On the $\mathrm{SL}(m,\mathbf{C})$-representation algebras of free groups and the Johnson homomorphisms (New transformation groups and its related topics)

On

the

_{\mathrm{S}\mathrm{L}(m\mathrm{C})}

_{‐representation}

_algebras

of free

groups

and the

Johnson

homomorphisms

東京理科大学理学部第二部数学科

佐藤隆夫

*

(Satoh, Takao)

Department

of

_Mathematics,

_Faculty

of

Science

Division

_{\mathrm{I}\mathrm{I}_{-}}

Tokyo University

of Science

Abstract

In this

_article,

we consider certain

_descending

filtrations of the

\mathrm{S}\mathrm{L}(m,

\mathrm{C})-representation

algebras

of free _{groups and} free abelian _groups.

_{By using}

_it, we

introduce

_analogs

of the Johnson

_{homomorphisms}

of the

_automorphism

_groups

of free groups. We show that thefirst

_{homomorphisms}

are extendedto the au‐

tomorphism

groups of free groups as crossed

homomorphisms.

Furthermore we

show that the extended crossed

_{homomorphisms}

induce Kawazumi’s

_cocycles

and

Morita’s

_cocycles.

This worksare

_{generalization}

ofour

_previous

results

[31]

and

[32]

for the

_{\mathrm{S}\mathrm{L}(2, \mathrm{C})}

_{‐representation}

_algebras.

For any

m\geq 2

and _{any group} G, let

R^{m}(G)

be the set

\mathrm{H}\mathrm{o}\mathrm{m}(G, \mathrm{S}\mathrm{L}(m, \mathrm{C}))

of

all

_{\mathrm{S}\mathrm{L}(m, \mathrm{C})}

_{‐representations}

of G. Let

\mathcal{F}(R^{m}(G), \mathrm{C})

be the set

\{ $\chi$ : R^{m}(G)\rightarrow \mathrm{C}\}

of all

_{complex‐valued}

functions on

R^{m}(G)

. Then

\mathcal{F}(R^{m}(G), \mathrm{C})

naturally

has the C‐

algebra

structure

_coming

fromthe

_pointwise

sum and

product.

For anyx\in Gand any

1\leq i, j\leq m

, wedefine anelement

a_{ij}(x)

of

\mathcal{F}(R^{m}(G), \mathrm{C})

tobe

(a_{ij}(x))( $\rho$) :=(i,j)

‐component

of

_$\rho$(x)

for _any

_{$\rho$\in R^{m}(G)}

. We call the map

a_{ij}(x)

the

(i, j)

‐component

function ofx

, or

simply

a

_component

function ofx. Let

\Re_{\mathrm{Q}}^{m}(G)

be the

\mathrm{Q}

‐subalgebra

of \mathcal{F}

(R^{m}(G)

_,

C)

generated by

all

_{a_{ij}(x)}

for x\in G and

_{1\leq i,}

j\leq m

. We call

\Re_{\mathrm{Q}}^{m}(G)

the

\mathrm{S}\mathrm{L}(m,

\mathrm{C})-representation rings

of Gover

Q.

In this

article,

we introduce a

descending

filtration

of

_{\Re_{\mathrm{Q}}^{m}(G)}

_consisting

of Aut G‐invariant

_ideals,

and

_study

the

_{graded quotients}

ofit.

To the best of our

knowledge,

the

study

of the

algebra

\Re_{\mathrm{Q}}^{m}(G)

has a not so

long

history. Classically,

the

_{\mathrm{Q}‐subalgebra}

of

_{\Re_{\mathrm{Q}}^{2}(F_{n})}

_{generated by}

characters of

_{F_{n}}

was

actively

studied. For _any

_{x\in F_{n}}

, the map trx

:=a_{11}(x)+a_{22}(x)

is called the Fricke

character of x. Fricke and Klein

[6]

used the Flricke characters for the

study

of the

classification of Riemann surfaces. In the 1970\mathrm{s}, Horowitz

[11]

and

[12]

investigated

several

_algebraic

_properties

of the

_ring

of Frricke characters

_by

_using

the combinatorial

* _‐address:

group

theory.

