PDF Satoh, Takao Tokyo University of Science

自由群のFricke_指標環とJohnson_{準同型について}

Satoh, Takao

Tokyo University of Science

Automorphism groups of free groups

• F_n := ⟨x₁, . . . , x_n⟩ : Free group of rank n ≥ 2

• H := F_n/[F_n, F_n] ∼= Z^⊕n : Abelianization of F_n

ρ : Aut F_n −−−→^surj. Aut(H)

=

Nielsen, 1924 GL(n, Z)

• IA_n := Ker(Aut F_n → GL(n, Z))

Free group analogue of the Torelli group

Mapping class groups of surfaces

• Σ_g,1 :=

· · ·

∗

· · ·

π₁(Σ_g,1, ∗) ∼= F_2g

• Mg,1 := Diff⁺(Σ_g,1, ∂)/isotopy

Theorem (Dehn, Nielsen) g ≥ 1

∃ι : Mg,1 ,→ Aut F_2g s.t.,

Im(ι) = {σ ∈ Aut F_2g | ζ^σ = ζ}

• Torelli group

H₁(Σ_g,1, Z) ↶ Ig,1 := IA_2g ∩ Mg,1

trivially

1 −→ IA_2g −→ Aut F_2g −→^ρ GL(2g, Z) −→ 1 x

 ^ιx x

1 −→ Ig,1 −→ Mg,1 −→ Sp(2g, Z) −→ 1

Andreadakis-Johnson filtration of Aut F_n

• Lower central series of F_n

F_n = Γ_n(1) ⊃ Γ_n(2) ⊃ Γ_n(3) ⊃ · · · [Γ_n(k), Γ_n(l)] ⊂ Γ_n(k + l)

Fact (Magnus, Witt, Hall)

Ln(k) := Γ_n(k)/Γ_n(k + 1) is a free abelian group of finite rank.

• Andreadakis-Johnson filtration k ≥ 1

An(k) := Ker(Aut F_n → Aut(F_n/Γ_n(k + 1)))

IA_n = An(1) ⊃ An(2) ⊃ · · ·

Theorem (Andreadakis, 1965) (1) [An(k), An(l)] ⊂ An(k + l) (2) ∩

k≥1

An(k) = 1

Johnson homomorphisms

• H^∗ := Hom_Z(H, Z)

• The k-th Johnson homomorphism of Aut F_n

τ_k : gr^k(An) → Hom_Z(H, Ln(k + 1)) = H^∗ ⊗ Ln(k + 1) σ mod(An(k + 1)) 7→ (x 7→ x⁻¹x^σ)

Fact.

Each of τ_k is injective and GL(n, Z)-equivariant.

S. Morita, R. Hain, F. Cohen, B. Farb, . . .

The first Johnson homomorphisms

Theorem. (Cohen-Pakianathan, Farb, Kawazumi)

τ₁ : gr¹(An) → H^∗ ⊗_Z Ln(2)

is surjective, and the abelianization of IA_n.

Theorem. (Kawazumi) For n ≥ 3,

IA_n → gr¹(An) −→^τ1 (H^∗ ⊗_Z Ln(2)) ⊗_Z Q extends to Aut F_n as a crossed homomorphism.

• Cf. (Day, 2009) An extension of τ_k for k ≥ 1.

• Johnson filtration : Mg,1(k) := A2g(k) ∩ Mg,1

• gr^k(Mg,1) := Mg,1(k)/Mg,1(k + 1)

Theorem. (Johnson, 1983)

τ₁ : gr¹(Mg,1) → Λ³H

detects the free part of the abelianization of Ig,1.

Theorem. (Morita, 1993) For g ≥ 3,

Ig,1 → gr¹(Mg,1) −→^τ1 Λ³H ⊗_Z Q

extends to Mg,1 as a crossed homomorphism.

The first cohomology groups

Theorem. (Morita, 1989) g ≥ 3

H¹(Mg,1, Λ³H) = Z^⊕2

Theorem. (S., 2009) n ≥ 6,

H¹(Aut F_n, H^∗ ⊗_Z Λ²H) = Z^⊕2

Fricke characters of F_n

• R(F_n) := Hom(F_n, SL(2, C))

= {

(a_i, b_i, c_i, d_i)_1≤i≤n ∈ C⁴ⁿ | a_id_i − b_ic_i = 1 }

• F(n, C) := {χ : R(F_n) → C} : C-algebra χ, χ^′ ∈ F(n, C), ρ ∈ R(F_n), λ ∈ C

(χ + χ^′)(ρ) := χ(ρ) + χ^′(ρ) (χχ^′)(ρ) := χ(ρ)χ^′(ρ)

(λχ)(ρ) := λχ(ρ)

• σ ∈ Aut F_n, ρ ∈ R(F_n),

(ρ · σ)(x) := ρ(x^σ−1), x ∈ F_n R(F_n) and F(n, C) ↶ Aut F_n

• Define a Fricke character tr x ∈ F(n, C) by (tr x)(ρ) := tr ρ(x)

for any ρ ∈ R(F_n).

