ( X ) is a finitenormal subgroup. Let us take the cover of X corresponding to N , say X . Then, X would be isometric to a quotient of Y by a subgroup of Isom ( Y ) isomorphic to N . If Y is H or E , by Corollary 2.8 of [1], such a finite subgroup would have afixed point. We know that in the isometry groups of both the Euclidean and theHyperbolic plane, such subgroups are generated by a single rotation. Since π orb ( X )acts on Y as a subgroup of Isom ( Y ), normality of N inside π orb ( X ) would implythat there is a fixed point of the action of π orb ( X ) on Y . This implies that π orb ( X )is itself finite, since otherwise X would have a cone point of infinite order, which isa clearly impossibility. (cid:3) Corollary 2.5.
Let X be a 2-dimensional good, non-spherical orbifold. Then, π orb ( X ) has no finite subnormal subgroups. JORDAN A. SAHATTCHIEVE
Proof.
Suppose we have
N ⊳ N ⊳ N ⊳ ... ⊳ π orb ( X ), with N finite. By the coveringspace theory of orbifolds, see [7], we construct a tower of orbifold covers X → X → X → ... → X k → X . Next, because X is a good 2-dimensional manifold, wesee that X k must also be a good 2-dimensional orbifold: if X k were a bad orbifold,it must be of types enumerated as (i) - (iv) in Theorem 2.3 in [7]; this would implythat X is itself bad - a contradiction. Inductively, we see that each X i is a goodorbifold. Finally, we have a finite normal subgroup N ⊳ π orb ( X ). From Lemma 2.4we conclude that N is trivial. (cid:3) Lemma 2.6.
Let X be a good closed, non-spherical, 2-dimensional orbifold. If U isa finitely generated, infinite index subgroup of π orb ( X ) and U contains a non-trivial N ⊳ s π orb ( X ) , then X has a finite orbifold cover X which is a S -bundle over either S or the orbifold S / Z . Also, the fiber group is commensurable with U .Proof. We proceed in a manner identical to Moon’s arguments in [6] mutatis mu-tandis to obtain the desired conclusion, see Lemma 1.2: Since X is a good closed2-dimensional orbifold, it has a finite cover which is a closed surface. Using thecovering space theory of orbifolds, we conclude that π orb ( X ) contains the funda-mental group of a closed surface Γ as a finite index subgroup and consequently | Γ : Γ ∩ U | = ∞ . Now, in view of Corollary 2.5, we must have Γ ∩ N = 1: Toprove this, let Core (Γ) = ∩ g Γ g − denote the normal core of Γ in π orb ( X ). We notethat | π orb ( X ) : Core (Γ) | < ∞ and Core (Γ) ⊳ π orb ( X ), and further that Γ ∩ N = 1would imply that N would embed into the finite quotient π orb ( X ) /Core (Γ), thuscontradicting Corollary 2.5. Now, Theorem 2.2 implies that Γ is the fundamentalgroup of the torus, and Γ ∩ N is now normal in Γ and the proof of Theorem 2.1 in[4] shows that Γ ∩ N is of finite index in the fiber group which is equal to Z , andalso that Γ ∩ U = Z . Thus the fiber group is commensurable with Γ ∩ U . (cid:3) Lemma 2.7.
Let X be a 2-dimensional compact orbifold with nonempty boundarywhose singular points are cone points in Int ( X ) . If a finitely generated subgroup U of π orb ( X ) contains a nontrivial subnormal subgroup N of π orb ( X ) , then U is offinite index in π orb ( X ) .Proof. As X has nonempty boundary, π orb ( X ) is a free product of cyclic groups.Suppose by way of obtaining contradiction that the index of U in π orb ( X ) is infi-nite. The hypotheses of Theorem 2.3 are satisfied and we conclude that π orb ( X )is indecomposable, hence cyclic. Since U is non-trivial, U must be of finite index:contradiction. (cid:3) Lemma 2.8.
Suppose that G is a cyclic extension of H : −→ < t > −→ G −→ H −→ . Suppose that U is subgroup of G , then the centralizer of < t > in U , C U ( t ) = u ∈ U : [ u, t ] = 1 is normal in U and is of index at most 2.Proof. The conjugation of < t > by elements of G , g → ψ g , where ψ g ( t ) = gtg − ,defines a homomorphism of G to Aut ( Z ) ∼ = Z / (2). Therefore, the kernel of thishomomorphism, which is precisely K = { g ∈ G : [ g, t ] = 1 } is of index at most 2 in G . The observation C U ( t ) = G ∩ K concludes the proof. (cid:3) Proposition 2.9.