In

1980,

Magnus

[19]

studied some relations among Fricke characters of

free groups

systematically.

As is found in Acuna Maria Montesinos’s_paper

_[1],

_today

Magnus’s

research has been

_developed

tothe

_study

of the

_{\mathrm{S}\mathrm{L}(2, \mathrm{C})}

‐character varieties

of free _groups

_{by quite}

_many authors. Let

_{\mathfrak{X}_{\mathrm{Q}}^{2}(F_{n})}

be the

_\mathrm{Q}

_{‐subalgebra}

of

_{\Re_{\mathrm{Q}}^{2}(F_{n})}

generated

by

alltrxfor

x\in F_{n}

. The

ring

\mathfrak{X}_{\mathrm{Q}}^{2}(F_{n})

is called the

ring

of Fricke characters

of

F_{n}

. Let \mathrm{C} be the ideal of

X_{\mathrm{Q}}^{2}(F_{n})

generated

by

trx-2 for any

x\in F_{n}

. In our

previous

papers

[9]

and

[30]

,weconsideredan

application

of the

theory

ofthe Johnson

homomorphisms

of Aut

F_{n}

by using

the Frickecharacters. In

_particular,

wedetermined

the structure of the

_{graded quotients}

_{\mathrm{g}\mathrm{r}^{k}(\mathrm{C}) :=\mathrm{C}^{k}/C^{k+1}}

for

_{1\leq k\leq 2}

, introduced

analogs

of the Johnson

_{homomorphisms,}

and showed that the first

_homomorphism

extendstoAut

_{F_{n}}

as a crossed

homomorphism.

We

_briefly

reviewthe

_history

ofthe Johnson

_{homomorphisms.}

In

_1965,

Andreadakis

[2]

introduced acertain

descending

centralfiltration of Aut

F_{n}

by

using

the naturalac‐

tionof Aut

_{F_{n}}

onthe

nilpotent quotients

of

F_{n}

. Wecallthis filtrationtheAndreadakis‐

Johnsonfiltration of Aut

F_{n}

. In the 1980\mathrm{s}_, Johnsonstudied suchfiltration for

mapping

classgroupsofsurfaces inorderto

_investigate

_{the group}structure_{of the Torelli groups in}

aseriesofworks

[13], [14], [15]

and

[16].

In

particular,

hedetermined the abelianization

of_{the Torelli group}

_by

_introducing

acertain

homomorphism.

Today,

his

homomorphism

is called the first Johnson

_{homomorphism,}

and it is

_generalized

to

_higher

_degrees.

Over

the last two

_decades,

the Johnson

_{homomorphisms}

of the

_mapping

class groups have

been

_actively

studied from various

_viewpoints

_by

_{many authors}

_including

Morita

_[21],

Hain

_[8]

and others.

The Johnson

_{homomorphisms}

are

naturally

defined for Aut

F_{n}

:

\tilde{ $\tau$}_{k}

:

\mathcal{A}_{F_{n}}(k)\mapsto \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Z}}(H, \mathcal{L}_{F_{n}}(k+1))

where H

_{:=\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Z}}(H, \mathrm{Z})}

. So

far,

we concentrate on the

study

of the cokernels of

Johnson

_{homomorphisms}

ofAut

F_{n}

inaseries ofourworks

[25], [26], [28]

and

[5]

with

combinatorial_group

_theory

and

_{representation}

_theory.

SinceeachoftheJohnsonhomo‐

morphisms

is

_{\mathrm{G}\mathrm{L}(n, \mathrm{Z})}

_{‐equivariant injective,}

it is an

important

problem

to determine

its

_images

and cokernels. On the other

_hand,

the

_study

ofthe

_{extendability}

of the

Johnson

_{homomorphisms}

have been received attentions. Morita

_[22]

showed that the

first Johnson

_homomorphism

of the

_mapping

_{class group, which initial domain is the}

Torelli group, canbeextendedtothe

_mapping

classgroupas a crossed

homomorphism

by

using

the extension

_theory

of_groups.

