RECOGNITION OF BEING FIBRED FOR COMPACT 3-MANIFOLDS

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1 RECOGNITION OF BEING FIBRED FOR COMPACT 3-MANIFOLDS ANDREI JAIKIN-ZAPIRAIN Abstract Let M be a compact orientable 3-manifold We show that if the profinite completion of π 1 (M) is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group, then M fibres over the circle with compact fibre 1 Introduction Throughout this paper, we allow 3-manifolds to have non-empty boundary, but we assume that any spherical boundary components have been removed by cappingoff with a 3-ball All manifolds are connected By a free-by-cyclic group we mean a group F r Z, where F r is a finitely generated free group and a surface-by-cyclic group is the group S g Z, where S g is the fundamental group of a compact closed surface Our result shows that the profinite completion of the fundamental group of a 3-manifold determines whether the manifold fibers over circle with compact fibre Theorem 11 Let M be a compact orientable 3-manifold Assume that the profinite completion of π 1 (M) is isomorphic to the profinite completion of a free-bycyclic group or to the profinite completion of a surface-by-cyclic group Then M fibres over the circle with compact fibre Under some additional conditions, this result was proved in [3, 4, 5] Our proof and all the previous proofs use in an essential way results of I Agol and D Wise on separability of 3-manifold groups ([1, 2, 9, 11]) Corollary 12 Let M and N be two compact orientable 3-manifolds such that π 1 (M) = π 1 (N) Then M fibers over the circle if and only if N does Acknowledgments This paper is partially supported by the grant MTM C2-01 of the Spanish MINECO, the grant PRX16/00179 of the mobility program Salvador de Madariaga of the Spanish MECD and by the ICMAT Severo Ochoa project SEV I would like to thank Pavel Zalesskii for bringing this problem to my attention and to Martin Bridson, Alan Reid and Henry Wilton for very useful discussions This work was done while the author was visiting the Mathematical Institute of the University Oxford I would like to thank everyone involved for their fine hospitality Date: June

2 2 ANDREI JAIKIN-ZAPIRAIN 2 Rank functions For a commutative ring R we denote by Q(R) its ring of fractions Let Γ be a finitely generated group and G = ˆΓ its profinite completion Let H be an open subgroup of G and N a closed normal subgroup of H such that H/N = Z k p is an abelian and torsion-free pro-p group We put D = Q(F p [[H/N]]) and define a rank function rk H,N on matices over Z[Γ] in the following way For A Mat n m (Z[Γ]), we put rk H,N (A) = rk D(φ A H,N ), G : H where φ A H,N is a D-morphism D Z[Γ H] (Z[Γ]) n D Z[Γ H] (Z[Γ]) m of D-vector spaces defined by means of (x 1,, x n ) (x 1,, x n )A Proposition 21 Let H be an open subgroup of G and let N 1 and N 2 be two normal closed subgroups of H such that H/N 1 and H/N 2 are abelian and torsionfree pro-p groups Assume that Γ N 2 Γ N 1 Then for any matrix A over Z[Γ] rk H,N1 (A) rk H,N2 (A) Proof For simplicity of exposition, let us assume that A Z[Γ] (that is A is an 1 by 1 matrix) Let {g 1,, g k } be a right transversal of H Γ in Γ Consider a k by k matrix M = (m lj ) over Z[H Γ] such that k g i A = m ij g j j=1 Let M l (l = 1, 2) be a matrix over F p [H Γ/(N l Γ)] which is obtained from M by reducing the elements modulo the ideal generated by {g 1 : g N l Γ} and p Let D l = Q(F p [[H/N l ]]) and E l = Q(F p [H Γ/(N l Γ)]) (l = 1, 2) Since F p [H Γ/(N l Γ)] F p [[H/N l ]], E l is a subfield of D l Since the coefficients of M l are in E l, we obtain that rk H,Nl (A) = rk D l (M l ) = rk E l (M l ) k k Since Γ N 2 Γ N 1, the ring Z[H Γ/(N 1 Γ)] is a quotient of the ring Z[H Γ/(N 2 Γ)] Therefore, and we are done rk E1 (M 1 ) rk E2 (M 2 ), 3 Preliminaries on 3-manifolds 31 Cohomologically good groups Let Γ be a group, Γ its profinite completion The group Γ is called cohomologically good if the homomorphism of cohomology groups i n (M) : H n ( Γ, M) H n (Γ, M) induced by the natural homomorphism i : Γ Γ is an isomorphism for every finite Z[Γ]-module M This notion was introduced by Serre (see [10, I26]) and has been studied recently in several papers (see, for example, [7, 8])

