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

Σg,n : a closed orientable surface of genus g with arbitrarily chosen n points. we call punctures P = {x1, x2, · · · , xn} Diff+(Σg,n) := {f : Σg,n → Σg,n |orientation preserving differomorphism, f(P) = P} Diff0(Σg,n) := {f ∈ Diff+(Σg,n) | f is isotopic to identity } Mod(Σg,n) := Diff+(Σg,n)/Diff0(Σg,n) : the mapping class group of Σg,n bg b3 b2 b1 ag a3 a2 a1 c1 c2 x1 x2 xn closed orientable surface

Dehn twist

a : simple closed curve on Σg,n. ta := the Dehn twist along a a ta

→

The Dehn twist along a

relation for Mod(Σg,n)

Lemma 1.1

a : a simple closed curve on Σg,n For f ∈ Mod(Σg,n), ftaf −1 = tf(a). an ordered set of c1, c2, . . . , cn of simple closed curves on Σg forms n-chain ⇐ ⇒ ci and ci+1 intersect transversely at one point for i = 1, 2, . . . , n − 1 and ci is disjoint from cj if | i − j |≥ 2. If n is odd, the boundary of regular neighborhood of n-chain has two components d1 and d2.

Lemma 1.2

{c1, c2, c3, c4, c5} : chain on Σg,n we have following relation. (tc1tc2tc3tc4tc5)6 = td1td2

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σg,0) is generated by finitely many Dehn twists.

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σg,0) is generated by finitely many Dehn twists.

Theorem 1.2 (Lickorish, 1961)

Mod(Σg,0) is generated by 3g − 1 Dehn twists ta1, ta2, . . . , tag, tb1, tb2, . . . , tbg, tc1, tc2, . . . , tcg−1.

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σg,0) is generated by finitely many Dehn twists.

Theorem 1.2 (Lickorish, 1961)

Mod(Σg,0) is generated by 3g − 1 Dehn twists ta1, ta2, . . . , tag, tb1, tb2, . . . , tbg, tc1, tc2, . . . , tcg−1.

Theorem 1.3 (Humphries, 1979)

Mod(Σg,0) is generated by 2g + 1 Dehn twists ta1, ta2, tb1, tb2, . . . , tbg, tc1, tc2, . . . , tcg−1. This is the minimum number of Dehn twists generating Mod(Σg,0).

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987)

Mod(Σg,0) is generated by infinitely many involutions.

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987)

Mod(Σg,0) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σg,n) is generated by finitely many involutions.

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987)

Mod(Σg,0) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σg,n) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σg,n) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987)

Mod(Σg,0) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σg,n) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σg,n) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1)

Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σg,n) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σg,n) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σg,n) is generated by 6 involutions. (g ≥ 4)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987)

Mod(Σg,0) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σg,n) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σg,n) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1)

Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σg,n) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σg,n) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σg,n) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σg,n) is generated by 4 involutions. (g ≥ 7) (2) Mod(Σg,n) is generated by 5 involutions. (g ≥ 5)

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σg,0) is generated by three elements of order 2g + 2, 4g + 2, 2.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σg,0) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σg,0) is generated by two elements of order 4g + 2.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σg,0) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σg,0) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012)

When g ≥ 3, (1) Mod(Σg,0) is generated by three elements of order 3. (2) Mod(Σg,0) is generated by four elements of order 4.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σg,0) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σg,0) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012)

When g ≥ 3, (1) Mod(Σg,0) is generated by three elements of order 3. (2) Mod(Σg,0) is generated by four elements of order 4.

Theorem 1.12 (Du, 2015)

(1) When g ≥ 4, Mod(Σg,0) is generated by three involutions and a element of order 3. (2) When g ≥ 3, Mod(Σg,0) is generated by four involutions and a element of order 3.

Torsion generator

Theorem 1.13 (Y)

(1) When g ≥ 10, Mod(Σg,0) is generated by three elements of order 6. (2) When g ≥ 5, Mod(Σg,0) is generated by four elements of order 6.

Torsion generator

Theorem 1.13 (Y)

(1) When g ≥ 10, Mod(Σg,0) is generated by three elements of order 6. (2) When g ≥ 5, Mod(Σg,0) is generated by four elements of order 6.

