ON GENERA OF LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS

Transcription

1 Kobayash, R. Osaka J. Math. 53 (2016), ON GENERA OF LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS RYOMA KOBAYASHI (Receved May 14, 2014, revsed January 7, 2015) Abstract It s known that every fntely presented group s the fundamental group of the total space of a Lefschetz fbraton. In ths paper, we gve another proof whch mproves the result of Korkmaz. In addton, Korkmaz defned the genus of a fntely presented group. We also evaluate upper bounds for genera of some fntely presented groups. 1. Introducton Gompf [5] proved that every fntely presented group s the fundamental group of a closed symplectc 4-manfold. Donaldson [4] proved that every closed symplectc 4- manfold admts a Lefschetz pencl. By blowng up the base locus of a Lefschetz pencl, we obtan a Lefschetz fbraton over S 2. In addton, blowng up does not change the fundamental group of a 4-manfold. Therefore, t mmedately follows that every fntely presented group s the fundamental group of the total space of a Lefschetz fbraton. Amoros Bogomolov Katzarkov Pantev [1] and Korkmaz [8] also constructed Lefschetz fbratons whose fundamental groups are a gven fntely presented group. In partcular, Korkmaz [8] provded explctly a genus and a monodromy of such a Lefschetz fbraton. Let F n g 1,, g n be the free group of rank n. For x ¾ F n, the syllable length l(x) of x s defned by l(x) mn { s x g m(1) (1) g m(s) (s), 1 ( j) n, m( j) ¾ }. For a fntely presented group ¼ wth a presentaton ¼ g 1,, g n r 1,, r k, Korkmaz [8] proved that for any g 2 n È 1k l(r ) k there exsts a genus-g Lefschetz fbraton f Ï X S 2 such that the fundamental group 1 (X) s somorphc to ¼, provdng explctly a monodromy. In ths paper, we mprove ths result. Theorem 1.1. Let ¼ be a fntely presented group wth a presentaton ¼ g 1,, g n r 1,, r k, and let l max 1k {l(r )}. Then for any g 2n l 1, there 2010 Mathematcs Subject Classfcaton. Prmary 57N13; Secondary 57M05.

2 352 R. KOBAYASHI Fg. 1. The Dynkn dagram. exsts a genus-g Lefschetz fbraton f Ï X S 2 such that the fundamental group 1 (X) s somorphc to ¼. In ths theorem, f k 0, we suppose l 1. We wll prove the theorem by provdng an explct monodromy. In addton, Korkmaz [8] defned the genus g(¼) of a fntely presented group ¼ to be the mnmal genus of a Lefschetz fbraton wth sectons whose fundamental group s somorphc to ¼. The Lefschetz fbratons constructed n Theorem 1.1 have sectons. Hence the defnton of the genus of a fntely presented group s well-defned. We wll also prove the followng theorem. Theorem 1.2. (1) Let B n denote the n-strands brad group. Then for n 3, we have 2 g(b n ) 4. (2) Let H g be the hyperellptc mappng class group of a closed connected orentable surface of genus g 1. Then we have 2 g(h g ) 4. (3) Let M 0,n denote the mappng class group of a sphere wth n punctures. Then for n 3, we have 2 g(m 0,n ) 4. (4) Let S n denote the n-symmetrc group. Then for n 3, we have 2 g(s n ) 4. (5) Let A n denote the n-artn group assocated to the Dynkn dagram shown n Fg. 1. Then for n 6, we have 2 g(a n ) 5. (6) Let n, k 0 be ntegers wth n k 3, and let m 1,, m k 2 be ntegers. Then we have (n k 1)2 g( n m1 mk ) n k A Lefschetz fbraton and prelmnares 2.1. A Lefschetz fbraton and ts monodromy. Here, we revew brefly the theory of Lefschetz fbratons. Let X be a closed connected orentable smooth 4-manfold. A smooth map f Ï X S 2 s a genus-g Lefschetz fbraton over S 2 f t satsfes followng propertes: All regular fbers are dffeomorphc to a closed connected orented surface of genus g. Each crtcal pont of f has an orentaton-preservng chart on whch f (z 1, z 2 ) z 2 1 z2 2 relatve to a sutable smooth chart on S2. Each sngular fber contans only one crtcal pont.

3 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 353 Fg. 2. The rght Dehn twst about c. Fg. 3. f s relatvely mnmal, that s, no fber contans an embedded sphere wth the self-ntersecton number 1. Let M g be the mappng class group of a closed connected orented surface g of genus g, that s, the group of sotopy classes of orentaton-preservng dffeomorphsms g g. In ths paper, for elements x and y of a group, the composton xy means that we frst apply x and then y. So for f, g ¾ M g, the composton f g means that we frst apply f and then g. For a smple closed curve c on g, let t c be the sotopy class of the rght Dehn twst about c (see Fg. 2). For a genus-g Lefschetz fbraton whch has n sngular fbers, there are smple closed curves c 1,, c n on g, each of whch s called the vanshng cycle, such that each sngular fber F s obtaned by collapsng c to a pont to create a transverse self-ntersecton, and t c1 t cn 1. Ths equaton s called the monodromy of a Lefschetz fbraton. Conversely, f there are smple closed curves c 1,, c n on g such that t c1 t cn 1, then we can construct a genus-g Lefschetz fbraton wth the monodromy t c1 t cn 1. For a Lefschetz fbraton f Ï X S 2, a smooth map s Ï S 2 X s a secton of f f f Æ s Ï S 2 S 2 s the dentty map. For a closed connected orentable surface g of genus g, let a 1,, a g, b 1,, b g and c 1,, c g be loops on g as shown n Fg. 3. Then the fundamental group 1 ( g )

