Generating the mapping class group of a surface by torsion

Generating the mapping class group of a surface by torsion

Kazuya Yoshihara

2016/12/22

Orientable surface

Σ

: a closed orientable surface of genus g with arbitrarily chosen n points.

we call punctures P = {x

, x

, · · · , x

} Diff

(Σ

) := { f : Σ

→ Σ

| orientation preserving differomorphism, f(P ) = P } Diff

(Σ

) := {f ∈ Diff

(Σ

) | f is isotopic to identity }

Mod(Σ

) := Diff

(Σ

)/Diff

(Σ

) : the mapping class group of Σ

b

b

b

b

a

a

a

a

c

c

x

x

x

closed orientable surface

Dehn twist

a : simple closed curve on Σ

. t

:= the Dehn twist along a

a

t

→

The Dehn twist along a

relation for Mod(Σ

)

Lemma 1.1

a : a simple closed curve on Σ

For f ∈ Mod(Σ

),

f t

f

= t

.

an ordered set of c

, c

, . . . , c

of simple closed curves on Σ

forms n-chain

⇐⇒ c

and c

intersect transversely at one point for i = 1, 2, . . . , n − 1 and c

is disjoint from c

if | i − j |≥ 2.

If n is odd, the boundary of regular neighborhood of n-chain has two components d

and d

.

Lemma 1.2

{ c

, c

, c

, c

, c

} : chain on Σ

we have following relation.

(t

t

t

t

t

)

= t

t

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σ

) is generated by finitely many Dehn twists.

Theorem 1.2 (Lickorish, 1961)

Mod(Σ

) is generated by 3g − 1 Dehn twists t

, t

, . . . , t

, t

, t

, . . . , t

, t

, t

, . . . , t

. Theorem 1.3 (Humphries, 1979)

Mod(Σ

) is generated by 2g + 1 Dehn twists t

, t

, t

, t

, . . . , t

, t

, t

, . . . , t

.

This is the minimum number of Dehn twists generating Mod(Σ

).

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σ

) is generated by finitely many Dehn twists.

Theorem 1.2 (Lickorish, 1961)

Mod(Σ

) is generated by 3g − 1 Dehn twists t

, t

, . . . , t

, t

, t

, . . . , t

, t

, t

, . . . , t

.

Theorem 1.3 (Humphries, 1979)

Mod(Σ

) is generated by 2g + 1 Dehn twists t

, t

, t

, t

, . . . , t

, t

, t

, . . . , t

.

This is the minimum number of Dehn twists generating Mod(Σ

).

Dehn twist generators

Theorem 1.1 (Dehn,1938)

Mod(Σ

) is generated by finitely many Dehn twists.

Theorem 1.2 (Lickorish, 1961)

Mod(Σ

) is generated by 3g − 1 Dehn twists t

, t

, . . . , t

, t

, t

, . . . , t

, t

, t

, . . . , t

. Theorem 1.3 (Humphries, 1979)

Mod(Σ

) is generated by 2g + 1 Dehn twists t

, t

, t

, t

, . . . , t

, t

, t

, . . . , t

.

This is the minimum number of Dehn twists generating Mod(Σ

).

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ

) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σ

) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σ

) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1) Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σ

) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σ

) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 7)

(2) Mod(Σ

) is generated by 5 involutions. (g ≥ 5)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ

) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σ

) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σ

) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1) Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σ

) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σ

) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 7)

(2) Mod(Σ

) is generated by 5 involutions. (g ≥ 5)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ

) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σ

) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σ

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

Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σ

) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σ

) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 7)

(2) Mod(Σ

) is generated by 5 involutions. (g ≥ 5)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ

) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σ

) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σ

) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1) Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σ

) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σ

) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 7)

(2) Mod(Σ

) is generated by 5 involutions. (g ≥ 5)

Involution generators

Theorem 1.4 (MacCarthy-Papadopoulus, 1987) Mod(Σ

) is generated by infinitely many involutions.

Theorem 1.5 (Luo, 1998)

Mod(Σ

) is generated by finitely many involutions.

Theorem 1.6 (Brendle-Farb, 2004)

Mod(Σ

) is generated by 6 involutions. (g ≥ 3, n = 0 or g ≥ 4, n ≤ 1) Theorem 1.7 (Kassbov, 2003)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 8) (2) Mod(Σ

) is generated by 5 involutions. (g ≥ 6) (3) Mod(Σ

) is generated by 6 involutions. (g ≥ 4)

Theorem 1.8 (Monden, 2008)

(1) Mod(Σ

) is generated by 4 involutions. (g ≥ 7)

(2) Mod(Σ

) is generated by 5 involutions. (g ≥ 5)

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σ

) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σ

) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012) When g ≥ 3,

(1) Mod(Σ

) is generated by three elements of order 3. (2) Mod(Σ

) is generated by four elements of order 4.