_{Inspired by}

Morita’s

_work,

Kawazumi

_[17]

obtained a

corresponding

results for Aut

F_{n}

by using

the

Magnus

_expansion

of

F_{n}.

Furthermore,

he constructed

_higher

twisted

_cohomology

classes with the extendedfirst

Johnson

_homomorphism

and the_cup

_product.

_{By restricting}

them tothe

_mapping

class

group, he

investigated

relations between the

higher cocycles

and the Morita‐Mumford

classes.

_Recently,

_Day

_[4]

showed that each ofJohnson

_{homomorphisms}

ofAut

F_{n}

can

be extended to acrossed

homomorphism

from Aut

_{F_{n}}

into acertain

finitely generated

free abelian _group.

As mentioned

above,

we

[9]

constructed

analogs

of the Johnson

homomorphisms

with

extended to Aut

F_{n}

in

_[30].

It

_is,

_however,

difficult to

_push

forward with our research

since the structures of the

_{graded quotients}

_{\mathrm{g}\mathrm{r}^{k}(C)}

are too

complicated

to handle. In

[31]

, we consideredasimilarsituationfor the

\mathrm{S}\mathrm{L}(2, \mathrm{C})

‐representation

algebra

\Re_{\mathrm{Q}}^{2}

(Fn).

Inthis

_article,

we

generalize

our

previous

works to the

_{\mathrm{S}\mathrm{L}(m, \mathrm{C})}

_{‐representation}

case.

Set

_{s_{ij}(x) :=a_{ij}(x)-$\delta$_{ij}}

for _any

_{1\leq i,j\leq m}

and

_{x\in F_{n}}

where $\delta$ meansKronecker’s

delta. Let

_{\mathrm{J}_{F_{n}}}

be the ideal of

_{\Re_{\mathrm{Q}}^{m}(F_{n})}

_generated

_by

_{s_{ij}(x_{l})}

for any

1\leq i,j\leq m

and

1\leq l\leq n

. Then the

products

of

\mathrm{J}_{F_{n}}

define a

descending

filtration of

\Re_{\mathrm{Q}}^{m}

(Fn):

\mathrm{J}_{F_{n}}\supset \mathrm{J}_{F_{n}}^{2}\supset \mathrm{J}_{F_{n}}^{3}\supset\cdots,

whichconsists of Aut

_{F_{n}}

‐invariantideals. Set

_{\mathrm{g}\mathrm{r}^{k}(\mathrm{J}_{F_{n}})}

_{:=\mathrm{J}_{F_{n}}^{k}/\mathrm{J}_{F_{n}}^{k+1}}

forany

k\geq 1

. Set

T_{k}:=\displaystyle \{ \prod_{1\leq i,j\leq m,(i,j)\neq(m,m)}\prod_{ $\iota$=1}^{n}s_{i\hat{g}}(x_{l})^{e_{ij,l}}|e_{ij,l}\geq 0, (i,j)\neq(m,m)\sum_{1\leq i,j\leq m}\sum_{l=1}^{n}e_{i\dot{},l}=k\}\subset_{1}\tilde{\int}_{F_{n}}^{k}.

Theorem 1. For each

_{k\geq 1}

, the set

T_{k}

(mod

\mathrm{J}_{F_{n}}^{k+1}

)

forms

a basis

of

\mathrm{g}\mathrm{r}^{k}(\mathrm{J}_{F_{n}})

as a

\mathrm{Q}

‐vector space.

