COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES

prceedings f the american mathematical sciety Vlume 105, Number 3, March 1989 COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES MARK D. BAKER (Cmmunicated by Frederick R. Chen) Abstract. Let M be a cmpact, rientable 3-manifld that fibers ver S1 with fiber a nce-punctured trus, T0, and characteristic hmemrphism h : T0 7". We prve that fr certain characteristic hmemrphisms, mst Dehn fillings n M yield maniflds with virturally Z-representable fundamental grups. 1. Intrductin and statement f results Let M be a cmpact, rientable 3-manifld that fibers ver S with fiber a nce-punctured trus, T0, and characteristic hmemrphism h : T0 TQ. In this paper we investigate the maniflds btained frm M by Dehn filling attaching a slid trus t the trus bundary f M t btain a clsed 3- manifld. We prve that fr certain characteristic hmemrphisms, mst Dehn fillings n M yield maniflds with virtually Z-representable fundamental grups. Let Dx (resp. D ) dente the left-handed Dehn twist abut the curve x (resp. curve v ) in TQ (see Figure 1). Then any rientatin preserving hmemrphism h: T0 > T0 can be represented up t istpy by a cmpsitin f the Dehn twists Dx and D. We prve: Received by the editrs May 20, 1988 and, in revised frm, Octber 16, 1988. 1980 Mathematics Subject Classificatin (1985 Revisin). Primary 57M10. 57N10. 747 1989 American Mathematical Sciety 0002-9939/89 $1.00+ $.25 per page License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

748 MARK D. BAKER Therem. Let M be a nce-punctured trus bundle ver S1 with characteristic hmemrphism h f the frm Dr' Ds' D[k Z)** and let m = x y x y g.c.d.(rx,...,rk), n = g.c.d.(sx.sk). Then (a) If m > 2, n>2, mn > 8 but mn ^ 9, then all Dehn fillings n M f the frm (ß, X), \X\ ^ 1 yield maniflds with virtually Z-representable fundamental grups. (b) If m > 2, «> 2 and mn = 6 r 9, then all Dehn fillings n M f the frm (p, X) with \X\/ 1 r 2 yield maniflds with virtually Z-representable fundamental grups. Remarks. 1. The (0,1) filling n any punctured surface bundle ver 5 yields a manifld with psitive first Betti number. Hence the cases nt cvered by ur therem are (p, X) = (n, 1), n ^ 0, as well as (ß, X) = (n, 2), n ^ 0 in part (b). 2. Other results n the virtual Z-representability f 3-manifld grups can be fund in [B, ], [B 2 ], [H, ], [H 2 ], and [M]. Dehn filling n hyperblic trus bundles gave the first example f hyperblic nn-haken 3-maniflds. Indeed, it fllws frm [F, H], [C, J, R] and [T] that all but finitely many Dehn fillings n hyperblic nce-punctured trus bundles yield maniflds that are hyperblic but cntain n incmpressible surfaces. Hwever fr the hyperblic trus bundles in ur therem, mst Dehn fillings yield maniflds that are virturally Haken. 2. Definitins and utline f prf Relevant backgrund material n trus bundles and Dehn twist hmemrphisms can be fund in [H 2 ] and [R]. By Dehn filling n a 3-manifld M with respect t a simple lp in a bundary trus we mean attaching a slid trus t dm s that this lp bunds a meridinal disk in the slid trus. We say that M has a virtually Z-representable fundamental grup if nx(m) cntains a finite index subgrup that maps epimrphically t Z, r equivalently if there is a finite cver M M with rank 77, (M) > 1. In particular, if M is cmpact, rientable and irreducible, then the virtual Z-representability f nx(m) implies that the abve cver M is a Haken manifld. Nw let M be a nce-punctured trus bundle with characteristic hmemrphism h: TQ * T0 which is the identity n dt0. Chse a base pint b0 in dt0 and cnsider the lps a = b0 x [0,1]/ ~ and ß = dt0 in dm (see Figure 1). Dente by M(ß,X) the manifld btained frm Dehn filling n M with respect t the lp a? ß in dm. Fr a trus bundle, M, with characteristic hmemrphism, h, and Dehn filling parameters (ß, X) as in the therem, we prve ur result by cnstructing a finite cver M M with the fllwing tw prperties: (i) The cver M M extends t a cver N M(ß, X) by Dehn filling n M and n M. License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