Theorem 1.12 (Du, 2015)

(1) When g ≥ 4, Mod(Σ

) is generated by three involutions and a element of order 3.

(2) When g ≥ 3, Mod(Σ

) is generated by four involutions and a element of order 3.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σ

) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σ

) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012) When g ≥ 3,

(1) Mod(Σ

) is generated by three elements of order 3. (2) Mod(Σ

) is generated by four elements of order 4.

Theorem 1.12 (Du, 2015)

(1) When g ≥ 4, Mod(Σ

) is generated by three involutions and a element of order 3.

(2) When g ≥ 3, Mod(Σ

) is generated by four involutions and a element of order 3.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σ

) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σ

) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012) When g ≥ 3,

(1) Mod(Σ

) is generated by three elements of order 3.

(2) Mod(Σ

) is generated by four elements of order 4.

Theorem 1.12 (Du, 2015)

(1) When g ≥ 4, Mod(Σ

) is generated by three involutions and a element of order 3.

(2) When g ≥ 3, Mod(Σ

) is generated by four involutions and a element of order 3.

Torsion generator

Theorem 1.9 (Brendle-Farb, 2004)

When g ≥ 3, Mod(Σ

) is generated by three elements of order 2g + 2, 4g + 2, 2.

Theorem 1.10 (Korkmaz, 2004)

Mod(Σ

) is generated by two elements of order 4g + 2.

Theorem 1.11 (Monden, 2012) When g ≥ 3,

(1) Mod(Σ

) is generated by three elements of order 3.

(2) Mod(Σ

) is generated by four elements of order 4.

Theorem 1.12 (Du, 2015)

(1) When g ≥ 4, Mod(Σ

) is generated by three involutions and a element of order 3.

(2) When g ≥ 3, Mod(Σ

) is generated by four involutions and a element of order 3.

Torsion generator

Theorem 1.13 (Y)

(1) When g ≥ 10, Mod(Σ

) is generated by three elements of order 6.

(2) When g ≥ 5, Mod(Σ

) is generated by four elements of order 6.

Theorem 1.14 (Lanier)

For k ≥ 5 and g ≥ (k − 1)(k − 3), Mod(Σ

) 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.

Torsion generator

Theorem 1.13 (Y)

(1) When g ≥ 10, Mod(Σ

) is generated by three elements of order 6.

(2) When g ≥ 5, Mod(Σ

) is generated by four elements of order 6.

Theorem 1.14 (Lanier)

For k ≥ 5 and g ≥ (k − 1)(k − 3), Mod(Σ

) 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

and x

be simple closed curves as shown in below. Then we have

t

t

t

t

= t

t

t

. a

a

a

c

c

x

x

Then rewrite lantern relation as follow,

t

= (t

t

)(t

t

)(t

t

).

Generating Dehn twist

Suppose that we can find elements of order six f and h such that f

(a

) = x

, f

(a

) = x

, f

(c

) = c

, f

(c

) = a

and h(c

) = a

. Let k be t

h

t

. k has order six.

Then we have

t

t

= t

t

= ht

h

t

= hk.

t

t

= t

t

= f

t

t

f

= f

hkf

.

t

t

= t

t

= f

t

t

f

= f

hkf

. By Lantern relation,

t

= (f

hkf

)(f

hkf

)(hk).

Hence t

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

along the curves

a

, c

, c

, ϵ

, c

, c

, a

, c

, c

, c

, c

, a

(i = 2, 3, . . . ,

), and δ

as shown in below.

ϵ

a

a

a

b

b

b

b

b

b

b

b

b

b

b

c

c

c

c

c

c

c

c

a

a

a

a

a

a

b

b

b

b

b

b

b

b

b

c

c

c

c

c

c

δ

δ

Construct element of order six I

S

:= Σ

S

:= Σ

s.t. ∂S

= a

∪ c

∪ c

∪ c

∪ c

∪ a

(j = 2, 3, . . . ,

) S

:= Σ

s.t ∂S

= δ

Let f

, f

, . . . , f

be

rotation as shown in below.

Construct element of order six I

Remark that (f

)

= t

.

f

= (t

t

t

t

t

)

(t

t

t

t

t

).

a

a

a

a

c

c

δ

δ

b

b

b

b

Note that (f

)

= t

g−4

. f

, f

, f

, . . . , f

5

define an element f of order six.