Furthermore, for

any

n\geq 2

and

k\geq 1

, we have

\displaystyle \mathrm{g}\mathrm{r}^{k}(\mathrm{J}_{F_{n}})\cong\oplus' \bigotimes_{1\leq i,\dot{}\leq m,(i,j)\neq(mm)},S^{\mathrm{e}_{ij}}H_{\mathrm{Q}}

as a

\mathrm{G}\mathrm{L}(n, \mathrm{Z})

‐module. Here the sum runs over all

tuples

(e_{ij})

for

1\leq i,

j\leq m

and

(i,j)\neq(m, m)

such that the sum

of

the e_{ij} is

equal

tok.

This theorem isa

_{generalization}

ofour

previous

resultforthe casewhere k=2 in

[31].

Now,

set

D_{F_{n}}^{m}(k) :=\mathrm{K}\mathrm{e}\mathrm{r}(\mathrm{A}\mathrm{u}\mathrm{t}F_{n}\rightarrow \mathrm{A}\mathrm{u}\mathrm{t}(\mathrm{J}_{F_{n}}/\mathrm{J}_{F_{n}}^{k+1}))

.

The groups

_{\mathcal{D}_{F_{n}}^{m}(k)}

define a

descending

central filtration of Aut

F_{n}

. Let

\mathcal{A}_{F_{n}}(1)\supset

\mathcal{A}_{F_{n}}(2)\supset\cdots

be the Andreadakis‐Johnson filtration of Aut

_{F_{n}}

. We show a relation

between

_{\mathcal{A}_{F_{n}}(k)}

and

_{\mathcal{D}_{F_{n}}^{m}(k)}

, and among

\mathcal{D}_{F_{n}}^{m}(k)\mathrm{s}

as follows.

Theorem 2.

(1)

For any

k\geq 1,

_{\mathcal{A}_{F_{n}}(k)\subset \mathcal{D}_{F_{n}}^{m}(k)}

.

(2)

For any

k\geq 1

and

m\geq 2

, we have

D_{F_{n}}^{m+1}(k)\subset \mathcal{D}_{F_{n}}^{m}(k)

.

In

_[31],

weshowed that

\mathcal{D}_{F_{n}}^{2}(k)=\mathcal{A}_{F_{n}}(k)

for

1\leq k\leq 4

.

Hence,

we have

\mathcal{D}_{F_{n}}^{m}(k)=

\mathcal{A}_{F_{n}}(k)

for any

1\leq k\leq 4

.

By

referring

tothe

theory

oftheJohnson

homomorphisms,

we can construct

_analogs

ofthem:

\tilde{ $\eta$}_{k}

:

D_{F_{n}}^{m}(k)\rightarrow \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Q}} (grl

(\mathrm{J}_{F_{n}}),

\mathrm{g}\mathrm{r}^{k+1}(\mathrm{J}_{F_{n}})

).

defined

_by

the

_{corresponding f\mapsto f^{ $\sigma$}-f}

for _any

_{f\in \mathrm{J}_{F_{n}}}

. Inthis

article,

weconsider

an extension of thefirst

homomorphism \tilde{ $\eta$}_{1}

, and

study

somerelations tothe extension

Set

_{H_{\mathrm{Q}}}

:=H\otimes \mathrm{z}

Q.

In

_[24],

we

computed

H^{1}(\mathrm{A}\mathrm{u}\mathrm{t}F_{n}, H_{\mathrm{Q}})=\mathrm{Q}

, and showed that

it is

generated

by

Morita’s

_{cocycle f_{M}}

. On the other

hand,

we

[27]

also

computed

H^{1}

_(Aut

_{F_{n},}

_{H_{\mathrm{Q}}^{*}\otimes_{\mathrm{Q}}$\Lambda$^{2}H_{\mathrm{Q}}}

_{) =\mathrm{Q}^{\oplus 2}}

, and showed that it is

generated by

Kawazumi’s

cocycle f_{K}

and the

_cocycle

induced from

_{f_{M}}

. Kawazumi’s

cocycle

f_{K}

is anextension

of

_{\tilde{ $\tau$}_{1}}

. Thenwe can constructthe crossed

homomorphism

$\theta$_{F_{n}}

:Aut

F_{n}\rightarrow \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Q}}

(grl

(\mathrm{J}_{F_{n}}),

\mathrm{g}\mathrm{r}^{2}(\mathrm{J}_{F_{n}})

)

whichisanextensionof

\tilde{ $\eta$}_{1}

.