Theorem 1.14 (Lanier)

For k ≥ 5 and g ≥ (k − 1)(k − 3), Mod(Σg,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 x1 and x2 be simple closed curves as shown in below. Then we have ta1tc1tc2ta3 = tx1tx2ta2. a1 a2 a3 c1 c2 x1 x2 Then rewrite lantern relation as follow, ta1 = (tx1t−1

c1 )(tx2t−1 a3 )(ta2t−1 c2 ).

Generating Dehn twist

Suppose that we can find elements of order six f and h such that f 4(a2) = x1, f 2(a2) = x2, f 4(c2) = c1, f 2(c2) = a3 and h(c2) = a2. Let k be tc2h−1t−1

c2 . k has order six.

Then we have ta2t−1

c2 = th(c2)t−1 c2 = htc2h−1t−1 c2 = hk.

tx1t−1

c1 = tf4(a2)t−1 f4(c2) = f 4ta2t−1 c2 f −4 = f 4hkf −4.

tx2t−1

a3 = tf2(a2)t−1 f2(c2) = f 2ta2t−1 c2 f −2 = f 2hkf −2.

By Lantern relation, ta1 = (f 4hkf −4)(f 2hkf −2)(hk). Hence ta1 is a product of elments of order six.

Construct element of order six I

Construct elements f which has order six. Cut the surface Σg along the curves a3, c1, c2, ϵ1, c4, c5, a5i−3, c5i−3, c5i−2, c5i−1, c5i, a5i+1 (i = 2, 3, . . . , g−5

5 ), and δg−4 as

shown in below. ϵ1 a3 a7 a11 b1 b2 b3 b4 b5 b6 b7 b8 b9 b10 b11 c1 c2 c4 c5 c7 c8 c9 c10 ag−8 ag−4 ag−3 a′

g−2

ag−1 ag bg−8 bg−7 bg−6 bg−5 bg−4 bg−3 bg−2 bg−1 bg cg−8 cg−7 cg−6 cg−5 cg−3 cg−1 δg−4 δg−2

Construct element of order six I

S1 := Σ0, 6g−18

5

Sj := Σ0,6 s.t. ∂Sj = a5j−3 ∪ c5j−3 ∪ c5j−2 ∪ c5j−1 ∪ c5j ∪ a5j+1 (j = 2, 3, . . . , g−5

5 )

S′

1 := Σ4,1 s.t ∂S′ 1 = δg−4

Let f ′

1, f2, . . . , f g−5

5

be π

3 rotation as shown in below.

Construct element of order six I

Remark that (f ′

1)6 = tδg−4.

f ′′

1 = (tag−3tbg−3tcg−3tbg−2ta′

g−2)−1(tag−1tbg−1tcg−1tbgtag).

ag−3 a′

g−2

ag−1 ag cg−3 cg−1 δg−4 δg−2 bg−3 bg−2 bg−1 bg Note that (f ′′

1 )6 = t−1 δg−4.

f ′

1, f ′′ 1 , f2, . . . , f g−5

5

define an element f of order six.

Construct element of order six I

note that f act the curves as follows. f 4(a2) = x1, f 2(a2) = x2, f 4(c2) = c1, f 2(c2) = a3. magnification →

Construct element of order six II

Construct elements h which has order six. cut the surface Σg along the curves a1, a2, c2, c3, ϵ2, ϵ3, a5i−5, c5i−5, c5i−4, c5i−3, c5i−2,and a5i−1 (i = 2, 3, . . . , g

5) as shown

in below.

Construct element of order six II

T1 := Σ0, 6(g−5)+12

5

Tj := Σ0,6 s.t. ∂Tj = a5j−3 ∪ c5j−3 ∪ c5j−2 ∪ c5j−1 ∪ c5j ∪ a5j+1 (j = 2, 3, . . . , g−5

5 )

Let h1, h2, . . . , h g+5

5

be π

3 rotation as follows.

Construct element of order six II

h1, h2, . . . , h g+5

5

define an element h of order six. note that h(c2) = a2.

Non-orientable surface

Ng,n : a closed non-orientable surface of genus g with n punctures P = {x1, x2, · · · , xn}. Diff(Ng,n) := {f : Ng,n → Ng,n |differomorphism, f(P) = P} Diff0(Ng,n) := {f ∈ Diff(Ng,n) | f is isotopic to identity } Mod(Ng,n) := Diff(Ng,n)/Diff0(Ng,n) : the mapping class group of Ng,n PMod(Ng,n) := {f ∈ Mod(Ng,n) | f(xi) = xi (i = 1, 2, · · · , n)} : the pure mapping class group of Ng,n Symn := symmetric group on n letters We have the exact sequence 1 → PMod(Ng,n) → Mod(Ng,n)

π

→ Symn → 1.