4 354 R. KOBAYASHI of g has a followng presentaton Fg ( g ) a 1, b 1,, a g, b g r, where r b 1 g b 1 1 (a 1b 1 a 1 1 ) (a gb g a 1 g ). Let B 0,, B g and a, b, c be smple closed curves on g as shown n Fg. 4. In ths paper, let W denote the followng W (tc t Bg t B0 ) 2 when g s even, (t 2 a t 2 b t B g t B0 ) 2 when g s odd. It was shown n [7] that W 1 n the mappng class group M g of g. In addton, the Lefschetz fbraton f W Ï X W S 2 wth the monodromy W 1 has a secton (see [7] and [8]) Prelmnares. We now state the way to obtan the presentaton of the fundamental group of a Lefschetz fbraton wth a secton. For a group ¼ and {x 1,,x n } ¼, let x 1,, x n denote the normal closure of {x 1,, x n } n ¼. Proposton 2.1 (cf. [6]). Let f Ï X S 2 be a genus-g Lefschetz fbraton wth the monodromy t c1 t cn 1. Suppose that f has a secton. Then we have 1 (X) 1 ( g )c 1,, c n, where we regard c 1,, c n as elements n 1 ( g ).

5 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 355 For x, y ¾ M g, let x y y 1 xy. For example, for smple closed curves c 1,, c n on g and h ¾M g, we have (t c1 t cn ) h (h 1 t c1 h) (h 1 t cn h) t (c1 )h t (cn )h, where (c )h means the mage of c by h. Proposton 2.2 ([8]). Let f Ï X S 2 be a genus-g Lefschetz fbraton wth the monodromy V t c1 t cn 1. Suppose that f has a secton. Let d be a smple closed curve on g whch ntersects some c transversely at only one pont. Let f ¼ Ï X ¼ S 2 be the genus-g Lefschetz fbraton wth the monodromy V V t d 1. Then we have 1 (X ¼ ) 1 ( g )c 1,, c n, d, where we regard c 1,, c n and d as elements n 1 ( g ). In ths paper, we denote the Lefschetz fbraton wth the monodromy V 1 by f V Ï X V S 2. For example, n the above proposton, f f V, X X V and f ¼ f V V tc, X ¼ X V V tc. We next state results of Korkmaz [8]. Theorem 2.3 ([8]). (1) Let g be a closed connected orentable surface of genus g 0. Then we have g( 1 ( g )) g. (2) Let m(¼) denote the mnmal number of generators for ¼. Then we have m(¼)2 g(¼), wth the equalty f and only f ¼ s somorphc to 1 ( g ). (3) For the mappng class group M 1 of 1, we have 2 g(m 1 ) 4. (4) Let B n denote the n-strands brad group. Then for n 3, we have 2 g(b n ) 5. (5) Let n, k 0 be ntegers wth n k 3, and let m 1,, m k 2 be ntegers. Then we have (n k 1)2 g( n m1 mk ) 2(n k) 1. Theorem 1.2 mproves Theorem 2.3 (4) and (5). 3. Proof of Theorem 1.1 Frst of all, we show a proposton used n proofs of Theorem 1.1 and 1.2. For elements x and y n a group, let [x, y] xyx 1 y 1. For a real number a, [a] s the maxmal nteger less than or equal to a. Proposton 3.1. Let f W Ï X W S 2 be the genus-g Lefschetz fbraton wth the monodromy W 1, where W s as above, and let a 1, b 1,, a g, b g be the generators of 1 ( g ) as shown n Fg. 3. Then we have followngs:

6 356 R. KOBAYASHI (1) (See [8].) Let U W W t b 1 W t bg, then the fundamental group 1 (X U ) of the Lefschetz fbraton X U has the followng presentaton b 1,, b g, a 1, b 1,, a g, b g when g s even, a 1 a g,, a g2 a (g2)2 1 (X U ) b 1,, b g, a 1, b 1,, a g, b g a 1 a g,, a (g 1)2 a (g3)2, when g s odd, a (g1)2 and, the group 1 (X U ) s somorphc to the free group of rank [g2]. (2) Let U ¼ W W t b 2 W t b g 1, then the fundamental group 1 (X U ¼) of the Lefschetz fbraton X U ¼ has the followng presentaton [a 1, b 1 ], b 2,, b g 1, a 1, b 1,, a g, b g when g s even, b 1 b g, a 1 a g,, a g2 a (g2)2 1 (X U ¼) a 1, b 1,, a g, b g [a 1, b 1 ], b 2,, b g 1, b 1 b g, a 1 a g,, a (g 1)2 a (g3)2, a (g1)2 when g s odd, and, the group 1 (X U ¼) s somorphc to the free product of the free group of rank ([g2] 1) wth. Proof. Smple closed curves B 0,, B g and a, b, c as shown n Fg. 4 can be descrbed n 1 ( g ), up to conjugaton, as follows B 2k a k b k1 b k2 b g k 1 b g k c g k a g k1, where 0 k g2, B 2k1 a k1 b k1 b k2 b g k 1 b g k c g k a g k, where 0 k g2, a a (g1)2, b c (g 1)2 a (g1)2 and c c g2, where let a 0 a g1 1. In addton, note that c b 1 b 1 1(a 1b 1 a 1 1) (a b a 1 ) up to conjugaton, for 1 g. Snce X W has a secton, by Proposton 2.1, we frst