Construct element of order six I

note that f act the curves as follows.

f

(a

) = x

, f

(a

) = x

, f

(c

) = c

, f

(c

) = a

.

magnification

→

Construct element of order six II

Construct elements h which has order six. cut the surface Σ

along the curves

a

, a

, c

, c

, ϵ

, ϵ

, a

, c

, c

, c

, c

,and a

(i = 2, 3, . . . ,

) as shown

in below.

Construct element of order six II

T

:= Σ

0,^6(g−5)+12₅

T

:= Σ

s.t. ∂T

= a

∪ c

∪ c

∪ c

∪ c

∪ a

(j = 2, 3, . . . ,

) Let h

, h

, . . . , h

5

be

rotation as follows.

Construct element of order six II

h

, h

, . . . , h

define an element h of order six.

note that h(c

) = a

.

Non-orientable surface

N

: a closed non-orientable surface of genus g with n punctures P = { x

, x

, · · · , x

} . Diff(N

) := {f : N

→ N

|differomorphism, f(P) = P }

Diff

(N

) := {f ∈ Diff(N

) | f is isotopic to identity } Mod(N

) := Diff(N

)/Diff

(N

)

: the mapping class group of N

PMod(N

) := { f ∈ Mod(N

) | f(x

) = x

(i = 1, 2, · · · , n) } : the pure mapping class group of N

Sym

:= symmetric group on n letters We have the exact sequence

1 → PMod(N

) → Mod(N

) →

Sym

→ 1.

Non-orientable surface

a

a

a

b

b

b

c

c

c

d

d

d

e

e

x

x

x

For g = 2r + 1, surface N

a

a

a

b

b

b

b

c

c

c

c

d

d

d

e

e

x

x

x

For g = 2r + 2, surface N

simple closed curve on N

c : a simple closed curve on 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¨ obius band.

one and two-sided simple closed curves on N

Dehn twist of Mod(N

)

a : two-sided simple closed curve on N

. Then we can define the Dehn twist t

along a.

Lemma 2.1

a : a two-sided simple closed curve on N

. For f ∈ Mod(N

), t

= f t

f

Where, N

:= the regular neighborhood of a.

f | N

is orientation preserving ⇒ ϵ = 1.

f | N

is orientation reversing ⇒ ϵ = − 1.

Y-homeomorphism

m : one-sided simple closed curve on N

a : two-sided simple closed curve on N

K := the regular neighborhood of m ∪ a ( ∼ = (the Klein bottle with one hole) ) Y

:= the Y-homeomorphism.

→ ^Y

a m

∂K

Y-homeomorphism on K note that Y

= t

.

Lemma 2.2 (1) Y

= Y

. (2) Y

= Y

.

(3) For f ∈ Mod(N

), f Y

f

= Y

.

Puncture slide

α : one-sided simple closed curve on N

, based at the puncture x M := the regular neighborhood of α ( ∼ = M¨ obius band with one puncture)

→ ^v

α x

puncture slide along α on M Lemma 2.3

For f ∈ Mod(N

), f v

f

is the puncture slide of f(x) along f(α).

Generator for Mod(N

)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(N

) is generated by Dehn twists and Y-homeomorphism.

(2) Mod(N

) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(N

) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(N

) is generated by finite generating set (g ≥ 3).

Theorem 2.4 (Szepietowski, 2013 , Hirose, 2016)

Mod(N

) is generated by g Dehn twists and a Y-homeomorphism.

Moreover, this generator set is minimal generator set by Dehn twists and

Y-homeomorphisms.

Generator for Mod(N

)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(N

) is generated by Dehn twists and Y-homeomorphism.

(2) Mod(N

) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(N

) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(N

) is generated by finite generating set (g ≥ 3).

Theorem 2.4 (Szepietowski, 2013 , Hirose, 2016)

Mod(N

) is generated by g Dehn twists and a Y-homeomorphism.

Moreover, this generator set is minimal generator set by Dehn twists and

Y-homeomorphisms.

Generator for Mod(N

)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(N

) is generated by Dehn twists and Y-homeomorphism.

(2) Mod(N

) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(N

) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(N

) is generated by finite generating set (g ≥ 3).

Theorem 2.4 (Szepietowski, 2013 , Hirose, 2016)

Mod(N

) is generated by g Dehn twists and a Y-homeomorphism.

Moreover, this generator set is minimal generator set by Dehn twists and

Y-homeomorphisms.

Generator for Mod(N

)

Theorem 2.1 (Lickorish, 1963)

(1) Mod(N

) is generated by Dehn twists and Y-homeomorphism.

