THE FINITENESS OF THE MAPPING CLASS GROUP FOR ATOROIDAL 3-MANIFOLDS WITH GENUINE LAMINATIONS

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1 j. differential gemetry 50 (1998) THE FINITENESS OF THE MAPPING CLASS GROUP FOR ATOROIDAL 3-MANIFOLDS WITH GENUINE LAMINATIONS DAVID GABAI & WILLIAM H. KAZEZ Essential laminatins were intrduced in [7] in rder t generalize the ntin f taut fliatin and incmpressible surface. The papers [7], [4], [5], [1], [3] have partially realized the hpe that maniflds with essential laminatins share many f the prperties f Haken maniflds. The main result f this paper establishes the finiteness f the mapping class grup, fr the class f atridal 3-maniflds cntaining genuine laminatins. A genuine laminatin A is an essential laminatin such that sme clsed cmplementary regin is nt an I-bundle. These are laminatins which are genuinely nt fliatins, that is, they are nt btained by splitting pen alng leaves (if A has n islated leaves). See [2] fr an extensive list f cnstructins f essential laminatins in 3-maniflds, many f which als yield genuine laminatins. See [7] fr backgrund infrmatin n essential laminatins. 0 Ntatin. E dentes the interir f E, and \E\ dentes the number f cmpnents f E. lx is the identity map f X. Hme(M) is the grup f hmemrphisms f M, and Hme(M) is the grup f hmemrphisms f M which are istpic t 1M- The mapping class grup, Hme(M)/Hme(M), is dented M(M). If a C M is a simple clsed curve, then I a is the subset f M(M) defined by I a = {[g] G M(M)\3f G [g] with f\a = lm\a}. Therem 1.1. If M is an atridal 3-manifld that cntains a genuine laminatin, then M(M) is finite. Received February 19, The first authr was partially supprted by NSF Grant DMS and the MSRI. 123

2 124 david gabai & william h. kazez Outline. We will assume that M is nt Haken since this case was prved by Jhannsn [8]. It therefre suffices t cnstruct a curve a C M such that I a is finite (Lemma 1.2) and such that the set f istpy classes {[f(a)] f G Hme(M)} in M is finite (Lemma 1.4). Fr the remainder f the paper we shall assume that M cntains a genuine laminatin A and is therefre irreducible [7]. We shall als assume that M is nt Haken. Lemma 1.2. finite. If a C M is nt hmtpically trivial, then I a is Prf. Given an element g f I a we may chse a representative f that preserves a neighbrhd N(a) f a, setwise. Since M is nt Haken, H2 (M) = 0 and it fllws that f : dn(a) > dn(a) is istpic t the identity map. Therefre we may assume that f\n(a) = ln( a )- M a is irreducible since a is hmtpically nntrivial. Cnsider f\(m - N(a)). By [8] the qutient f M(M - N(a)) by the subgrup generated by Dehn twists alng annuli and tri is finite. If A is a prperly embedded annulus in M N(a) whse restrictin t dn(a) is hmlgically trivial, then a Dehn twist alng A is equal, in M{M N(a)), t a Dehn twist alng a trus. There is n prperly embedded annulus in M N(a) whse restrictin t dn(a) is nt hmlgically trivial, fr such an annulus wuld imply the existence f a 1-sided Klein bttle in M N(a) and since M is nt Haken, this wuld imply that 7Ti(M) is finite (it wuld be 2-fld cvered by a lens space), and this cntradicts the fact that -KI(M) is infinite [7]. It fllws that up t Dehn twists abut tri in M N(a), there are nly finitely many pssibilities fr f\(m N(a)). Since M is atridal, Dehn twists alng tri are istpic t IM and hence I a is finite. q.e.d. We will use the fllwing unique decmpsitin therem frm [6] t define a. This therem is a natural versin f a very easy case f the Jac-Shalen, Jhannsn [9], [8] Characteristic Submanifld Therem. Lemma 1.3. Let V be the disjint unin f the clsed cmplementary regins f an essential laminatin A. There exists a unique (up t istpy in V) finite cllectin A = A\ U U A n f prperly embedded annuli in V such that the fllwing hld: (1) V = G U X where G n X = dv G = dvl = A.