Let M be a compact 3-manifold whose fundamental group π ( M ) has the property that all of its subgroups are finitely generated. If π ( M ) contains a FIBERING THEOREM FOR 3-MANIFOLDS 5 finitely generated, infinite index subgroup U which contains a non-trivial subgroup N = Z subnormal in π ( M ) , then a finite cover of M fibers over S with fiber acompact surface F , and π ( F ) is commensurable with U .Proof. Let N = N ⊳ N ⊳ ... ⊳ N k − ⊳ N k = π ( M ). By assumption, each N i isfinitely generated. Let i be the largest index such that | π ( M ) : N i | = ∞ and | π ( M ) : N j | < ∞ , for all j > i . Then, N i is a normal infinite index subgroupof N i +1 . Note that since M is compact, Theorem 2.1 in [8] shows that N i isfinitely presented. Let M N i be the finite cover of M whose fundamental groupis N i +1 . We now have 1 −→ N i −→ π ( M N i ) −→ Q −→
1, where N i isfinitely presented and Q infinite. Applying Theorem 3 of Hempel and Jaco in [5],we conclude that M N i has a finite cover which is a bundle over S with fiber acompact surface F and that N i is subgroup of finite index in π ( F ). If i = 0,we are done since in that case π ( M ) is virtually an extension of π ( F ) by Z ,hence | π ( M ) : U | = ∞ , | π ( F ) : N | < ∞ , and N < U together imply that | U : N | < ∞ showing that U is commensurable with π ( F ) as desired. If i = 0,we have π ( F ) > N i ⊲ ... ⊲ N , where F is a compact surface; in this case we shallshow that N is of finite index in π ( F ), which will reduce the argument to the case i = 0.If F is an orientable surface other than the 2-torus, since π ( F ) is infinite, notabelian, and is not isomorphic to Z ∗ Z or < a , b , a , b : a = 1 , b = 1 , a =1 , b = 1 , a b a b = 1 > , and since N is finitely generated, we can apply Griffiths’Theorem 2.2 to conclude that N is of finite index in π ( F ).If F is the 2-torus, then π ( F ) = Z × Z and N is either of finite index in π ( F ),or N is trivial or infinite cyclic. The hypotheses of the theorem rule out the lattercases, hence we conclude that N is again of finite index in π ( F ).If F is a non-orientable surface, consider its orientable double cover F ′ . We have π ( F ) > π ( F ′ ) > π ( F ′ ) ∩ N i ⊲ ... ⊲ π ( F ′ ) ∩ N . First, suppose that F ′ is not the2-torus. If π ( F ′ ) ∩ N j = { } for all 0 ≤ j ≤ i , then N j ֒ → π ( F ) /π ( F ) = Z ,and we must have N i = N = Z thus proving that N is of finite index in π ( F ). If π ( F ′ ) ∩ N j = { } , for some 0 ≤ j ≤ i , let j be the least index with this property.In other words, π ( F ′ ) ∩ N j = { } while π ( F ′ ) ∩ N j − = { } . If j = 0, that is tosay π ( F ′ ) ∩ N = { } , because π ( F ′ ) ∩ N is finitely generated as a subgroup of π ( M ), we can apply Griffiths’ Theorem 2.2 to deduce that | π ( F ′ ) : π ( F ′ ) ∩ N | < ∞ .Since, | π ( F ) : π ( F ′ ) || π ( F ′ ) : π ( F ′ ) ∩ N | = | π ( F ) : N || N : π ( F ′ ) ∩ N | , wesee that | π ( F ) : N | < ∞ and we deduce that U is commensurable with π ( F ).If 0 < j ≤ i , we have N j = Z for j < j , while | π ( F ) : N j | < ∞ as above.If N j − = N = Z is of finite index in N j , we conclude that N is of finite indexin π ( F ) as before. If | N j : N j − | = ∞ , since N j − = Z is an infinite index,finitely generated normal subgroup of N j , the cover F N j of F corresponding to N j is either the 2-torus or the Klein bottle by Griffiths’ Theorem 2.1. Since thefundamental groups of the 2-torus and the Klein bottle are both torsion free, wehave a contradiction. This concludes the argument in all cases where F ′ is not the2-torus.If F ′ is the 2-torus, computing the Euler characteristics of F and F ′ and applyingthe classification theorem of closed non-orientable surfaces, we see that F is theKlein bottle, π ( F ) = < a, b : aba − b = 1 > ∼ = Z ⋊ Z . Now, since N is a subgroup ofthe Klein bottle group π ( F ), N is either trivial, infinite cyclic, or | π ( F ) : N | < JORDAN A. SAHATTCHIEVE ∞ . The first two cases are contrary to the hypotheses of the theorem, while thelatter case finishes the proof. (cid:3) The following theorem was first proved in [6] in the context of N being a normalsubgroup of π ( M ), the proof given below is a modification of the proof therein. Theorem 2.10.