(2) Mod(N

) is not generated by Dehn twists.

Theorem 2.2 (Chillingworth, 1969)

Mod(N

) is generated by finite generating set. (g ≥ 3)

Theorem 2.3 (Korkmaz, 2002)

Mod(N

) is generated by finite generating set (g ≥ 3).

Theorem 2.4 (Szepietowski, 2013 , Hirose, 2016)

Mod(N

) 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(N

)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(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(N

) is generated by 4 involutions.

Involution generator for Mod(N

)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(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(N

) is generated by 4 involutions.

Involution generator for Mod(N

)

Theorem 2.5 (Szepietowski, 2004)

For g ≥ 1, Mod(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(N

) is generated by 4 involutions.

Involution generator for Mod(N

)

Theorem 3.1 (Y)

Mod(N

) is generated by 8 involutions. (g ≥ 13 and g is odd) Mod(N

) is generated by 11 involutions. (g ≥ 14 and g is even) Suppose that g = 2r + 1, r = 2k and n = 2l + 1.

v

:= the puncture slide of x

along α

.

x

α

Generator for PMod(N

)

y := the Y-homeomorphism s.t. y

= t

ξ

S := { a

, a

, · · · , a

, b

, b

, c

, c

, · · · , c

, d

, d

, e

, e

, · · · , e

− 1 } Theorem 3.2 (Korkmaz, 2002)

PMod(N

) is generated by following elements.

(1) t

for l ∈ S . (2) v

for 1 ≤ j ≤ n.

(3) y.

involution σ

the next figure gives the involution σ.

mirror x

x

x

x

x

x

x

b

b

b

b

b

b

b

b

a

a

a

a

a

a

a

a

c

c

c

c

c

The mirror image σ

involution τ

the next figure gives the involution τ.

mirror x

x

x

x

x

x

x

b

b

b

b

b

b

b

b

b

a

a

a

a

a

a

a

a

a

a

c

c

c

c

c

c

The mirror image τ

involution I

We will construct the third involution.

Cut the surface along a

∪ b

∪ c

∪ c

∪ x.

a

x

c

c

b

S

:= the five holed sphere bounded by a

∪ b

∪ c

∪ c

∪ x.

S

:= N

bounded by a

∪ b

∪ c

∪ c

∪ x.

involution I

the next figure gives the involution I on S

. a

c

c

b

mirror x

The mirror image I on S

the next figure gives the involution I e on S

. a

c

c

b

mirror x

d

d

b

b

e

e

e

e

e

e

The mirror image I e on S

I and I e define the involution I on N

.

Generating Dehn twist and puncture slide

ρ

:= τ t

. Since τ t

τ = t

,

ρ

= τ t

τ t

= t

t

= id.

∴ ρ

is involution.

ρ

:= τ v

.

Since τ(α

) = α

,

τ v

τ is the puncture slide of puncture τ(x

) = x

along τ(α

) = α

.

∴ τ v

τ = v

ρ

= τ v

τ v

= v

v

= id.

∴ ρ

is the involution.

α

α

α

α

α

α

Generating Y-homeomorphism

ξ x₁ xn

x2 x_l

x_l+1

x_l+2 x_n−1

→

ma

x₁ x_n x2

x_l

x_l+1

x_l+2 x_n−1

mirror

diffeo Φ : N

→ N

s.t. ΦyΦ

= Y

.

w := the reflection of the right model in above figure.

w(m) = m

and w(a) = a

.

wY

w = Y

= Y

= Y

. W := Φ

wΦ.

ρ

:= W y.

W yW = Φ

wΦyΦ

wΦ = Φ

wY

wΦ = Φ

Y

Φ = y

. Hence, ρ

= W yW y = y

y = id.

y = W · W y.

Involution J

the next figure gives the involution J.

x

x

x

x

x

e

e

b

e

e

mirror

The mirror image J on N

Generator for Sym

Lemma 3.1

Sym

is generated by

r

= (1, n)(2, n − 1) · · · (l, l + 2)(l + 1) r

= (2, n)(3, n − 1) · · · (l + 1, l + 2)(1)

r

= (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 := ⟨ g

, g

, . . . , g

| (g

g

)

= 1 ⟩ we call the group G Coxter group.

where m

= 1 and m

≥ 2 if i ̸= j.

m

= ∞ means no relation of the form (g

g

)

. Cor 3.1

For g ≥ 13 and g is odd,

Mod(N

) can be realized as a quotient of a Coxter group on 8 generators.

For g ≥ 14 and g is even,

Mod(N

) can be realized as a quotient of a Coxter group on 11 generators.

Thank you for your attention.