Generating the mapping class group of a surface by torsion Kazuya - PowerPoint PPT Presentation

Generating the mapping class group of a surface by torsion Kazuya Yoshihara 2016/12/22

Orientable surface Σ g,n : a closed orientable surface of genus g with arbitrarily chosen n points. we call punctures P = { x 1 , x 2 , · · · , x n } Diff + (Σ g,n ) := { f : Σ g,n → Σ g,n | orientation preserving differomorphism, f ( P ) = P } Diff 0 (Σ g,n ) := { f ∈ Diff + (Σ g,n ) | f is isotopic to identity } Mod(Σ g,n ) := Diff + (Σ g,n ) / Diff 0 (Σ g,n ) : the mapping class group of Σ g,n a 1 a 2 a 3 a g x 1 x 2 c 1 c 2 b 1 b 2 b 3 b g x n closed orientable surface

Dehn twist a : simple closed curve on Σ g,n . t a := the Dehn twist along a a t a → The Dehn twist along a

relation for Mod(Σ g,n ) Lemma 1.1 a : a simple closed curve on Σ g,n For f ∈ Mod(Σ g,n ) , ft a f − 1 = t f ( a ) . an ordered set of c 1 , c 2 , . . . , c n of simple closed curves on Σ g forms n -chain ⇐ ⇒ c i and c i +1 intersect transversely at one point for i = 1 , 2 , . . . , n − 1 and c i is disjoint from c j if | i − j |≥ 2 . If n is odd, the boundary of regular neighborhood of n -chain has two components d 1 and d 2 . Lemma 1.2 { c 1 , c 2 , c 3 , c 4 , c 5 } : chain on Σ g,n we have following relation. ( t c 1 t c 2 t c 3 t c 4 t c 5 ) 6 = t d 1 t d 2

Dehn twist generators Theorem 1.1 (Dehn,1938) Mod(Σ g, 0 ) is generated by finitely many Dehn twists.

Dehn twist generators Theorem 1.1 (Dehn,1938) Mod(Σ g, 0 ) is generated by finitely many Dehn twists. Theorem 1.2 (Lickorish, 1961) Mod(Σ g, 0 ) is generated by 3 g − 1 Dehn twists t a 1 , t a 2 , . . . , t a g , t b 1 , t b 2 , . . . , t b g , t c 1 , t c 2 , . . . , t c g − 1 .

Dehn twist generators Theorem 1.1 (Dehn,1938) Mod(Σ g, 0 ) is generated by finitely many Dehn twists. Theorem 1.2 (Lickorish, 1961) Mod(Σ g, 0 ) is generated by 3 g − 1 Dehn twists t a 1 , t a 2 , . . . , t a g , t b 1 , t b 2 , . . . , t b g , t c 1 , t c 2 , . . . , t c g − 1 . Theorem 1.3 (Humphries, 1979) Mod(Σ g, 0 ) is generated by 2 g + 1 Dehn twists t a 1 , t a 2 , t b 1 , t b 2 , . . . , t b g , t c 1 , t c 2 , . . . , t c g − 1 . This is the minimum number of Dehn twists generating Mod(Σ g, 0 ) .

Involution generators Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ g, 0 ) is generated by infinitely many involutions.

Involution generators Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ g, 0 ) is generated by infinitely many involutions. Theorem 1.5 (Luo, 1998) Mod(Σ g,n ) is generated by finitely many involutions.

Involution generators Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ g, 0 ) is generated by infinitely many involutions. Theorem 1.5 (Luo, 1998) Mod(Σ g,n ) is generated by finitely many involutions. Theorem 1.6 (Brendle-Farb, 2004) Mod(Σ g,n ) is generated by 6 involutions. ( g ≥ 3 , n = 0 or g ≥ 4 , n ≤ 1)

Involution generators Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ g, 0 ) is generated by infinitely many involutions. Theorem 1.5 (Luo, 1998) Mod(Σ g,n ) is generated by finitely many involutions. Theorem 1.6 (Brendle-Farb, 2004) Mod(Σ g,n ) is generated by 6 involutions. ( g ≥ 3 , n = 0 or g ≥ 4 , n ≤ 1) Theorem 1.7 (Kassbov, 2003) (1) Mod(Σ g,n ) is generated by 4 involutions. ( g ≥ 8) (2) Mod(Σ g,n ) is generated by 5 involutions. ( g ≥ 6) (3) Mod(Σ g,n ) is generated by 6 involutions. ( g ≥ 4)