(ii) COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES 749 RankHX(M) > number f cmpnents f dm. Prperty (ii) guarantees that any manifld btained by Dehn filling n M (hence TV ) has psitive first Betti number. Hence M(ß, X) has virtually Z- representable fundamental grup. The cver M is gtten by cnstructing a cver n: F T0 t which h lifts t h: F F. Then the mapping trus, M, f the pair (F,h) is the desired cver f M. 3. Prf f case (a) f therem Recall that the characteristic hmemrphism, h, f M is f the frm Dx Dy - Drxk Dsyk where g.c.d.(rx,...,rk) = m and g.c.d.(sx,...,sk) = n. We can assume that m > 2 and n > 4. Dente by n : S T0 the m«-fld cver crrespnding t the kernel f the map 6: nx(t0) = ZxZ^Z/mxZ/n where 6([x]) = (1,0) and 6([y]) = (0,1). S is a trus with mn punctures, {/?;}, pictured in Figure 2. *in*- * ß ß9 -> l p - y3...y m s\ Figure 2 Lemma 3.1. The hmemrphism h: Tn T0 lifts t a hmemrphism g: Prf. The lp x (resp. y ) in T0 lifts t n lps xx,...,xn (resp. m lps yx,...,ym) in S that prject m t 1 nt x (resp. n t 1 nt y ). Since m\ri and n\s, the Dehn twists Dx lift t Dehn twists abut the lifts, {x(}, f x and the Dehn twists D* lift t Dehn twists abut the lifts, {y }, f y. Hence h lifts t a hmemrphism g : S 5. We begin with the ( 1,0) Dehn filling n M the case X = 0. Let M dente the mapping trus f (S, g). By cnstructin, the "meridian" lp a = b0 x [0,1]/ ~ in dm lifts t mn lps à,,. àm mn indexed s that the lps ä(, ßi lie n the i th bundary trus f M (cf. Figure 2) License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

750 MARK D. BAKER Lemma 3.2. The cver M -> M extends t a cver N -+ M (1,0) by Dehn filling n M and M with respect t the curves {à;} in dm and a in dm. Lemma 3.3. Rank 77, (N) > 1. Prf. There are relatins in HX(M) between the lifts {à(} f a. Fr example [à2] - [à,] = [âm+2] - [àm+1] Thus Dehn filling with respect t the lps {a(} yields a manifld N with psitive first Betti number. Indeed, ne cmputes [à ] [<*,-] as fllws. Let <t. be a simple path in S frm the pint bi = à/n.s t the pint 6y = a; n S(bi e ßt is the lift f the basepint b0 e ß in T0). Then the disk ct.. x 7 c S x I prvides the relatin [a ] - [à.] = [g(t..) * ctj1] in 77, (M), where * dentes cmpsitin f paths. Nw cr,2 and am+x m+2 can be chsen s that they bth intersect the Dehn twist curve y2 exactly nce and have zer intersectin with all ther Dehn twist curves xi, yi f the hmemrphism g : S -* S. Thus it fllws that (g(n) * (T"1] = [g(m+x m+2) * ~m\x m+2] in HX(M). U Nw cnsider the (p, X) Dehn filling n M where \X\ = d >2. We cnstruct the desired cver n: F T0 by gluing tgether d cpies f the surface 51 alng certain cuts. Let Sx,...,Sd be cpies f S and make fur vertical cuts {t.} n each St as pictured in Figure 3. Glue the left edges f the cuts t,, t3 in 5, t the right edges f the same cuts in Si+X (md d) and glue the left edges f the cuts t2, t4 in S t the right edges f the same cuts in 5(_, (md d). Dente this surface by F. F has genus 4d - 3 and dmn -Sd + S punctures. Then Figure 3 Lemma 3.4. n : F *rs(-+t0. T0 is a (nn-regular) cvering, where n\s is defined t be License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES 751 Lemma 3.5. The hmemrphism h: T0 TQ lifts t a hmemrphism h : F -F. Prf. F was cnstructed s that the lp x (resp. y) in T0 lifts t dn lps x,,...,xdn (resp. dm lps y,,...y m) in F which prject m t 1 nt x (resp. n t 1 nt y). Thus the Dehn twists Dx (resp. Z)*') in T0 lift t Dehn twists abut the lps {x } (resp. {y,}) in j*. Cnsequently h lifts t h:f^f. U Dente by M the mapping trus f (F, h). By cnstructin the lps a and ß in dm lift t lps in the bundary tri f M, s the lp f1 ß lifts t lps in dm. Chse ne lift in each cmpnent f dm and label them Lemma 3.6. The cver M M extends t a cver N > M(p, X) by Dehn filling n M and M with respect t the curves {c(} in dm and afß in dm. Nw all that remains t shw is Lemma 3.7. Rank 77, (TV) > 1. Prf. As pinted ut in sectin 2, it suffices t shw that rank HX(M) > number f bundary tri f M. Since M is the mapping trus f (F,h), we have 77, (M) = Z Cker (h\ - Id: HX(F) 77,(F)) where the first factr is generated by any lp (e.g. a lift f a) transverse t F. As rank Cker (ht - Id) is equal t the rank f the subgrup f 77, (F) fixed by ht and the bundary cmpnents f F are fixed by h, it fllws that rank 77, (M) > number f bundary cmpnents f M since the bundary cmpnents f F and M are in 1-1 crrespndence and there is ne relatin in HX(F) between the cmpnents f df given by the surface, F. Thus we need nly exhibit ne fixed class in HX(F) that is nt hmlgus t a sum f bundary curves f F. A prtin f the surface F is shwn in Figure 4. (Recall the gluing f the surfaces 5( alng the cuts {t^}). The class [y] + [ö] in HX(F) crrespnding t the curves y, 8 is fixed by ht, since y and S each intersect with ppsite rientatins the same tw Dehn twist curves. Finally [y] + [a] is nt hmlgus in HX(F) t a sum f bundary curves f F since y and S have nntrivial intersectin with the curves a and p while all the bundary cmpnents f F have null intersectin with a and p. This cncludes the prf case (a) f the therem. Remark. The abve arguments generalize t shw that rank HX(N) > [f ][f ]. License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