7 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 357 obtan a presentaton of 1 (X W ) as follows. 1 (X W ) c g, c g2, a1, b1,, ag, bg a 1 a g,, a g2 a (g2)2, b 1 a g b g a g 1,, b g2a (g2)2 b (g2)2 a (g2)2 1 when g s even, c g, a (g1)2, b (g1)2, c (g 1)2, a1, b1,, ag, bg a 1 a g,, a (g 1)2 a (g3)2, b 1 a g b g a g 1,, b (g 1)2a (g3)2 b (g3)2 a (g3)2 1 when g s odd. (We have that 1 (X W ) s somorphc to 1 ( [g2] ).) Snce each b ntersects some B j transversely at only one pont, by Proposton 2.2, we obtan the clam. REMARK. From Proposton 3.1, we have followngs. For n 1, there are genus-2n and (2n 1) Lefschetz fbratons whose fundamental groups are somorphc to the free group of rank n. For n 2, there are genus-(2n 2) and (2n 1) Lefschetz fbratons whose fundamental groups are somorphc to the free product of the free group of rank (n 2) wth. Let ¼ be a fntely presented group wth a presentaton ¼ g 1,, g n r 1,, r k and let l max 1k {l(r )}. For g n l 1 and r, we construct a smple closed curve R on g as below. At frst, we construct a smple closed curve R n the case n 4 and r g 2 g 1 g2 2g 4 1g 3 2 as an example. Note that l(r) 5. Let x 1, x 2, x 3, x 4, x 5 be loops on g whch are homotopc to a 2, a 1, a 2, a 4 and a 3, respectvely, as shown n Fg. 5 (a). Let y 1, y 2, y 3, y 4 be loops on g whch are homotopc to a 5, a 6, a 7, a 8, respectvely, and let z 1, z 2, z 3, z 4 be loops on g whch are homotopc to a 5, a 6, a 7, a 8, respectvely, as shown n Fg. 5 (a). Frst we deform g around y 1, z 1,, y 4, z 4 as shown n Fg. 5 (b). Then let D be a subsurface contanng y t and z t whch s surrounded by a smple closed curve on g as shown n Fg. 5 (b). Next, for 1 t 4, we move y t to the rght sde of x t n D, and z t to the left sde of x t1 n D, as shown n Fg. 5 (c). Let R Æ be the loop as shown n Fg. 6 (a), and let R ( R)t Æ 1 x 1 t x 1 2 t x 2 3 t x4 tx 2 5, as shown n Fg. 6 (b). Fnally, we deform the surface so that y 1,, y 4 and z 1,, z 4 go back to ther orgnal poston as shown n Fg. 6 (c). In general, a loop R s constructed as follows. Let r g m(1) j(1) gm(l(r )) j(l(r )). For 1 t l(r ), let x t be a loop on g whch s homotopc to a j(t). If j(s) j(s ¼ ) for some s s ¼, we put x s ¼ to the rght sde of x s. For 1 t l(r ) 1, let y t and z t be loops on g whch are homotopc to a nt, such that z t s n the rght sde of y t.

8 358 R. KOBAYASHI Fg. 5. The loop R n the case n 4, r g 2 g 1 g 2 2 g 1 4 g 2 3.

9 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 359 Fg. 6. The loop R n the case n 4, r g 2 g 1 g 2 2 g 1 4 g 2 3.

10 360 R. KOBAYASHI Fg. 7. The loop c where s l(r ) 1. Frst we deform g around y 1,z 1,, y l(r ) 1,z l(r ) 1, smlarly to the above example. Let c be a smple closed curve whch s descrbed n 1 ( g ) as follows c (a n1 b n1 a 1 n1 ) (a nl(r ) 1b nl(r ) 1a 1 nl(r ) 1 )b 1 nl(r ) 1 b 1 n1, and ntersects each of a 1,, a n at two ponts, as shown n Fg. 7. Then let D be a subsurface whose boundary s c, and whch contans y t and z t. Next we deform D as follows. For 1 t l(r ) 1, we move y t to just rght sde of x t n D, and z t to just left sde of x t1 n D as shown n Fg. 5 (c). We regard that ths moton does not affect on loops a, b and c. Hence x 1,, x l(r ) also do not deform, as shown n Fg. 5 (c). After that, we defne a smple closed curve as shown n Fg. 6 (a). More precsely, we construct arcs L and L ¼ as follows. The arc L s n D. L begns from the pont at the left sde of x 1 on the loop c, crosses x 1, y 1, z 1, x 2, y 2, z 2,, n ths order, fnally crosses x l(r ), and stops at the rght sde of x l(r ) on the loop c. Let L ¼ be an arc whose base pont s the end pont of L, end pont s the base pont of L, and whch does not ntersect the nteror of D and loops a 1, b 1,, a n, b n and c n. Note that the surface whch s obtaned by removng loops c, a 1, b 1,, a n, b n and c n from g, and whch contans L ¼ s a dsk. Hence the arc L ¼ s unque up to homotopy relatve to the base pont and the end pont. Let L L ¼ denote the composton of L and L ¼. We now defne R (L L ¼ m(1) )t x 1 t m(l(r )) x l(r. Fnally, we deform the surface so ) that y 1, z 1,, y l(r ) 1, z l(r ) 1 go back to ther orgnal poston. Note that the loop R s descrbed n 1 ( g ), up to conjugaton, as follows: ( ) R x,1,t a j(1) ÉL, 1tm(1) 1tm(l(r )) x,l(r ),ta j(l(r ))