Let Y be a compact Seifert fibered space, whose base orbifold isnon-spherical, and let U be a finitely generated, infinite index subgroup of π ( Y ) .Suppose, further, that U contains a non-trivial subnormal subgroup N = Z of π ( Y ) .Then, Y is finitely covered by a compact 3-manifold Y , which is a bundle over S with fiber a compact surface F , and π ( F ) is commensurable with U .Proof. It is a well-known fact that Seifert fibered spaces admit an action of the fun-damental group of a regular fiber which is equal to Z = < t > , such that the quotientof this action is an orbifold. Now, consider the short exact sequence of this action:1 −→ < t > −→ π ( Y ) −→ π orb ( X ) −→
1. The proof, as in [6] proceeds in two cases:
Case 1:
The orbifold X is non-spherical orbifold covered by an orientable sur-face other than the torus.First, we show that | π ( Y ) : φ − ( φ ( U )) | < ∞ . Since N = Z , π orb ( X ) ⊲ s φ ( N ) = 1.Since we also have φ ( N ) < φ ( U ) < π orb ( X ), and φ ( U ) is finitely generated, wecan apply Lemma 2.6 to first infer that X must have non-empty boundary, thenapply Lemma 2.7 to conclude that | π orb ( X ) : φ ( U ) | < ∞ . Therefore, | π ( Y ) : φ − ( φ ( U )) | < ∞ as desired. Next, we show that φ | U is a monomorphism:To this end, we prove that U ∩ < t > = 1. If some power of t is in U , then U willbe of finite index in the ”closure” φ − ( φ ( U )). Since φ − ( φ ( U )) was shown to be offinite index in π ( Y ), we conclude that U must be of finite index in π ( Y ) contraryto the assumptions in the statement of the theorem. Since φ | U is a monomorphism, φ ( U ) is torsion free and therefore the fundamental group of a compact surface(take the orbifold cover corresponding to φ ( U ), it has no cone points or cornerreflectors, therefore it is a surface). Next, consider the centralizer of < t > in U , C U ( t ) = { u ∈ U : [ u, t ] = 1 } , and the subgroup G of π ( Y ) generated by C U ( t ) and t . Note that C U ( t ) ⊳ U , and the index of C U ( t ) in U is at most 2 by Lemma 2.8,hence | φ − ( φ ( U )) : G | ≤
2. Thus G is a finite index subgroup of π ( Y ). Now, takethe cover of M of M corresponding to G ≤ π ( Y ). Since G = < C U ( t ) , t > with C U ( t ) ∩ < t > = 1, we conclude that G ∼ = C U ( t ) × Z and also that M ∼ = F × S ,were F is a compact surface whose fundamental group is isomorhic to C U ( t ). Thisconcludes Case 1. Case 2 : The orbifold X is orbifold covered by the torus.Let Γ ∼ = Z be the fundamental group of the torus which orbifold covers X . Weproceeding as in Moon [6]. In view of Proposition 2.9, we only need to show thatevery subgroup H of π ( Y ) is finitely generated.We have the short exact sequence 1 −→ < t > ∩ H −→ H −→ φ ( H ) −→
1. SinceΓ is of finite index in π orb ( X ), φ ( H ) ∩ Γ is of finite index in φ ( H ). The intersec-tion φ ( H ) ∩ Γ is finitely generated as it is a subgroup of Γ, hence φ ( H ) is finitelygenerated. Because the kernel of φ | H is a subgroup of Z and is therefore trivially FIBERING THEOREM FOR 3-MANIFOLDS 7 finitely generated, we conclude that H is itself finitely generated as needed. Case 3 : The orbifold X is a spherical orbifold. This case is ruled out by thehypothesis of the theorem. (cid:3) Non-SFS Geometric Manifolds
The proof of Theorem 3.2 below is implicit in Moon’s work in [6] as his argumentsextend verbatim. We record the result here for completeness.
Theorem 3.1.
Let M be a closed Sol manifold, such that π ( M ) contains a finitelygenerated, infinite index subgroup U which contains a non-trivial subgroup N = Z subnormal in π ( M ) . Then, a finite cover of M is a fiber bundle over S whose fiberis a compact surface F and π ( F ) is commensurable with U .Proof. The fundamental group π ( M ) of a closed Sol manifold satisfies1 −→ Z −→ π ( M ) φ −→ Z −→
1. Suppose H is a subgroup of π ( M ); then, wehave 1 −→ Z ∩ H −→ H −→ φ ( H ) −→
1. Since Z ∩ H and φ ( H ) are finitelygenerated, H is also finitely generated. Now, Proposition 2.9 yields the desiredconclusion. (cid:3) Theorem 3.2.
Let M be a complete hyperbolic manifold of finite volume whosefundamental group G contains a finitely generated subgroup of infinite index U which contains a non-trivial subgroup N subnormal in G . Then, M has a finitecovering space M which is a bundle over S with fiber a compact surface F , and π ( F ) is a subgroup of finite index in U .Proof. See Theorem 1.10 in [6]. (cid:3)
The above results are summarized in:
Theorem 3.3.