3 the finiteness f the mapping class grup 125 (2) (I, dh I) is an I-bundle ver a pssibly nncmpact r discnnected surface. N cmpnent f (I, h I) is an I-bundle ver a cmpact surface with nn-empty bundary. (3) (G,h G) is cmpact, has n cmpnents hmemrphic t (D 2 x I, D 2 x di) and cntains n essential prduct disks. Since A is genuine, the gut, G, is nnempty. Let Z be a cmpnent f A and let a be its cre. Recall that a is hmtpically nn-trivial by [7]- Lemma 1.4. The set f istpy classes {[f(ce)]\f G Hme(M)} in M is finite. Prf. Let A be a fixed triangulatin f M. By [3] we may assume that A is maximal gut number laminatin and hence, after pssibly splitting A alng finitely many leaves, f is istpic t a map, als called f, such that f(a) is in nrmal frm with respect t A. In the usual way, put an I-bundle structure near f(a) s that in each 3-simplex a f A, every maximal cllectin f parallel nrmal disks f f(a) n a lies in an I-fibred D 2 x I. Thus each a is decmpsed int an I-bundle regin and a nn-prduct regin. Let C be the clsed cmplementary regin f A cntaining a. Then f(c) can be decmpsed as an I-bundle, K, and a space H which is a finite unin f nn-prduct prtins f each a G A. dv H = dv K is a finite unin f annuli. We need t take care f thse cmpnents which are cmpressible in f(c). If sme cmpnent A f dv K bunds a D 2 xi C V, with D 2 xdi C dh{f{c)) and if {D 2 xi)nh ^ 0, then define Ki = KU{D 2 x I) and H 1 = H-(D 2 xi). Nte that K x is an I-bundle and H\ is a unin f cmpnents f H. By cntinuing in this manner we btain an I-bundle K n such that H n = V K n is a unin f cmpnents f H. Furthermre if A is a cmpnent f dv H n, then either A is 7ri-injective in f(c) r A bunds a D 2 x I cmpnent f K n. It is imprtant t nte that n matter which hmemrphism f was initially chsen r hw f(a) was put int nrmal psitin, by the finiteness f A, up t p.l. istpies f simplices in A, there are nly

4 126 david gabai & williamh. kazez finitely many H n that can arise in this cnstructin. Figure 1 Let H' U K' be the decmpsitin f f(c) given by deleting the D 2 x I cmpnents frm K n (if there are any) and adding them t H n. The thickened disks that are added t H n t prduce H' depend n f, but since M is irreducible, there is a unique, up t istpy f M, way t add such disks. Thus, up t istpy f M, there are nly finitely many H' that can arise in this cnstructin. Since H' cmpact and dv K' is incmpressible, K' extends by [9], [8], t a maximal I-bundle L C f(c). Let L be the I-bundle which is a unin f thse cmpnents f L' which cntain an end f f(c). L is maximal in the sense that it satisfies (2), (3) f Lemma 1.3. (The G f (3) is f(c)-l and dh G = G n df(c).) Let G UÎ be the decmpsitin f the clsed cmplementary regins f A given by Lemma 1.3. If J = X n C, then f(j) is istpic t a cre f a cmpnent f dv L'. Thus, up t istpy, f(a) is the cre f ne f finitely many characteristic annuli in the JSJ splitting f H'. Since there are nly finitely many pssibilities fr H', Lemma 1.4 fllws. q.e.d. References [1] M. Brittenham, Essential laminatins and Haken nrmal frm. I, Pacific J. 168 (1995) [2] D. Gabai, Prblems in the gemetric thery f fliatins and laminatins n 3- maniflds, Gemetric Tplgy (ed. W. H. Kazez), Amer. Math. Sc. Internat. Press 2 (1997) 1-33.

5 the finiteness f the mapping class grup 127 [3], Essential laminatins and Kneser nrmal frm, Preprint. [4] D. Gabai & W.H. Kazez, 3-Maniflds with essential laminatins are cvered by slid tri, J. Lndn Math. Sc. 47 (1993) [5], Hmtpy, istpy and genuine laminatins f S-maniflds, Gemetric Tplgy (ed. W.H. Kazez), Amer. Math. Sc. Internat. Press 1 (1997) [6], Grup negative curvature fr S-maniflds with genuine laminatins, Gem. Tplgy 2 (1998) [7] D. Gabai & U. Oertel, Essential laminatins in S-maniflds, Ann. f Math. (2)130 (1989) [8] K. Jhannsn, Hmtpy equivalences f S-maniflds with bundary, Lecture Nte in Math., Springer, Berlin, Vl. 761, 1983, [9] W. Jac & P. Shalen, Seifert Fibered Spaces in 3-Maniflds, Mem. Amer. Math. Sc. 21 (1979). Califrnia Institute f Technlgy University f Gergia