11 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 361 where x,s,t s a loop whch s some products of a n1, b n1,, a l(r ) 1, b l(r ) 1 and c n1, and É L s a loop whch s descrbed n 1 ( g ) as follows: ÉL We now prove Theorem b j(l(r )) b j(l(r 1 )) 1 b j(1)1 1 j(1) 1 when j(1) j(l(r )), b j(l(r ))1b j(l(r )) b j(1) b j(1) 1 when j(1) j(l(r )). Proof of Theorem 1.1. For g 2n l 1, let V be the followng V U W t a n1 W t a[g2], where U W W t b 1 W t bg. In addton, let V ¼ be the followng V ¼ V V t R 1 V t Rk, where R s the loop constructed prevously. We show that the fundamental group 1 (X V ¼) s somorphc to ¼. Snce each of b 1,, b g and a n1,, a [g2] ntersects some B transversely at only one pont, by Proposton 2.2, we have 1 (X V ) 1 ( g )b 1,, b g, a n1,, a [g2] 1 (X U )a n1,, a [g2]. In addton, by the presentaton of (1) of Proposton 3.1, we have 1 (X U ) a 1,, a [g2]. Therefore we have 1 (X V ) a 1,, a [g2] a n1,, a [g2] a 1,, a n, Because of the presentaton of 1 (X U ) n (1) of Proposton 3.1, we assume g 2n l 1 n place of g n l 1. For any 1 k, consder the vanshng cycle ((B 0 )t an1 )t R of X V ¼. Note that (B 0 )t an1 and (a n1 )t R are descrbed n 1 ( g ), up to conjugaton, as follows: (B 0 )t an1 a n1 (b 1 b g ), (a n1 )t R a n1 (z R z 1 ) for some z ¾ 1 ( g ). Then, we have that ((B 0 )t an1 )t R s descrbed n 1 ( g ) as follows: ((B 0 )t an1 )t R (x a n1 (b 1 b n ) x 1 )t R (x)t R (a n1 )t R (b 1 b n )t R (x 1 )t R (x)t R (y a n1 (z R z 1 ) y 1 )(Û (B 0 )t R Û 1 )((x)t R ) 1,

12 362 R. KOBAYASHI for some elements x, y and Û n 1 ( g ). Snce a n1 (B 0 )t R 1 n 1 (X V ¼), we have R 1 from ((B 0 )t an1 )t R 1, n 1 (X V ¼). For a vanshng cycle c of X V, f R ntersects c transversely at s ponts, then the vanshng cycle (c)t R of X V ¼ s descrbed n 1 ( g ), up to conjugaton, as follows: (c)t R x 1 R 1 x s R s x s1, where j 1 and x 1,, x s1 are elements n 1 ( g ) such that c x 1 x s1. Snce R 1 and c 1 n 1 (X V ¼), we can delete the relaton (c)t R 1 of 1 (X V ¼). We now defne Çr a m(1) j(1) a m(l(r )) j(l(r )) for r g m(1) j(1) g m(l(r )) j(l(r )). Snce x,s,t and L É n the descrpton ( ) of R are 1 n 1 (X V ¼), the natural epmorphsm 1 ( g ) 1 (X V ¼) sends R to Çr. Note that the vanshng cycles of X V ¼ consst of c and (c)t R for all vanshng cycles c of X V and 1 k. Therefore, we have 1 (X V ¼) a 1,, a n Çr 1,, Çr k ¼. Thus, the proof of Theorem 1.1 s completed. 4. Proof of Theorem 1.2 In ths secton, we prove Theorem Proof of (1) of Theorem 1.2. For n 2, let B n denote the n-strands brad group. The group B n has a presentaton wth generators 1,, n 1 and wth relatons j 1 j 1 1, where 1 j 1 n 2, , where 1 n 2. Let x 1 and y 1 n 1. Then B n can be presented wth generators x, y and wth relatons xy k xy k x 1 y k x 1 y k 1, where 2 k n 2, xyxy 1 xyx 1 y 1 x 1 yx 1 y 1 1, (xy) n 1 y n 1. A correspondence between the frst presentaton and the second presentaton s gven by y 1 xy 1 for 1 n 1. See [8] for ths presentaton. We now prove (1) of Theorem 1.2. Proof of (1) of Theorem 1.2. For n 3, snce B n s generated by two generators x, y, we have g(b n ) 2 from (2) of Theorem 2.3 (cf. [8]). Therefore, we prove g(b n ) 4 for n 3. Let R 1,k, R 2 and R 3,n be smple closed curves on 4 as shown n Fg. 8, where 2 k n 2. Note that R 1,k, R 2 and R 3,n ntersect B 4 transversely at only one pont, for 2 k n 2. Loops R 1,k, R 2 and R 3,n can be descrbed n 1 ( 4 ), up to conjugaton, as follows

13 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 363 Fg. 8.