Let M be a compact geometric manifold and let G = π ( M ) . Sup-pose that U is a finitely generated subgroup of G with | G : U | = ∞ , and supposethat U contains a subnormal subgroup N ⊳ s G . If N is not infinite cyclic, then M isfinitely covered by a bundle over S with fiber a compact surface F such that π ( F ) is commensurable with U . Torus sums
Lemma 4.1.
Suppose G is a finitely generated 3-manifold group. Suppose N = N n ⊳ N n − ⊳ ... ⊳ N ⊳ N ⊳ N = G , is a subnormal subgroup such that N = 1 and N = Z . Then, there is an index ≤ i ≤ n , such that N i is finitely generated forall i ≤ i , and no N i is finitely generated for any i > i .Proof. Suppose, for the purpose of obtaining a contradiction that there exists anoccurrence of an ”inversion”: N i ⊳ N i − with N i is finitely generated while N i − not finitely generated. Since every subgroup of a 3-manifold group is obviouslyitself a 3-manifold group, we can apply Proposition 2.2 in [2] to conclude that N i must be isomorphic to Z , and thus N n = Z contrary to assumption. Therefore,such an inversion is not possible and the conclusion follows. (cid:3) JORDAN A. SAHATTCHIEVE
The following lemma is standard and its proof is often left as an exercise. Weinclude a proof of it here, which we borrowed from Henry Wilton’s unpublishednotes titled
Group actions on trees , for completeness:
Lemma 4.2.
Let χ be a tree and let α, β ∈ Aut ( χ ) be a pair of elliptic automor-phisms. If F ix ( α ) ∩ F ix ( β ) = ∅ , then αβ is a hyperbolic element of Aut ( χ ) .Proof. It suffices to construct an axis for αβ on which it acts by translation, whichwe now do.First, note that F ix ( α ) and F ix ( β ) are closed subtrees of χ . Let x ∈ F ix ( α ) bethe unique point in χ closest to F ix ( β ) and y ∈ F ix ( β ) be the unique point in χ closest to F ix ( α ). Then, the unique geodesic from y to αβ · y is [ x, y ] ∪ α · [ x, y ] sincethere are no point in the interior of [ x, y ] fixed by α so that the concatenation of thegeodesic segments above is a geodesic segment. Hence, d ( y, αβ · y ) = d ( y, α · y ) =2 d ( x, y ). Similarly, the geodesic from y to ( αβ ) · y is [ x, y ] ∪ α · [ x, y ] ∪ αβ · [ x, y ] ∪ αβα · [ x, y ], so that d ( y, ( αβ ) · y ) = 4 d ( x, y ) = 2 d ( y, αβ · y ) thus establishing theexistence of an axis for αβ , which finishes the proof. (cid:3) Proposition 4.3.
Suppose G is direct product with amalgamation G = A ∗ C B , G = A , and G = B , and suppose χ is the Bass-Serre tree for G . Then, if H ≤ G is a subgroup of finite index in G , the action of H on χ is minimal and χ is theBass-Serre tree for a graph of groups which carries H .Proof. Since G acts simplicially on χ , G acts by isometries on the CAT(0) space χ . Therefore, every g ∈ G acts by an elliptic or hyperbolic isometry accordingto whether g fixes a vertex or realizes a non-zero minimal translation distance g min = inf { d ( x, g · x ) : x ∈ χ } . In the latter case, g leaves a subspace of χ isometricto R invariant, and acts on this subspace, called an axis for g , as translation bya g min . See [1] for an account of the classification of the isometries of a CAT(0)space.First, we show that there exists an element g ∈ G which acts on χ as a hyperbolicisometry - such an isometry is sometimes called loxodromic in the context of groupactions on trees. Since every automorphism of χ is either elliptical or hyperbolic,in view of Lemma 4.2 it suffices to show that there are two distinct elements of G which do not fix the same vertex. This, however, is obvious as the vertext stabilizersin a Bass-Serre tree are precisely conjugates of A and B . As G was assumed to bedifferent from A and from B , it follows that G cannot equal a conjugate of either A or B . Now, if g · v = v for some v ∈ χ and all g ∈ G , then G ⊆ Stab ( v ), while Stab ( v ) equals a conjugate of A or B - a contradiction.Next, we show that every point of χ lies on the axis for some hyperbolic isometry g ∈ G . Suppose that g ∈ G is an element which acts as a hyperbolic isometry on χ , whose axis is γ . Suppose e is any edge of γ . Since g · γ is an axis for gg g − ,it follows that g · e is contained in the axis for gg g − . But the action of G istransitive on the set of edges of χ , therefore χ = S g ∈ G g · γ . However, the union onthe right-hand side is exactly the union of the axes for the elements of G which areconjugates of g , which proves our first claim.Finally, we show that every axis γ for a hyperbolic element g ∈ G is an axis forsome h ∈ H as follows: Consider the cosets H, gH, g H, ..., g n H , where | G : H | = n .These cannot be all distinct, therefore, we conclude that g i ∈ H for some 1 ≤ i ≤ n .However, γ is an axis for g i as well, hence our second claim has been established. FIBERING THEOREM FOR 3-MANIFOLDS 9