• σ ∈ Aut F_n, (tr x)^σ = tr x^σ

• ρ ∈ R(F_n), (tr 1_Fn)(ρ) = 2

Formulae for tr x

• tr x⁻¹ = tr x

• tr xy = tr yx,

• tr xy + tr xy⁻¹ = (tr x)(tr y)

• tr xyz + tr yxz = (tr x)(tr yz) + (tr y)(tr xz)

+(tr z)(tr xy) − (tr x)(tr y)(tr z)

• 2tr xyzw = (tr x)(tr yzw) + (tr y)(tr zwx) +(tr z)(tr wxy) + (tr w)(tr xyz)

+(tr xy)(tr zw) − (tr xz)(tr yw) + (tr xw)(tr yz)

−(tr x)(tr y)(tr zw) − (tr y)(tr z)(tr xw)

−(tr x)(tr w)(tr yz)

−(tr z)(tr w)(tr xy) + (tr x)(tr y)(tr z)(tr w)

• X_Q(F_n) := ⟨tr x | x ∈ F_n⟩_Q ⊂ F(n, C) : Q-subalgebra

Theorem (Horowitz, 1972) For n ≥ 1, As a ring, X_Q(F_n) is generated by n + (_n

2

) + (_n

3

) elements

• tr x_i, 1 ≤ i ≤ n

• tr x_ix_j, 1 ≤ i < j ≤ n

• tr x_ix_jx_k, 1 ≤ i < j < k ≤ n

• Q-polynomial ring

Q[t] := Q[t_i, t_pq, t_stu | 1 ≤ i ≤ n, 1 ≤ p < q ≤ n,

1 ≤ s < t < u ≤ n]

• π : Q[t] → F(n, C) : ring homomorphism 1 7→ 1

2(tr 1_Fn), t_i1···i

l 7→ tr x_i1 · · · x_i

l

Im(π) = X_Q(F_n)

• I := Ker(π)

= {f ∈ Q[t] | f(tr ρ(x_i1 · · · x_i

l)) = 0, ∀ρ ∈ R(F_n)}

• The ring of Fricke characters of F_n over Q X_Q(F_n) ∼= Q[t]/I

Theorem (Horowitz, 1972) (1) For n = 1, 2, I = (0)

(2) For n = 3, I = (t²₁₂₃ − P₁₂₃(t)t₁₂₃ + Q₁₂₃(t))

P_abc(t) := t_abt_c + t_act_b + t_bct_a,

Q_abc(t) := t²_a + t²_b + t²_c + t²_ab + t²_ac + t²_bc

− t_at_bt_ab − t_at_ct_ac − t_bt_ct_bc + t_abt_bct_ac − 4

• t^′_i

1···i_l := t_i1···i

l − 2 ∈ Q[t]

∀ f ∈ Q[t], f is considered as a polynomial of t^′_i

1···i_ls.

• J₀ := (t^′_i, t^′_pq, t^′_stu | i; p < q; s < t < u) ⊂ Q[t]

I ⊂ J₀, and J := J₀/I ⊂ Q[t]/I.

Lemma. (For n = 3, Magnus)

The ideal J is Aut F_n-invariant.

• A descending filtration

J ⊃ J² ⊃ J³ ⊃ · · ·

of Aut F_n-invariant ideals of Q[t]/I

• gr^k(J) := J^k/J^k+1 : Q-vector space of finite dim.

We want to extract group theoretic properties of Aut F_n from

gr^k(J) := J^k/J^k+1 ↶ Aut F_n.

Theorem (Hatakenaka-S., 2012) (1) A set

T := {t^′_i | 1 ≤ i ≤ n} ∪ {t^′_pq | 1 ≤ p < q ≤ n}

∪ {t^′_stu | 1 ≤ s < t < u ≤ n} is a basis of gr¹(J).

(2) We have obtained a basis of gr²(J).

It seems too hard to write down a basis of gr^k(J) explicitly for k ≥ 3.

New filtration of Aut F_n

• k ≥ 1,

En(k) := Ker(Aut F_n → Aut(J/J^k+1)) Then we have a descending filtration

En(1) ⊃ En(2) ⊃ · · · ⊃ En(k) ⊃ · · ·

Theorem (Hatakenaka-S., 2012)

(1) [En(k), En(l)] ⊂ En(k + l) for any k, l ≥ 1.

(2) En(1) = Inn F_n · An(2).

(3) An(2k) ⊂ En(k).

Graded quotients

• gr^k(En) := En(k)/En(k + 1)

Theorem (Hatakenaka-S., 2012) (1) gr^k(En) is torsion-free.

(2) dim_Q(gr^k(En) ⊗_Z Q) < ∞.

In order to show this, we construct and use a Johnson homomorphism like homomorphism:

η_k : gr^k(En) → Hom_Q(gr¹(J), gr^k+1(J)) σ 7→ (f 7→ f^σ − f)

The main theorem

Theorem (S., 2013) For n ≥ 3,

En(1) → gr¹(En) −→^η1 Hom_Q(gr¹(J), gr²(J)) extends to Aut F_n as a crossed homomorphism.

• We showed that η is non-trivial in H¹.

H¹(Aut F_n, Hom_Q(gr¹(J), gr²(J))) =?

The keypoint of the proof

• We show that there exists a split exact sequence 0 →Hom_Q(gr¹(J), gr²(J))

→ Aut (J/J³) → Aut (J/J²) → 1.

• We obtain a crossed homomorphism

Aut F_n → Aut (J/J³) → Hom_Q(gr¹(J), gr²(J))