14 364 R. KOBAYASHI R 1,k a 3 1a 4 k(b 3b 4 ) 1 a 2 a 1 k(b 1)a 2 1(b 1b 2 ) 1 a1 ka 2 1(b 3b 4 )a4 k, where 2 k n 2, R 2 a 3 1a 4 1(b 4 1)a 3 1a 4a 3 1a 4 1(b 2b 3 b 4 ) 1 a 2 1(b 3b 4 )a 4 a 3 a 4 1a 3(b 4 )a 4, R 3,n (a 3 1a 4 1(b 4 1))n 1 (b 1 b 3 ) 1 a 1 n. Let V 1 be the followng: V 1 W W t b 1 W t b2 W t b3 W t b4 2kn 2 W t R 1,k W t R 2 W t R3,n. Then, from Proposton 2.2 and (1) of Proposton 3.1, the fundamental group 1 (X V1 ) can be presented wth generators a 2, a 1 and wth relatons a 2 a1 ka 2a 1 ka 2 1ak 1 a 2 1a 1 k 1, where 2 k n 2, a 2 a 1 a 2 a 2a 1 a 2 1a 2 1a 1a 2 1a 1 1 1, (a 2 a 1 ) n 1 a 1 n 1. Let a 2 x and a 1 y. Then t follows that 1 (X V1 ) s somorphc to B n. Therefore, for n 3 we have g(b n ) 4. Thus, the proof of (1) of Theorem 1.2 s completed Proof of (2) of Theorem 1.2. For g 1, let H g be the hyperellptc mappng class group of g, that s, a subgroup of the mappng class group M g whch conssts of elements commutatve wth a hyperellptc nvoluton. It s well known that there s the natural epmorphsm B 2g2 H g. For g 2, Brman and Hlden [2] gave a presentaton of the group H g wth generators 1,, 2g1 and wth relatons j 1 j 1 1, where 1 j 1 2g, , where 1 2g, ( 1 2g1 ) 2g2 1, ( 1 2g1 2g1 1 ) 2 1, [ 1 2g1 2g1 1, 1 ] 1. Smlarly to Subsecton 4.1, let x 1 and y 1 2g1. Then, note that y 2g2 1. We calculate 1 2g1 2g1 1 y(y 2g xy 2g ) (yxy 1 )x y 2g1 (xy 1 ) 2g x y 1 (xy 1 ) 2g x (y 1 x) 2g1. Then we have ( 1 2g1 2g1 1 ) 2 (y 1 x) 4g2. In addton, we have [ 1 2g1 2g1 1, 1 ] (y 1 x) 2g1 x(x 1 y) 2g1 x 1 (y 1 x) 2g1 (yx 1 ) 2g1. Therefore, H g can be presented wth generators x, y and wth relatons

15 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 365 Fg. 9. The loop R 6. xy k xy k x 1 y k x 1 y k 1, where 2 k 2g, xyxy 1 xyx 1 y 1 x 1 yx 1 y 1 1, (xy) 2g1 y 2g 2 1, y 2g2 1, (y 1 x) 4g2 1, (y 1 x) 2g1 (yx 1 ) 2g1 1. We now prove (2) of Theorem 1.2. Proof of (2) of Theorem 1.2. For g 2, snce H g s generated by two generators x, y, we have g(h g ) 2 from (2) of Theorem 2.3 (cf. [8]). Therefore, we prove g(h g ) 4 for g 2. Let R 4, R 5 and R 6 be smple closed curves on 4 descrbed n 1 ( 4 ), up to conjugaton, as follows R 4 a 2g2 1 (b 1 1), R 5 (a 2) 4g2 (b 1 1), R 6 (a 2) 2g1 (b 2 b 3 b 4 )(a 4 1a 3) 2g1 (b 3 1). For the loop R 6, see Fg. 9. Note that R 4, R 5 and R 6 ntersect B 2, B 1 and B 4 transversely at only one pont, respectvely. Let V 2 be the followng: V 2 W W t b 1 W t b2 W t b3 W t b4 2k2g W t R 1,k W t R 2 W t R3,2g2 W t R4 W t R5 W t R6. Then, from Proposton 2.2 and (1) of Proposton 3.1, the fundamental group 1 (X V2 ) can be presented wth generators a 2, a 1 and wth relatons a 2 a1 ka 2a 1 ka 2 1ak 1 a 2 1a 1 k 1, where 2 k 2g, a 2 a 1 a 2 a 2a 1 a 2 1a 2 1a 1a 2 1a 1 1 1, (a 2 a 1 ) 2g1 a 2g 2 1 1, a 2g2 1 1, (a 1 1 a 2) 4g2 1, (a 2) 2g1 (a 1 a 1 2 )2g1 1. Let a 2 x and a 1 y. Then t follows that 1 (X V2 ) s somorphc to H g. Therefore, for g 2 we have g(h g ) 4. In partcular, snce the group H 1 s somorphc to M 1,