Now, we see that χ is a union of the axes for the hyperbolic elements of H . Fromthis we deduce that every edge of χ lies on an axis of a hyperbolic element of H andtherefore H cannot leave invariant any subtree of χ . This shows that the action of H on χ is minimal and that χ is, therefore, the Bass-Serre tree for H . (cid:3) Given a graph of groups Γ, we can form a different graph of groups Γ ′ by col-lapsing an edge in Γ to a single vertex with vertex group G v ∗ G e G v , or G v ∗ G e ifthe edge collapsed joins a single vertex v . To obtain a graph of groups structure,we define the inclusion maps of all the edges meeting the new vertex in Γ ′ to be thecompositions of the injections indicated by Γ followed by the canonical inclusion of G i ֒ → G v ∗ G e G v for i = 1 ,
2, or G v ֒ → G v ∗ G e as appropriate. It is obvious that π (Γ) ∼ = π (Γ ′ ). If Γ is a finite graph of groups, given an edge e ∈ Edge (Γ), we canproceed to collapse all the edges of Γ in this way to obtain a new graph of groupsΓ e which has a single edge. Thus, Γ e gives the structure of a free product withamalgamation or an HNN extension to G . We shall call this process of obtainingΓ e from Γ collapsing around e . Proposition 4.4.
Let Γ be a finite graph of groups with fundamental froup π (Γ) = G . Let T be the Bass-Serre tree of Γ and let U be a finitely generated subgroup of G such that U \ T has infinite diameter. Then, for some edge e ∈ Edge (Γ) the quotientof the Bass-Serre tree of the graph Γ e by U has infinite diameter.Proof. For e ∈ Edge (Γ), let Q e be the graph obtained from T by collapsing everyedge which is not a preimage of e under the quotient map T → G \ T = Γ. One easilyverifies that Q e is a tree and that the action of G on T descends to an action of G on Q e . The quotient of Q e by G has a single edge and thus gives G the structureof an amalgamated free product or an HNN extension. Further, the stabilizersof the vertices of Q e are precisely conjugates of the vertex groups of Γ e , and thestabilizers of edges are conjugates of the edge group of Γ e : this can easily be verifiedby observing that for every v ′ ∈ T , the quotient map T → Q e collapses the tree T v ′ ,which consists of all edge paths starting at v ′ not containing preimages of e . Thus, Stab ( w ′ ) ⊆ G for w ′ ∈ Q e is precisely the subgroup Inv ( T v ′ ) ⊆ G which leaves T v ′ invariant. It is easy to see that Inv ( T v ′ ) contains every vertex stabilizer subgroupof G for vertices in T v ′ and that G splits as a direct product with amalgamation of Inv ( T v ′ ) ∗ Stab ( e ′ ) Inv ( T v ′ ) for two adjacent vertices v ′ , v ′ ∈ T joined by the edge e ′ (or an HNN extension). Thus, we conclude that Q e is the Bass-Serre tree for Γ e .Finally, to prove that U \ Q e has infinite diameter for some e , we argue by con-tradiction. Suppose U \ Q e has finite diameter for every e ∈ Edge (Γ). First, observethat there is a bijection between the edges of Q e and the edges of T which are notcollapsed. Because Q e is the Bass-Serre tree of an amalgamated free product or anHNN extension, Lemma 2.1 in [6] shows that since U \ Q e is assumed to have finitediameter, it must be a finite graph of groups. This, however, shows that there arefinitely many U -orbits of edges in Q e , hence there are finitely many U -orbits ofedges in T lying above e ∈ Edge (Γ). Because this is true by assumption for everyedge e ∈ Edge (Γ) and because
Edge (Γ) is finite, we conclude that there are finitelymany U -orbits of edges in T . This means that the quotient U \ T must be a finitegraph which contradicts the assumption that U \ T has infinite diameter. Therefore,there exists an edge e ∈ Edge (Γ) such that U \ Q e has infinite diameter. (cid:3) Propositions 4.3 and 4.4 allow us to prove the following generalization of Theorem2.4 in [6]:
Theorem 4.5.