Involution generators Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ g, 0 ) is generated by infinitely many involutions. Theorem 1.5 (Luo, 1998) Mod(Σ g,n ) is generated by finitely many involutions. Theorem 1.6 (Brendle-Farb, 2004) Mod(Σ g,n ) is generated by 6 involutions. ( g ≥ 3 , n = 0 or g ≥ 4 , n ≤ 1) Theorem 1.7 (Kassbov, 2003) (1) Mod(Σ g,n ) is generated by 4 involutions. ( g ≥ 8) (2) Mod(Σ g,n ) is generated by 5 involutions. ( g ≥ 6) (3) Mod(Σ g,n ) is generated by 6 involutions. ( g ≥ 4) Theorem 1.8 (Monden, 2008) (1) Mod(Σ g,n ) is generated by 4 involutions. ( g ≥ 7) (2) Mod(Σ g,n ) is generated by 5 involutions. ( g ≥ 5)

Torsion generator Theorem 1.9 (Brendle-Farb, 2004) When g ≥ 3 , Mod(Σ g, 0 ) is generated by three elements of order 2 g + 2 , 4 g + 2 , 2 .

Torsion generator Theorem 1.9 (Brendle-Farb, 2004) When g ≥ 3 , Mod(Σ g, 0 ) is generated by three elements of order 2 g + 2 , 4 g + 2 , 2 . Theorem 1.10 (Korkmaz, 2004) Mod(Σ g, 0 ) is generated by two elements of order 4 g + 2 .

Torsion generator Theorem 1.9 (Brendle-Farb, 2004) When g ≥ 3 , Mod(Σ g, 0 ) is generated by three elements of order 2 g + 2 , 4 g + 2 , 2 . Theorem 1.10 (Korkmaz, 2004) Mod(Σ g, 0 ) is generated by two elements of order 4 g + 2 . Theorem 1.11 (Monden, 2012) When g ≥ 3 , (1) Mod(Σ g, 0 ) is generated by three elements of order 3 . (2) Mod(Σ g, 0 ) is generated by four elements of order 4 .

Torsion generator Theorem 1.9 (Brendle-Farb, 2004) When g ≥ 3 , Mod(Σ g, 0 ) is generated by three elements of order 2 g + 2 , 4 g + 2 , 2 . Theorem 1.10 (Korkmaz, 2004) Mod(Σ g, 0 ) is generated by two elements of order 4 g + 2 . Theorem 1.11 (Monden, 2012) When g ≥ 3 , (1) Mod(Σ g, 0 ) is generated by three elements of order 3 . (2) Mod(Σ g, 0 ) is generated by four elements of order 4 . Theorem 1.12 (Du, 2015) (1) When g ≥ 4 , Mod(Σ g, 0 ) is generated by three involutions and a element of order 3 . (2) When g ≥ 3 , Mod(Σ g, 0 ) is generated by four involutions and a element of order 3 .

Torsion generator Theorem 1.13 (Y) (1) When g ≥ 10 , Mod(Σ g, 0 ) is generated by three elements of order 6 . (2) When g ≥ 5 , Mod(Σ g, 0 ) is generated by four elements of order 6 .

Torsion generator Theorem 1.13 (Y) (1) When g ≥ 10 , Mod(Σ g, 0 ) is generated by three elements of order 6 . (2) When g ≥ 5 , Mod(Σ g, 0 ) is generated by four elements of order 6 . Theorem 1.14 (Lanier) For k ≥ 5 and g ≥ ( k − 1)( k − 3) , Mod(Σ g, 0 ) is generated by four elements of order k . If k is also a multiple of three, then only three elements of order k are required.

Lantern relation The key idea generating a Dehn twist is to use lantern relation . Lemma 1.3 (lantern relation) Let x 1 and x 2 be simple closed curves as shown in below. Then we have t a 1 t c 1 t c 2 t a 3 = t x 1 t x 2 t a 2 . a 1 a 2 a 3 x 2 x 1 c 1 c 2 Then rewrite lantern relation as follow, t a 1 = ( t x 1 t − 1 c 1 )( t x 2 t − 1 a 3 )( t a 2 t − 1 c 2 ) .