16 366 R. KOBAYASHI we have 2 g(h 1 ) 4 from (3) of Theorem 2.3 (cf. [8]). Thus, the proof of (2) of Theorem 1.2 s completed Proof of (3) of Theorem 1.2. For n 3, let M 0,n denote the mappng class group of an n-punctured sphere, that s, the group of sotopy classes of orentatonpreservng dffeomorphsms S 2 Ò {p 1,, p n } S 2 Ò {p 1,, p n }. Magnus [9] gave a presentaton of the group M 0,n wth generators 1,, n 1 and wth relatons j 1 j 1 1, where 1 j 1 n 2, , where 1 n 2, ( 1 n 1 ) n 1, 1 n 1 n Smlarly to Subsecton 4.1 and 4.2, let x 1 and y 1 n 1. Then M 0,n can be presented wth generators x, y and wth relatons xy k xy k x 1 y k x 1 y k 1, where 2 k n 2, xyxy 1 xyx 1 y 1 x 1 yx 1 y 1 1, (xy) n 1 y n 1, y n 1, (y 1 x) n 1 1. We now prove (3) of Theorem 1.2. Proof of (3) of Theorem 1.2. For n 3, snce M 0,n s generated by two generators x, y, we have g(m 0,n ) 2 from (2) of Theorem 2.3 (cf. [8]). Therefore, we prove g(m 0,n ) 4 for n 3. Let R 7 and R 8 be smple closed curves on 4 descrbed n 1 ( 4 ), up to conjugaton, as follows R 7 a1 n(b 1 1), R 8 (a 1 1 a 2) n 1 (b 1 1 ). Note that R 7 and R 8 ntersect B 2 and B 1 transversely at only one pont, respectvely. Let V 3 be the followng: V 3 V 1 W t R 7 W t R8. Then, from Proposton 2.2 and (1) of Proposton 3.1, the fundamental group 1 (X V3 ) can be presented wth generators a 2, a 1 and wth relatons a 2 a1 ka 2a 1 ka 2 1ak 1 a 2 1a 1 k 1, where 2 k n 2, a 2 a 1 a 2 a 2a 1 a 2 1a 2 1a 1a 2 1a 1 1 1, (a 2 a 1 ) n 1 a 1 n 1, a1 n 1, (a 2) n 1 1. Let a 2 x and a 1 y. Then t follows that 1 (X V3 ) s somorphc to M 0,n. Therefore, for n 3 we have g(m 0,n ) 4. Thus, the proof of (3) of Theorem 1.2 s completed.

17 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS Proof of (4) of Theorem 1.2. For n 3, let S n denote the n-symmetrc group. It s well known that the group S n has a presentaton wth generators 1,, n 1 and wth relatons j 1 j 1 1, where 1 j 1 n 2, , where 1 n 2, 2 1, where 1 n 1. Smlarly to Subsecton 4.1, let x 1 and y 1 n 1. Snce y 1 xy 1, 2 1 f and only f x 2 1. Therefore S n can be presented wth generators x, y and wth relatons xy k xy k x 1 y k x 1 y k 1, where 2 k n 2, xyxy 1 xyx 1 y 1 x 1 yx 1 y 1 1, (xy) n 1 y n 1, x 2 1. We now prove (4) of Theorem 1.2. Proof of (4) of Theorem 1.2. For n 3, snce S n s generated by two generators x, y, we have g(s n ) 2 from (2) of Theorem 2.3 (cf. [8]). Therefore, we prove g(s n ) 4 for n 3. Let R 9 be the smple closed curve on 4 descrbed n 1 ( 4 ), up to conjugaton, as follows R 9 a2 2(b 2 1). Note that R 9 ntersects B 4 transversely at only one pont. Let V 4 be the followng: V 4 V 1 W t R 9. Then, from Proposton 2.2 and (1) of Proposton 3.1, the fundamental group 1 (X V4 ) can be presented wth generators a 2, a 1 and wth relatons a 2 a1 ka 2a 1 ka 2 1ak 1 a 2 1a 1 k 1, where 2 k n 2, a 2 a 1 a 2 a 2a 1 a 2 1a 2 1a 1a 2 1a 1 1 1, (a 2 a 1 ) n 1 a 1 n 1, a Let a 2 x and a 1 y. Then t follows that 1 (X V4 ) s somorphc to S n. Therefore, for n 3 we have g(s n ) 4. Thus, the proof of (4) of Theorem 1.2 s completed Proof of (5) of Theorem 1.2. The Artn group s ntroduced by [3]. For n 6, the n-artn group A n assocated to the Dynkn dagram shown n Fg. 1 s defned by a presentaton wth generators 1,, n 1, and wth relatons j 1 j 1 1, where 1 j 1 n 2, , where 1 n 2, , 1 1 1, where 1 n 1 wth 4.

18 368 R. KOBAYASHI Fg. 10.