Let M be a compact 3-manifold with π ( M ) = G , and suppose that M splits along an incompressible torus T , M = X ∪ T X , or M = X ∪ T . Supposethat G contains a non-trivial subnormal subgroup N = N n ⊳ N n − ⊳ ...⊳ N = G suchthat N = Z , and such that all but one of the N i are assumed to be finitely generated.Suppose further that G contains a finitely generated subgroup U of infinite indexin G such that N < U . If the graph of groups U corresponding to U has infinitediameter, then M is finitely covered by a torus bundle over S with fiber T , and U and π ( T ) are commensurable.Proof. From Lemma 4.1 we conclude that our assumption on all but one of the N i being finitely generated is actually equivalent with the assumption that every N i for i < n is finitely generated. We consider two cases: Either | N i − : N i | < ∞ forall i < n , or there exists at least one occurrence of | N i − : N i | = ∞ where i < n . Case 1 : | N i − : N i | < ∞ for all i < n In this case, all the N i have finite index in G , except for N , in particular | G : N n − | < ∞ , and N ⊳ N n − < G . Consider the finite cover M N n − of M whosefundamental group is N n − . This cover will be made up of covers for the pieces X and X glued along covers of the splitting torus T . Let χ be the Bass-Serretree of the graph of groups induced by the splitting of M along T which gives G the structure of an amalgamated free product or HNN extension over Z . Inview of Proposition 4.3, χ is the Bass-Serre tree of N n − . Then, N n − acts on χ with quotient the graph of groups N n − \ χ , which is also the graph of groupsdecomposition of N n − obtained by the splitting of fundamental group of the cover M N n − along lifts of the splitting torus T of M .Consider now the subgroup U ∩ N n − ; we will show that | U : U ∩ N n − | < ∞ ,hence that U ∩ N n − is a finitely generated, infinite index subgroup of N n − , whichcontains N : There is a well-defined map ψ from the coset space U/U ∩ N n − to thecoset space G/N n given by ψ ( u U ∩ N n − ) = uN n − . This map is easily seen tobe injective, hence | U : U ∩ N n − | < | G : N n − | < ∞ .Note that U ∩ N n − acts on χ with an infinite diameter quotient: The map[ x ] U ∩ N n − \ χ → [ x ] U \ χ from U ∩ N n − \ χ to U \ χ is onto. Hence, if U ∩ N n − \ χ hadfinite diameter L , we would conclude that any two vertices v , v ∈ U \ χ would beat a distance at most L as we can find an edge path in U ∩ N n − \ χ of legth at most L between any two preimages of v and v . This path projects to a path of lengthat most L between v and v .Now, since the finite index subgroup U ∩ N n − of U also acts on χ with aninfinite diameter quotient, Proposition 4.4 shows that there is an edge in N n − \ χ ,such that the splitting of N n − along the corresponding edge group has a Bass-Serre tree whose quotient by U ∩ N n − has infinite diameter. The edge groups of N n − \ χ are all isomorphic to Z since they are the fundamental groups of lifts ofthe splitting torus T of M . Therefore, the cover M N n − is a torus sum and we canapply Theorem 2.4 in [6] to conclude that M N n − fibers in the desired way, and that U ∩ N n − is commensurable with π ( e T ), and therefore that U is commensurablewith π ( T ) as desired. FIBERING THEOREM FOR 3-MANIFOLDS 11
Case 2 : | N i − : N i | = ∞ for some i < n .Let i be the least integer index for which | N i − : N i | = ∞ . In this case, weconsider the finite cover M N i − whose fundamental group is N i − . Since N i isassumed to be finitely generated, it is also finitely presented by Theorem 2.1 in[8]. Now, M N i − fibers in the desired way by Theorem 3 of [5], and, further, N i is a subgroup of finite index in π ( T ). Finally, we show that U is commensurablewith π ( T ). Consider U ∩ N i ; this group is a subgroup of Z , therefore it is eithertrivial, Z or Z . Since U ∩ N i contains the non-trivial N = Z , we must have U ∩ N i ∼ = Z . Because the finite cover M N i − of M fibers over the circle, wehave | G : π ( T ) ⋊ Z | < ∞ . If U ∩ N i were not of finite index in U , then U would obviously be of finite index in G , which contradicts the assumptions on U .Therefore, we conclude that U is commensurable with π ( T ), as desired. (cid:3) The last step towards proving our main theorem is a restatement of Theorem2.9 in [6]. While the proof in [6] only treats the case of N being a normal subgroupof π ( M ), it is obvious that it applies verbatim to the case of N ⊳ s π ( M ). Theorem 4.6. (Moon) Let M be a closed 3-manifold with M = X ∪ T X or M = X ∪ T . Suppose that X i satisfies the following condition for i = 1 , : If π ( X i ) contains a finitely generated subgroup U i with | π ( X i ) : U i | = ∞ such that U i contains a nontrivial subnormal subgroup Z = N i ⊳ s π ( X i ) , then a finite coverof X i fibers over S with fiber a compact surface F i and π ( F i ) is commensurablewith U i . Suppose, further, that G = π ( M ) contains a finitely generated subgroup U of infinite index in G which contains a nontrivial subnormal subgroup N of G ,and that N intersects non-trivially the fundamental group of the splitting torus and N ∩ π ( X i ) = Z . If the graph of groups U corresponding to U is of finite diameter,then M has a finite cover f M which is a bundle over S with fiber a compact surface F , and π ( F ) is commensurable with U .Proof. See the proof of Theorem 2.9 in [6]: it generalizes verbatim under the hy-pothesis that N is only subnormal rather than normal in π ( M ). (cid:3) For the sake of brevity, let us introduce the following terminology: We shall saythat a compact manifold M has property (S) if whenever π ( M ) contains a finitelygenerated subgroup U of infinite index, and a subnormal subgroup Z = N ⊳ s G which has a subnormal series in which all but one terms are assumed to be finitelygenerated, and such that N < U , then M fibers over S with fiber a compact surface F such that π ( F ) is commensurable with U .Combining Theorem 4.5 and Theorem 4.6, we obtain Theorem 4.7.