752 MARK D. BAKER Figure 4 ßx h h \ß< \ß5Q iß 6 M y, ^2 ^3 'i Figure 5 4. Prf f Case (b) f therem We assume that m = 3 and n = 2. The prf when m = n = 3 is analgus. The punctured trus r has a 6-fld cver, S, crrespnding t the kernel f the hmmrphism 0: nx(t0) = ZxZ^Z/3x Z/2 where 0([x]) = (1,0) and 0([y]) = (0,1) (see Figure 5). The hmemrphism h: T0 > TQ lifts t a hmemrphism g : S S. Dente by Ai the mapping trus (S, g). As in sectin 3 we have: Lemma 4.1. The cver M -> M extends t a cver N > M (1,0) by Dehn filling n M with respect t the curve a and n M with respect t the lifts {a } f a in dm. Lemma 4.2. Rank 77, (/V) > 1. Nw cnsider the (p,x) Dehn filling n M where \X\ d > 3. We cnstruct the desired cver F > T0 by gluing tgether d cpies f S alng three vertical cuts {t^} as pictured in Figure 6. Glue the left edges f the cuts t, and x2 n S i t the right edges f the same cuts n S +x (md d). Glue the left edge f License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

COVERS OF DEHN FILLINGS ON ONCE-PUNCTURED TORUS BUNDLES 753 the cut t3 n S t the right edge f the same cut n St_2 (md d). As befre, call this surface F. The cver F > T0 is 6d-fld and the hmemrphism h : T0 T0 lifts t a hmemrphism h: F > F. Let A/" dente the mapping trus f (F,h). By cnstructin the lp <//?A in dm lifts t lps in the bundary tri f M. O Ó ó 3Ó?i vi y y\ Figure 6 Lemma 4.3. The cver M -+ M extends t a cver N * M(p, X) by Dehn filling n M with respect t amßx and n M with repsect t lifts f c/ ßk in dm. Lemma 4.4. Rank 77, (TV) > 1. Prf. Cnsider the curves y and S n F in Figure 7. The class [y] + [S] in HX(F) is fixed by hm and is nt hmlgus t a sum f bundary curves f f. m 0 Ô Ô Figure 7 References [B i ] M. Baker, On certain branched cyclic cvers f S3, Gemetry and Tplgy (Prc. Univ. Gergia Tp. Cnf., 1985), Marcel Dekker, New Yrk, 1987, 43-46. [B 2 ]_, The virtual Z-representability f certain 3-manifld grups, Prc. Amer. Math. Sc. 103 (1988), 996-998. License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use

754 MARK D. BAKER [C,J,R] M. Culler, W. Jac, and H. Rubinstein, Incmpressible surfaces in nce-punctured trus bundles, Prc. Lndn Math. Sc. (3) 45 (1982), 385-419. [F,H] W. Flyd and A. Hatcher, Incmpressible surfaces in punctured trus bundles, Tplgy and its Applicatins 13 (1982), 263-282. [H i ] J. Hempel, Orientatin reversing invlutins and the first Betti number fr finite cverings f i-maniflds, Invent. Math. 67 (1982), 133-142. [H 2 ]_, Cverings f Dehn fillings f surface bundles, Tplgy Appl. 24 (1986), 157-170. [M] S. Mrita, Finite cverings f punctured trus bundles and the first betti number, preprint. [R] D. Rlfsen, Knts and Links, Publish r Perish Inc., 1976. [T] W. Thurstn, The gemetry and tplgy fi-maniflds, Xerxed ntes, Princetn University. Department f Mathematics, Vanderbilt University, Nashville, Tennessee 37235 License r cpyright restrictins may apply t redistributin; see http://www.ams.rg/jurnal-terms-f-use