Non-orientable surface

ar ar−1 a1 br br−1 b1 cr−1 cr−2 c1 dr dr−1 d1 e1 eb−1 x1 x2 xn For g = 2r + 1, surface Ng,n ar ar−1 a1 br+1 br br−1 b1 cr cr−1 cr−2 c1 dr dr−1 d1 e1 eb−1 x1 x2 xn For g = 2r + 2, surface Ng,n

simple closed curve on Ng,n

c : a simple closed curve on Ng,n. c is a two-sided ⇔ the regular neighborhood of c is an annulus. c is a one-sided ⇔ the regular neighborhood of c is a M¨

• bius band.
• ne and two-sided simple closed curves on Ng,n

Dehn twist of Mod(Ng,n)

a : two-sided simple closed curve on Ng,n. Then we can define the Dehn twist ta along a.

Lemma 2.1

a : a two-sided simple closed curve on Ng,n. For f ∈ Mod(Ng,n), tϵ

f(a) = ftaf −1

Where, Na := the regular neighborhood of a. f | Na is orientation preserving ⇒ ϵ = 1. f | Na is orientation reversing ⇒ ϵ = −1.

Y-homeomorphism

m : one-sided simple closed curve on Ng,n a : two-sided simple closed curve on Ng,n K := the regular neighborhood of m ∪ a (∼ = (the Klein bottle with one hole) ) Ym,a := the Y-homeomorphism.

→

Ym,a

a m ∂K Y-homeomorphism on K note that Y 2

m,a = t∂K.

Lemma 2.2

(1) Ym−1,a = Ym,a. (2) Ym,a−1 = Y −1

m,a.

(3) For f ∈ Mod(Ng,n), fYm,af −1 = Yf(m),f(a).

Puncture slide

α : one-sided simple closed curve on Ng,n, based at the puncture x M := the regular neighborhood of α ( ∼ = M¨

• bius band with one puncture)

→

vα

α x puncture slide along α on M

Lemma 2.3

For f ∈ Mod(Ng,n), fvαf −1 is the puncture slide of f(x) along f(α).

Generator for Mod(Ng,n)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(Ng,0) is generated by Dehn twists and Y-homeomorphism. (2) Mod(Ng,0) is not generated by Dehn twists.

Generator for Mod(Ng,n)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(Ng,0) is generated by Dehn twists and Y-homeomorphism. (2) Mod(Ng,0) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(Ng,0) is generated by finite generating set. (g ≥ 3)

Generator for Mod(Ng,n)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(Ng,0) is generated by Dehn twists and Y-homeomorphism. (2) Mod(Ng,0) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(Ng,0) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(Ng,n) is generated by finite generating set (g ≥ 3).

Generator for Mod(Ng,n)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(Ng,0) is generated by Dehn twists and Y-homeomorphism. (2) Mod(Ng,0) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(Ng,0) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(Ng,n) is generated by finite generating set (g ≥ 3).

Theorem 2.4 (Szepietowski, 2013 , Hirose, 2016)

Mod(Ng,0) is generated by g Dehn twists and a Y-homeomorphism. Moreover, this generator set is minimal generator set by Dehn twists and Y-homeomorphisms.

Involution generator for Mod(Ng,b)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(Ng,n) is generated by involutions.

Involution generator for Mod(Ng,b)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(Ng,n) is generated by involutions. The cardinality of this set of generating involutions depends on g and n.

Involution generator for Mod(Ng,b)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(Ng,n) is generated by involutions. The cardinality of this set of generating involutions depends on g and n.

Theorem 2.6 (Szepietowski, 2006)

For g ≥ 4, Mod(Ng,0) is generated by 4 involutions.