Let M be a closed 3-manifold with M = X ∪ T X or M = X ∪ T ,such that each X i , for i = 1 , , has property (S). Then, if G = π ( M ) contains afinitely generated subgroup U of infinite index, which contains a nontrivial subnor-mal subgroup N of G , which has a subnormal series in which all but one terms areassumed to be finitely generated, and which intersects non-trivially the fundamentalgroup of the splitting torus T , and whose intersection with π ( Xi ) is not infinitecyclic, for i = 1 , , then M has a finite cover which is a bundle over S with fiber acompact surface F , and π ( F ) is commensurable with U . The main theorem
Let us recall first the Geometrization Theorem proved by Perelman in 2003:
Theorem 5.1. (Geometrization Theorem) Let N be an irreducible compact 3-manifold with empty of toroidal boundary. Then there exists a collection of dis-jointly embedded incompressible tori T , ..., T k such that each component of N cutalong T ∪ ... ∪ T k is geometric. Furthermore, any such collection with a minimalnumber of components is unique up to isotopy. Using the Geometrization Theorem, we are able to extend the work of Moon. Toimprove the exposition of the proof of the main theorem below, we define property( S ′ ): We shall say that a closed manifold M has property ( S ′ ) if whenever π ( M )contains a finitely generated subgroup U with | π ( M ) : U | = ∞ and a subnormalsubgroup Z = N ⊳ s π ( M ) which has a subnormal series in which all but one termsare assumed to be finitely generated, such that N < U , and such that N intersectsnon-trivially the fundamental groups of the splitting tori in a minimal geometricdecomposition of M and N ∩ π ( X i ) = Z for every geometric piece X i , then M fibers over S with fiber a compact surface F such that π ( F ) is commensurablewith U .Using this terminology, Theorem 4.7 yields Corollary 5.2. If M is a closed manifold which splits along an incompressibletorus T as M = X ∪ T X or M = X ∪ T , where X and X satisfy property ( S ′ ) ,then M satisfies property ( S ′ ) .Proof. The proof is immediate from Theorem 4.7. (cid:3)
Theorem 5.3.
Let M be a closed 3-manifold. Suppose that G = π ( M ) containsa finitely generated group U of infinite index in G which contains a subnormalsubgroup N of G , and suppose that N has a composition series of length n in whichat least n − terms are finitely generated. Suppose that N intersects nontrivially thefundamental groups of the splitting tori given by the Geometrization Theorem andthat the intersections of N with the fundamental groups of the geometric pieces arenon-trivial and not isomorphic to Z . Then, M has a finite cover which is a bundleover S with fiber a compact surface F such that π ( F ) and U are commensurable.Proof. Let M = M ♯M ♯...♯M p be the decomposition of M into prime manifolds.Then, we have G = G ∗ G ∗ ... ∗ G p , where G i = π ( M i ). In view of Theorem 1.5 in[2], we must have G i = 1 for i ≥
2, after possibly reindexing the terms. Therefore, M i ∼ = S for all i ≥
2, and we conclude that M is a prime manifold. Now, weproceed by induction on the number of geometric pieces n in a minimal geometricdecomposition of M . The conclusion is clearly true for n = 1, this follows fromTheorem 3.3. Suppose the conclusion of the theorem holds for all prime manifoldswhose minimal decomposition into geometric pieces has at most n − S ′ ). Suppose M has n geometricpieces in its decomposition. Let X , ..., X n and T , ..., T p be the geometric piecesand incompressible tori given by Theorem 5.1. Consider the graph of groups G whose fundamental group is π ( M ), induced by the geometric decomposition. Let M ′ be the manifold obtained by gluing n − X i along the T j ina manner given by the graph of groups G ′ obtained from G by removing a singlevertex corresponding to one of the geometric pieces and all the edges adjacent to FIBERING THEOREM FOR 3-MANIFOLDS 13 it. Thus, M = ( ... (( M ′ ∪ T i X k ) ∪ T i ) ∪ T i ... ) ∪ T iq . Since M ′ and X k have property( S ′ ), so does M ′ ∪ T i X k , by Corollary 5.2. By induction on the number of edgesadjacent to π ( X k ) applying Corollary 5.2, we conclude that M also has property( S ′ ). Since M was assumed to satisfy the premises of property ( S ′ ), it follows that M fibers in the required way. (cid:3) A corollary to Griffiths’ theorem
In this section we prove a corollary to Griffiths’ Theorem 2.2, which in a sensegeneralizes the following restatement of Griffiths’ Theorem 2.1:
Theorem 6.1. (Griffiths, 1962) Let S be a surface whose fundamental group π ( S ) contains a finitely generated subgroup U of infinite index, which contains a non-trivial normal subgroup N of π ( S ) . Then, S is the torus or the Klein bottle. Namely, we prove:
Theorem 6.2.