3 RECOGNITION OF BEING FIBRED FOR COMPACT 3-MANIFOLDS 3 It is known that free, surface, free-by-cyclic, surface-by-cyclic and 3-manifold groups are cohomologically good The result on 3-manifold groups (see, for example, [8, Remark 514])) uses the recent advances in the theory of 3-manifolds, including the solution of virtual Haken conjecture by I Agol We note that under some additional conditions on a 3-manifold the goodness of its fundamental group can be proved by more elementary arguments For example, let us mention the following result Proposition 31 Let M be a compact 3-manifold Assume that the profinite completion of π 1 (M) is isomorphic to the profinite completion of a free-by-cyclic group or to the profinite completion of a surface-by-cyclic group Then π 1 (M) is cohomologically good Proof When M has boundary, the result follows from [8, Corollary 54] The case where M does not have boundary can be proved similarly One useful application of goodness is the following Assume that Γ is a F P -group of finite cohomological dimension Then ˆΓ is also of finite cohomological dimension and, cd(ˆγ) = cd(γ) This follows from the following result Proposition 32 Let Γ is a F P -group and let di φi Z[Γ] Z[Γ] di 1 d1 φ1 Z[Γ] Z[Γ] Z 0 be a resolution of Z[Γ]-module Z Then Γ is cohomologically good if and only if the induced sequence is also exact Ẑ[[ˆΓ]] di ˆφi Ẑ[[ˆΓ]] di 1 Ẑ[[ˆΓ]] ˆφ d1 1 Ẑ[[ˆΓ]] Ẑ 0 32 Profinite completions of 3-manifold groups Proposition 33 Let M be a compact orientable 3-manifold and let Γ be a freeby-cyclic group or of a surface-by-cyclic group Assume that π 1 (M) = Γ Then the boundary of M is a union of incompressible tori or it is empty Proof If Γ is a free-by-cyclic group, the proposition follows from [3, Corollary 43] If Γ is a surface-by-cyclic group, then π 1 (M) is of cohomological dimension 3 Since π 1 (M) is cohomologically good, π 1 (M) is of cohomological dimension 3 as well, and so, M does not have boundary 33 A criterion for fibering Let X be a CW-complex, let φ : π 1 (X) Z be a non-trivial map and let V be a finite left π 1 (X)-module The tensor product V φ = F p [π 1 (X)/ ker φ] Fp V is a finitely generated left π 1 (X)-module The i-th twisted Alexander polynomial V X,φ,i is the order of F p[π 1 (X)/ ker φ]-module H i (π 1 (M), V φ ) (see [4] for details) Observe that V X,φ,i 0 if and only if H i(π 1 (M), V φ ) is finite The following criterion follows from [6, Theorem 11] Theorem 34 Let M be a compact orientable 3-manifold with toroidal or empty boundary and let φ : π 1 (M) Z be a non-trivial map Pick a prime p The class φ is fibred if and only if for every normal subgroup H of π 1 (M) of finite index the group H 1 (H ker φ, F p ) is finite

4 4 ANDREI JAIKIN-ZAPIRAIN Proof The only if part is clear, since if φ fibers, ker φ is finitely generated Now assume that for every normal subgroup H of π 1 (M) of finite index the group H 1 (H ker φ, F p ) = H 1 (π 1 (M), F p [π 1 (M)/(H ker φ)]) is finite Hence Fp[π1(M)/H] M,φ,1 0 Thus, [6, Theorem 11] implies that φ is fibered Let Γ be a finitely generated group and G a profinite group Let α : ˆΓ G be an isomorphism between the profinite completion of Γ and G A map φ : Γ Z induced a homomorphism φ k : Γ Z p /p k Z p We put ˆφ = lim k φ k α 1 : G Z p and say that ˆφ is associated to φ (with respect to α) Let γ : G Z p be a non-trivial homomorphism and let U be an open subgroup of G We denote by U γ the intersection U ker γ Note that U/U γ = Zp Corollary 35 Let M be a compact orientable 3-manifold with toroidal or empty boundary and let φ : π 1 (M) Z be a non-trivial map Denote by ˆφ : π 1 (M) Z p the homomorphism associated to φ Pick a prime p The class φ is fibered if and only if for every normal open subgroup U of π1 (M) the group H 1 (U ˆφ, F p ) is finite Proof The only if part is clear Let us proof the if part We write a projective resolution of Z as Z[π 1 (M)]-module Z[π 1 (M)] r m B Z[π1 (M)] d Z[π 1 (M)] Z 0, where r = d or r = d 1 (depending on whether M is closed or not) and m B is is a multiplication by a matrix B Since, π 1 (M) is cohomologically good, using Propositon 32 we obtain an exact sequence F p [[ π 1 (M)]] r m B Fp [[ π 1 (M)]] d F p [[ π 1 (M)]] F p 0 Let H be a normal subgroup of π 1 (M) of finite index We put U = H Since H 1 (U ˆφ, F p ) = H 1 ( π 1 (M), F p [[ π 1 (M)/U ˆφ]]) is finite we obtain that rk U,U ˆφ(B) = d 1 But this also implies that H 1 (H ker φ, F p ) = H 1 (π 1 (M), F p [π 1 (M)/H ker φ]) is finite Hence we are done by the previous theorem 4 A lemma on homomorphisms Proposition 41 Let V = Z k p and V = Hom(V, Z p ) Let L = K = Z k be two dense subgroups of V Let {v 1,, v k } and {u 1,, u k } be Z-bases of L and K respectively Define elements v 1, v k, u 1,, u k of V by means of v i (v j ) = u i (u j ) = δ ij For γ V we denote by γ L the restriction of γ on L Then for every φ Zv1 + Zvk there exists an element ψ Zu 1 + Zu k such that ker ψ L ker φ L