Involution generator for Mod(Ng,n)

Theorem 3.1 (Y)

Mod(Ng,n) is generated by 8 involutions. (g ≥ 13 and g is odd) Mod(Ng,n) is generated by 11 involutions. (g ≥ 14 and g is even) Suppose that g = 2r + 1, r = 2k and n = 2l + 1. vj := the puncture slide of xj along αj. xj αj

Generator for PMod(Ng,b)

y := the Y-homeomorphism s.t. y2 = tξ ξ S := {a1, a2, · · · , ar, b1, b2, c1, c2, · · · , cr−1, d1, d2, e1, e2, · · · , en − 1}

Theorem 3.2 (Korkmaz, 2002)

PMod(Ng,n) is generated by following elements. (1) tl for l ∈ S. (2) vj for 1 ≤ j ≤ n. (3) y.

involution σ

the next figure gives the involution σ. mirror x1 x2 xl xl+1 xn xn−1 xl+2 br br−1 bk+2 bk+1 bk bk−1 b2 b1 ar ar−1 ak+2 ak+1 ak ak−1 a2 a1 c1 ck−1 ck ck+1 cr−1 The mirror image σ

involution τ

the next figure gives the involution τ. mirror x2 x3 xl+1 x1 xn xn−1 xl+2 br br−1 bk+3 bk+2 bk+1 bk bk−1 b3 b2 ar ar−1 ak+3 ak+2 ak+1 ak ak−1 a3 a2 a1 c2 ck−1 ck ck+1 ck+2 cr−1 The mirror image τ

involution I

We will construct the third involution. Cut the surface along ak+3 ∪ bk ∪ ck ∪ ck+1 ∪ x. ak+3 x ck+1 ck bk S1 := the five holed sphere bounded by ak+3 ∪ bk ∪ ck ∪ ck+1 ∪ x. S2 := Ng−8,b bounded by ak+3 ∪ bk ∪ ck ∪ ck+1 ∪ x.

involution I

the next figure gives the involution I on S1. ak+3 ck ck+1 bk mirror x The mirror image I on S1 the next figure gives the involution I on S2. ak+3 ck ck+1 bk mirror x d2 d1 b2 b1 e1 e2 el el+1 en−1 en−2 The mirror image I on S2 I and I define the involution I on Ng,n.

Generating Dehn twist and puncture slide

ρ1 := τta1. Since τta1τ = t−1

a1 ,

ρ2

1 = τta1τta1 = t−1 a1 ta1 = id.

∴ ρ1 is involution. ρ2 := τv1. Since τ(α1) = α−1

1 ,

τv1τ is the puncture slide of puncture τ(x1) = x1 along τ(α1) = α−1

1 .

∴ τv1τ = v−1

1

ρ2

2 = τv1τv1 = v−1 1 v1 = id.

∴ ρ2 is the involution. α1 αl+1 αn α1 α2 αn

Generating Y-homeomorphism

ξ x1 xn x2 xl xl+1 xl+2 xn−1

Φ

→

m a x1 xn x2 xl xl+1 xl+2 xn−1 mirror

diffeo Φ : Ng,b → Ng,b s.t. ΦyΦ−1 = Ym,a . w := the reflection of the right model in above figure. w(m) = m−1 and w(a) = a−1. wYm,aw = Yw(m),w(a) = Ym−1,a−1 = Y −1

m,a.

W := Φ−1wΦ. ρ3 := Wy. WyW = Φ−1wΦyΦ−1wΦ = Φ−1wYm,awΦ = Φ−1Y −1

m,aΦ = y−1.

Hence, ρ2

3 = WyWy = y−1y = id.

y = W · Wy.

Involution J

the next figure gives the involution J. x1 x2 xl+1 xl+2 xn e1 e2 b1 en−1 el+1 mirror The mirror image J on Ng,n

Generator for Symn

Lemma 3.1

Symn is generated by r1 = (1, n)(2, n − 1) · · · (l, l + 2)(l + 1) r2 = (2, n)(3, n − 1) · · · (l + 1, l + 2)(1) r3 = (2, n − 1)(3, n − 2) · · · (l, l + 2)(1)(l + 1)(n). π(σ) = (1, n)(2, n − 1) · · · (l, l + 2)(l + 1). π(τ) = (2, n)(3, n − 1) · · · (l + 1, l + 2)(1). π(W) = (2, n − 1)(3, n − 2) · · · (l, l + 2)(1)(l + 1)(n).

Coxeter group

G := ⟨g1, g2, . . . , gn | (gigj)mij = 1⟩ we call the group G Coxter group. where mii = 1 and mij ≥ 2 if i ̸= j. mij = ∞ means no relation of the form (gigj)mij.

Cor 3.1

For g ≥ 13 and g is odd, Mod(Ng,n) can be realized as a quotient of a Coxter group on 8 generators. For g ≥ 14 and g is even, Mod(Ng,n) can be realized as a quotient of a Coxter group on 11 generators.

Thank you for your attention.