19 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 369 It s known that there s the natural epmorphsm A 2g1 M g. Smlarly to Subsecton 4.1, let x 1 and y 1 n 1. In addton, let z. Then the group A n can be presented wth generators x, y, z and wth relatons xy k xy k x 1 y k x 1 y k 1, where 2 k n 2, xyxy 1 xyx 1 y 1 x 1 yx 1 y 1 1, (xy) n 1 y n 1, (y 3 xy 3 )z(y 3 xy 3 )z 1 (y 3 x 1 y 3 )z 1 1, z(y 1 xy 1 )z 1 (y 1 x 1 y 1 ) 1, where 1 n 1 wth 4. We now prove (5) of Theorem 1.2. Proof of (5) of Theorem 1.2. Snce A n s generated by three generators x, y and z, we have g(a n ) 2 from (2) of Theorem 2.3 (cf. [8]). Therefore, we prove g(a n ) 5. Let R 1,k, R 2, R 3, R 4 and R 5, be smple closed curves on 5 as shown n Fg. 10, where 2 k n 2 and 2 n 1 wth 4. Note that we can not consder the loop R 5,1. Note that R 1,k, R 2 and R 3 ntersect a transversely at only one pont, for 2 k n 2, and that R 4 and R 5, ntersect b transversely at only one pont, for 2 n 1 wth 4. Loops R 1,k, R 2, R 3, R 4 and R 5, can be descrbed n 1 ( 5 ), up to conjugaton, as follows R 1,k b 5 1(b 2b 3 b 4 ) 1 a2 k(b 3b 4 )b 5 1(b 3b 4 ) 1 a 2 k(b 2b 3 b 4 )b 5 a 4 2k (b 3 1)a 2 kb k 2 a2k 4, where 2k n 2, R 2 b 1 a 2 (b 3 b 4 )b 5 1(b 3b 4 ) 1 a 2 1(b 3b 4 )b 5 1(b 2b 3 b 4 ) 1 a 2 (b 3 b 4 )b 5 (b 3 b 4 ) 1 a 2 1(b 2b 3 b 4 ) b 5 a 2 (b 3 b 4 )b 5 (b 3 b 4 ) 1 a 2 1, R 3 (b 1 (b 2 )a 2 ) n 1 (b 1 (b 2 b 3 b 4 )b 5 )a n2 4 a2 2, R 4 a2 3b 1(b 2 )a4 3a 5 1a 4 3(b 2 1)b 1(b 2 )a4 3a 5a 4 3(b 2 1)b 1 1(b 2)a4 3a 5(a 3 b 3 b 4 ) 1, R 5, a 1 a 1 2 (b 4 )b 1 5 (b 4)a 1 2 a 1 1 (b 1(b 2 b 4 )b 5 )a 1 4 (a 3 b 4 )b 5 (a 2 4 a 2 2 (b 2 ))a 1 2 a 2 (b 1 (b 2 b 3 b 4 )b 5 ) 1, where 2 n 1 wth 4. Let V 5 be the followng: V 5 W W t b 2 W t b3 W t b4 W t R 1,k 2kn 2 W t R 2 W t R3 W t R4 2n 1,4 W t R 5,. 4 Then, from Proposton 2.2 and (2) of Proposton 3.1, the fundamental group 1 (X V5 ) can be presented wth generators b 1, a 2, a 1 and wth relatons b 1 a2 kb 1a 2 kb k 2 b 2 k 1, where 2 k n 2, b 1 a 2 b 1 a 2 1b 1a 2 b 2 1b 2b 2 1 1, (b 1 a 2 ) n 1 a n 2 1, (a2 3b 1a 2 3)a 1(a2 3b 1a 2 3)a 1 1(a3 2 b 2 3)a 1 a 1 (a 1 2 b 1 a 1 2 )a 1 1 (a 1 2 b 1 1 a1 a 1 b 1 a 1 1b , 2 ) 1, where 2 n 1 wth 4, Let b 1 x, a 2 y and a 1 z. Then 1 (X V5 ) s somorphc to A n. Therefore, for n 6 we have g(a n ) 5. Thus, the proof of (5) of Theorem 1.2 s completed.

20 370 R. KOBAYASHI 4.6. Proof of (6) of Theorem 1.2. Proof of (6) of Theorem 1.2. Let n, k 0 be ntegers wth n k 3. At frst, we consder the case n k s even. We put n k 2r. Let A, j and B, j be smple closed curves on nk1 as shown n (a) and (b) of Fg. 11, respectvely, where 1 j r, and let C, j be the smple closed curve on nk1 as shown n (c), (d) and (e) of Fg. 11, where 1, j r. Note that each of A, j, B, j and C, j ntersects a r1 transversely at only one pont. Loops A, j, B, j and C, j can be descrbed n 1 ( nk1 ), up to conjugaton, as follows A, j a a 1 j a 2r 2 a 1 2r j2 (c 1 r1 b 1 r1 B, j b b j b 1 a 2r j2 b 2r j2 a 1 2r j2 (b 1 ), where 1 j r, C, j a b j 1 a 1 a 2r j2 b 2r 1 j2 a 2r 1 j2 (a r1b r1 1 C, b 1 a b a 1 (b r1 1 ), where 1 r. Let V 6 be the followng: V 6 W 1 jr W t A, j r1 c r), where 1 j r, ), where 1, j r and j, 1 jr W t B, j 1, jr W t C, j. Note that we have relatons a r1 1, b r1 1, c r 1 and c r1 1 n 1 (X W ). In addton, we have the relaton a 2r j2 b 2r j2 a 2r 1 j2 b j 1 n 1 (X W ) (see the presentaton of 1 (X W ) n the proof of Proposton 3.1). Then, from Proposton 2.2, the fundamental group 1 (X V6 ) can be presented wth generators a 1, b 1,, a r, b r and wth relatons a a j 1 a 1 a j, where 1 j r, b b j b 1 b 1 j, where 1 j r, a b 1 j a 1 b j, where 1, j r and j, b 1 a b a 1, where 1 r. Namely, 1 (X V6 ) s somorphc to 2r. We next consder the smple closed curve R m on nk1 as shown n Fg. 12, where 1 2r and m 2. Note that R m ntersects a r1 transversely at only one pont. Loops R m conjugaton, as follows R m a m (a 2r 2 b 1 2r 2 a 1 2r 2 a r1b 1 r1 b 1 b m r (a 1 a 1 2r 2 a r1b 1 r1 R m r r Let V 7 be the followng: can be descrbed n 1 ( nk1 ), up to ), where 1 r, ), where 1 r. V 7 V 6 1k W t R m Then, from Proposton 2.2, the fundamental group 1 (X V7 ) s somorphc to n m1 mk. Therefore, f n k s even, we have g( n m1 mk ) n k 1..