5 RECOGNITION OF BEING FIBRED FOR COMPACT 3-MANIFOLDS 5 Proof Since {v 1,, v k } and {u 1,, u k } are Z p -bases of V, {v 1,, v k } and {u 1,, u k } are Z p-bases of V Hence there exists a matrix A GL k (Z p ) such that (u 1,, u k) = (v 1,, v k)a Write A as A = s i=1 A iz i where A i Mat k (Z) and {z 1,, z s } are Z-linearly independent Let R = Q[t 1,, t s ] be a polynomial ring We put B = s i=1 A it i Mat k (R) Since A = s i=1 A iz i is invertible over Z p, det A 0 Hence det B 0 as well Since det B 0, there are q 1,, q s Z such that Let us put We can express φ as det( A i q i ) 0 i=1 C = A i q i i=1 φ = α 1 v α k v k, where α i Z Since C is invertible over Q, there are,, Z and 0 n Z such that C Now we put k ψ = β i u i = (v1,, vk)a i=1 Let us define = n = ψ j = (v 1,, v k)a j α 1 α k z j (v1,, vk)a j Then ker ψ L = ker(z 1 (ψ 1 ) L + +z k (ψ k ) L ) = s i=1 ker(ψ j ) L ker q j (ψ j ) L = ker (v1,, vk)c L j=1 = ker (v1,, vk) α 1 α k j=1 L = ker φ L

6 6 ANDREI JAIKIN-ZAPIRAIN 5 Proof of Theorem 11 Let Γ = S Z be a free-by-cyclic or a surface-by-cyclic group and let G = Ŝ Ẑ its profinite completion We write a projective resolution of Z as Z[Γ]-module Z[Γ] r m A Z[Γ] d Z[Γ] Z 0, where r = d 1 or r = d (depending on whether S is a free or surface group) and m A is is a multiplication by a matrix A Lemma 51 Let γ : G Z p be a non-trivial homomorphism and let U be an open normal subgroup of G Then H 1 (U γ, F p ) is finite if and only if rk U,Uγ (A) = d 1 Proof We may use the same argument as in the proof of Corollary 35, because Γ is cohomologically good Now we are ready to prove Theorem 11 First observe that, by Proposition 33, M has toroidal or empty boundary Denote by α : π 1 (M) G an isomorphism between π 1 (M) and G Let ˆf : G Z p be the map associated to f : Γ Z with ker f = S Denote by V the maximal torsion-free abelian pro-p quotient of G Then V = Z k p Let L be the image of Γ in V and K the image of α(π 1 (M)) in V Applying Proposition 41 in case φ = ˆf, we obtain that there exists γ : π 1 (M) Z such that ker ˆγ Γ ker f = S, where ˆγ is associated to γ with respect to α By Proposition 21, for every open normal subgroup U of G, rk U,Uˆγ (A) rk U,U ˆf (A) = d 1 Hence, by Lemma 51, H 1 (Uˆγ, F p ) is finite Therefore, by Corollary 35, γ is fibered References [1] I Agol, Criteria for virtual fibering J Topol 1 (2008), [2] I Agol, The virtual Haken conjecture With an appendix by Agol, Daniel Groves, and Jason Manning Doc Math 18 (2013), [3] M Bridson, A Reid, Profinite rigidity, fibering, and the figure-eight knot, preprint [4] M Boileau and S Friedl, The profinite completion of 3-manifold groups, fiberedness and the Thurston norm, preprint [5] M Bridson, A Reid, H Wilson, Profinite rigidity and surface bundles over the circle, preprint [6] S Friedl, S Vidussi, A vanishing theorem for twisted Alexander polynomials with applications to symplectic 4-manifolds J Eur Math Soc 15 (2013), [7] F Grunewald, A Jaikin-Zapirain, P Zalesskii, Cohomological goodness and the profinite completion of Bianchi groups Duke Math J 144 (2008), [8] F Grunewald, A Jaikin-Zapirain, A Pinto, P Zalesskii, Normal subgroups of profinite groups of non-negative deficiency, J Pure Appl Algebra 218 (2014), [9] P Przytycki and D Wise, Mixed 3-manifolds are virtually special, preprint [10] J-P Serre, Galois cohomology Translated from the French by Patrick Ion and revised by the author Corrected reprint of the 1997 English edition Springer Monographs in Mathematics Springer-Verlag, Berlin, 2002 [11] D Wise, Research announcement: the structure of groups with a quasiconvex hierarchy Electron Res Announc Math Sci 16 (2009), Departamento de Matemáticas, Universidad Autónoma de Madrid and Instituto de Ciencias Matemáticas, CSIC-UAM-UC3M-UCM address: andreijaikin@uames