Generating Dehn twist Suppose that we can find elements of order six f and h such that f 4 ( a 2 ) = x 1 , f 2 ( a 2 ) = x 2 , f 4 ( c 2 ) = c 1 , f 2 ( c 2 ) = a 3 and h ( c 2 ) = a 2 . Let k be t c 2 h − 1 t − 1 c 2 . k has order six. Then we have t a 2 t − 1 c 2 = t h ( c 2 ) t − 1 c 2 = ht c 2 h − 1 t − 1 c 2 = hk. c 2 f − 4 = f 4 hkf − 4 . t x 1 t − 1 c 1 = t f 4 ( a 2 ) t − 1 f 4 ( c 2 ) = f 4 t a 2 t − 1 c 2 f − 2 = f 2 hkf − 2 . t x 2 t − 1 a 3 = t f 2 ( a 2 ) t − 1 f 2 ( c 2 ) = f 2 t a 2 t − 1 By Lantern relation, t a 1 = ( f 4 hkf − 4 )( f 2 hkf − 2 )( hk ) . Hence t a 1 is a product of elments of order six.

Construct element of order six I Construct elements f which has order six. Cut the surface Σ g along the curves a 3 , c 1 , c 2 , ϵ 1 , c 4 , c 5 , a 5 i − 3 , c 5 i − 3 , c 5 i − 2 , c 5 i − 1 , c 5 i , a 5 i +1 ( i = 2 , 3 , . . . , g − 5 5 ) , and δ g − 4 as shown in below. a 3 a 7 a 11 b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 b 10 b 11 c 1 c 2 c 4 c 5 c 7 c 8 c 9 c 10 ϵ 1 a g − 8 a g − 4 a g − 3 δ g − 2 a g − 1 b g − 2 b g c g − 8 c g − 7 c g − 6 c g − 5 c g − 3 c g − 1 b g − 8 b g − 7 b g − 6 b g − 5 b g − 4 b g − 3 b g − 1 a ′ δ g − 4 a g g − 2

Construct element of order six I S 1 := Σ 0 , 6 g − 18 5 S j := Σ 0 , 6 s.t. ∂S j = a 5 j − 3 ∪ c 5 j − 3 ∪ c 5 j − 2 ∪ c 5 j − 1 ∪ c 5 j ∪ a 5 j +1 ( j = 2 , 3 , . . . , g − 5 5 ) S ′ 1 := Σ 4 , 1 s.t ∂S ′ 1 = δ g − 4 Let f ′ be π 1 , f 2 , . . . , f g − 5 3 rotation as shown in below. 5

Construct element of order six I 1 ) 6 = t δ g − 4 . Remark that ( f ′ g − 2 ) − 1 ( t a g − 1 t b g − 1 t c g − 1 t b g t a g ) . f ′′ 1 = ( t a g − 3 t b g − 3 t c g − 3 t b g − 2 t a ′ a g − 3 δ g − 2 a g − 1 b g − 2 b g c g − 3 c g − 1 b g − 3 b g − 1 δ g − 4 a ′ a g g − 2 1 ) 6 = t − 1 Note that ( f ′′ δ g − 4 . f ′ 1 , f ′′ 1 , f 2 , . . . , f g − 5 define an element f of order six. 5

Construct element of order six I note that f act the curves as follows. f 4 ( a 2 ) = x 1 , f 2 ( a 2 ) = x 2 , f 4 ( c 2 ) = c 1 , f 2 ( c 2 ) = a 3 . magnification →

Construct element of order six II Construct elements h which has order six. cut the surface Σ g along the curves a 1 , a 2 , c 2 , c 3 , ϵ 2 , ϵ 3 , a 5 i − 5 , c 5 i − 5 , c 5 i − 4 , c 5 i − 3 , c 5 i − 2 , and a 5 i − 1 ( i = 2 , 3 , . . . , g 5 ) as shown in below.

Construct element of order six II T 1 := Σ 0 , 6( g − 5)+12 5 T j := Σ 0 , 6 s.t. ∂T j = a 5 j − 3 ∪ c 5 j − 3 ∪ c 5 j − 2 ∪ c 5 j − 1 ∪ c 5 j ∪ a 5 j +1 ( j = 2 , 3 , . . . , g − 5 5 ) be π Let h 1 , h 2 , . . . , h g +5 3 rotation as follows. 5

Construct element of order six II h 1 , h 2 , . . . , h g +5 define an element h of order six. 5 note that h ( c 2 ) = a 2 .