21 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 371 Fg. 11.

22 372 R. KOBAYASHI Fg. 12. Next, we consder the case n k s odd. We put n k 2r 1. Let A, j and B, j be smple closed curves on nk1 as shown n (a) and (b) of Fg. 13, respectvely, where 1 j r, and let C, j be the smple closed curve on nk1 as shown n (c), (d) and (e) of Fg. 13, where 1, j r. In addton, let A,r1 and C r1, be smple closed curves on nk1 as shown n (a) and (b) of Fg. 14, where 1 r. Note that each of A, j, B, j and C, j ntersects B 2r2 transversely at only one pont. Loops A, j, B, j and C, j can be descrbed n 1 ( nk1 ), up to conjugaton, as follows A, j a a j 1 a 2r 3 a 2r 1 j3 (c r1 1 b r1 1 ), where 1 j r, A,r1 a a r1 1 (b r2)a 2r 3 (c r2 )a r1, where 1 r, B, j b b j b 1 (b r2 )a 2r j3 b 2r j3 a 2r 1 j3 (b r2 1 b r1c r1 ), where 1 j r, C, j a b j a 1 (b r2 )a 2r j3 b 2r j3 a 2r 1 j3 (b r2 1 b r1c r1 ), where 1, j r and j, C, b 1 a b a 1 (b r1 1 ), where 1 r, C r1, a r1 b a r1 1 (b r2)a 2r 3 b 2r 3 a 2r 1 3 (c r2), where 1 r. Let V 8 be the followng: V 8 W W t b r1 W t A, j W t B, j W t C, j. 1 jr1 1 jr 1r1,1 jr Snce b r1 ntersects B 2r2 transversely at only one pont, we have the relaton b r1 1

23 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 373 Fg. 13.

24 374 R. KOBAYASHI Fg. 14. n 1 (X t W W br1 ) from Proposton 2.2. Hence we have relatons b r2 1 and c r2 1 n 1 (X t W W br1 ). Then, from Proposton 2.2 and the presentaton of 1 (X W ) n the proof of Proposton 3.1, the fundamental group 1 (X V8 ) s somorphc to an abelan generated by a 1, b 1,, a r, b r and a r1. We next consder the smple closed curve R m on nk1 as shown n Fg. 15, where 1 2r 1 and m 2. Note that R m ntersects B 2r2 transversely at only one pont. Loops R m can be descrbed n 1 ( nk1 ), up to conjugaton, as follows R m R m r r a m (a 2r 3 b 1 2r 3 a 1 2r 3 c 1 r1 b 1 r1 b 1 b m r (a 1 a 1 2r 3 c 1 r1 b 1 r1 R m 2r1 2r1 am 2r1 r1 (b 1 Let V 9 be the followng: r1 ). ), where 1 r, ), where 1 r, V 9 V 8 1k W t R m Then, from Proposton 2.2, the fundamental group 1 (X V9 ) s somorphc to n m1 mk. Therefore, f nk s odd, we have g( n m1 mk ) nk1. Moreover, t s mmedately follows from Theorem 2.3 (2) or (5) (cf. [8]) that g( n m1 mk ) (n k 1)2. Thus, the proof of (6) of Theorem 1.2 s completed..

25 LEFSCHETZ FIBRATIONS AND FINITELY PRESENTED GROUPS 375 Fg. 15.

26 376 R. KOBAYASHI ACKNOWLEDGEMENT. The author would lke to express thanks to Susumu Hrose and Naoyuk Monden for ther valuable suggestons and useful comments. References [1] J. Amorós, F. Bogomolov, L. Katzarkov, T. Pantev: Symplectc Lefschetz fbratons wth arbtrary fundamental groups, J. Dfferental Geom. 54 (2000), [2] J.S. Brman and H.M. Hlden: On the mappng class groups of closed surfaces as coverng spaces; n Advances n the Theory of Remann Surfaces (Proc. Conf., Stony Brook, N.Y., 1969), Ann. of Math. Studes 66, Prnceton Unv. Press, Prnceton, NJ, 1971, [3] E. Breskorn: De Fundamentalgruppe des Raumes der regulären Orbts ener endlchen komplexen Spegelungsgruppe, Invent. Math. 12 (1971), [4] S.K. Donaldson: Lefschetz fbratons n symplectc geometry, Doc. Math. 1998, [5] R.E. Gompf: A new constructon of symplectc manfolds, Ann. of Math. (2) 142 (1995), [6] R.E. Gompf and A.I. Stpscz: 4-Manfolds and Krby Calculus, Graduate Studes n Mathematcs 20, Amer. Math. Soc., Provdence, RI, [7] M. Korkmaz: Noncomplex smooth 4-manfolds wth Lefschetz fbratons, Internat. Math. Res. Notces (2001), [8] M. Korkmaz: Lefschetz fbratons and an nvarant of fntely presented groups, Internat. Math. Res. Notces (2009), [9] W. Magnus: Über Automorphsmen von Fundamentalgruppen berandeter Flächen, Math. Ann. 109 (1934), Department of Mathematcs Faculty of Scence and Technology Tokyo Unversty of Scence 2641 Yamazak, Noda Chba Japan Current address: General Educaton Ishkawa Natonal College of Technology Tsubata, Ishkawa, Japan e-mal: kobayash_ryoma@shkawa-nct.ac.jp