Suppose S is a closed surface whose fundamental group contains afinitely generated subgroup U of infinite index, which contains a non-trivial subnor-mal subgroup N of π ( S ) . Then, S is the torus or the Klein bottle.Proof. Suppose that S is not the torus. If S is orientable, since π ( S ) is infinite,not abelian, and not isomorphic to Z ∗ Z or < a , b , a , b : a = 1 , b = 1 , a =1 , b = 1 , a b a b = 1 > , Griffiths’ Theorem 2.2 shows that U is of finite indexin π ( S ), which is a contradiction. Therefore, S is non-orientable. Let S ′ be itsorientable double cover. Suppose that S ′ is not the torus. We have π ( S ′ ) ⊲ π ( S ′ ) ∩ N k ⊲ ... ⊲ π ( S ′ ) ∩ N . Because the genus of S is at least 3, it follows that S ishyperbolic and therefore that π ( S ) has the finite generated intersection property,see Theorem 2 in [12]. Since π ( S ′ ) is a finitely generated subgroup of π ( S ), weconclude that π ( S ′ ) ∩ U is also finitely generated. Now, if π ( S ′ ) ∩ N = { } ,Griffiths’ Theorem 2.2 shows that π ( S ′ ) ∩ U must be of finite index in π ( S ′ ) andtherefore also in π ( S ). This, however, implies that U is itself of finite index in π ( S ), which yields a contradiction. On the other hand, if π ( S ′ ) ∩ N = { } , then N ֒ → π ( S ) /π ( S ′ ) ∼ = Z , hence N = Z . Note also that since N < U , N is ofinfinite index in π ( S ). Now, we can find and index j such that | N j : N j − | = ∞ and such that N j − is a finite subgroup of π ( S ). By Griffiths’ Theorem 2.1, thecover S N j of S corresponding to N j is the torus or the Klein bottle. As neitherthe fundamental group of the torus nor the fundamental group of the Klein bottlecontains torsion subgroups, we conclude that we have arrived to a contradiction.Therefore, S ′ must be the torus, but in this case computing the relevant Eulercharacteristics shows that S must be the Klein bottle. (cid:3) Acknowledgments
I wish to thank Prof. Peter Scott, who was my Ph.D. thesis adviser at TheUniversity of Michigan in Ann Arbor, for suggesting to me the problem of thefibering of compact 3-manifolds and for the many years of patient discussions andinvaluable advice. Notably, I am indebted to Prof. Scott for sketching an idea ofthe proof of Proposition 4.3 during one of our e-mail discussions. I, however, takefull responsibility for any errors or omissions in this pre-print.
References [1] M Bridson, A Haefliger,
Metric spaces of non-positive curvature , Springer Verlag (1999).[2] H Elkalla,
Subnormal subgroups in 3-manifolds groups , J. London Math. Soc. (2) 30 (1984),no. 2, 342–360.[3] H Griffiths,
A covering-space approach to theorems of Greenberg in Fuchsian, Kleinian andother groups , Communications on pure and applied mathematics, Vol. 20 (1967), 365 - 399.[4] H Griffiths,
The fundamental group of a surface, and a theorem of Schreier , Acta mathe-matica, vol. 110 (1963), 1 - 17.[5] J Hempel, W Jaco,
Fundamental Groups of 3-Manifolds which are Extensions , The Annalsof Mathematics, Second Series, Vol. 95, No. 1 (Jan., 1972), 86 - 98.[6] M Moon,
A generalization of a theorem of Griffiths to 3-manifolds , Topology and its appli-cations 149 (2005).[7] G P Scott,
The geometries of 3-manifolds , Bull. London Math. Soc. 15 (1983), 401 - 487.[8] G P Scott,
Finitely generated 3-manifold groups are finitely presented , J. London Math. Soc.(2), 6 (1973), 437 - 440[9] G P Scott, T C Wall,
Topological methods in group theory , Homological group theory (Proc.Sympos., Durham, 1977), 137 - 203.[10] J Stallings,
On fibering certain 3-manifolds , Topology of 3-manifolds, and related topics;proceedings of the University of Georgia Institute, GA (1961).[11] N Steenrod,
The topology of fibre bundles , Princeton University Press; Princeton, NJ (1957).[12] L Greenberg,
Discrete groups of motions , Canadian Journal of Mathematics, Vol. 12, pp. 415- 426 (1960).
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