By taking

suitable reductions of the

target

of

$\theta$_{F_{n}}

,weobtain

the crossed

_{homomorphisms}

f_{1}

:Aut

F_{n}\rightarrow H_{\mathrm{Q}}^{*}\otimes_{\mathrm{Q}}$\Lambda$^{2}H_{\mathrm{Q}},

f_{2}

:Aut

F_{n}\rightarrow H_{\mathrm{Q}}.

Then weshow the

following.

Theorem 3. For_any

_{n\geq 2,}

f_{K}=f_{1}, f_{M}=-f_{2}+$\delta$_{x}

for

x=x_{1}+x_{2}+\cdots+x_{n}\in H_{\mathrm{Q}}

as crossed

homomorphisms.

This shows that our crossed

homomorphism

induces both ofKawazumi’s

cocycle

and

Morita’s

_cocycle,

andthat

_{$\theta$_{F_{n}}}

defines the non‐trivial

_cohomology

class in

H^{1}

_(Aut

F_{n},

\mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Q}}

(grl

(\mathrm{J}_{F_{n}}),

\mathrm{g}\mathrm{r}^{2}(\mathrm{J}_{F_{n}})

))

.

In

_[10]

and

_[32],

we studied

\mathrm{X}_{\mathrm{Q}}^{2}(H)

and

\Re_{\mathrm{Q}}^{2}(H)

. In this paper, we

generalize

the

results in

_[32]

tothe

_{\mathrm{S}\mathrm{L}(m, \mathrm{C})}

_{‐representation}

cases.

By using

a

parallel

_argument,

we

candefine a

descending

filtration

\mathrm{J}_{H}\supset \mathrm{J}_{H}^{2}\supset \mathrm{J}_{H}^{3}\supset\cdots

of ideals in

\Re_{\mathrm{Q}}^{2}(H)

. In contrast

with thefree group case,

however,

it is a

_quite

hardto determine thestructureofthe

graded

quotients

\mathrm{g}\mathrm{r}^{k}(\mathrm{J}_{H})

.

Here,

wegavebasis of

\mathrm{g}\mathrm{r}^{k}(\mathrm{J}_{H})

for

1\leq k\leq 2

. In

particular,

we see

\mathrm{g}\mathrm{r}^{1}(\mathrm{J}_{H})\cong H_{\mathrm{Q}}^{\oplus m^{2}-1},

\mathrm{g}\mathrm{r}^{2}(\mathrm{J}_{H})\cong(S^{2}H_{\mathrm{Q}})^{\oplus\frac{1}{2}m^{2}(m^{2}-1)}\oplus($\Lambda$^{2}H_{\mathrm{Q}})^{\{\oplus\frac{1}{2}(m^{2}-1)(m^{2}-4)\}}.

Weremark that fora

general

_{m\geq 3}

_,the situation ofthe

\mathrm{S}\mathrm{L}(m, \mathrm{C})

_{‐representation}

case

ismuchmoredifferent and

complicated

thanthoseofthe

\mathrm{S}\mathrm{L}(2, \mathrm{C})

_{‐representation}

case.

At the

_{present stage,}

wehavenoideato

_give

aresultfor a

general

k\geq 3

.

By

using

the

above

_results,

we construct the crossed

homomorphism

$\theta$_{H}

: Aut

F_{n}\rightarrow \mathrm{H}\mathrm{o}\mathrm{m}_{\mathrm{Q}}

(grl

(\mathrm{J}_{H})

,

\mathrm{g}\mathrm{r}^{2}(\mathrm{J}_{H})

),

andshow that itinduces Morita’s

_{cocycle f_{M}.}

References

[1]

F.

_G.‐Acuna;

J. Mariaand

_{Montesinos‐Amilibia;}

Onthecharacter

_variety

ofgroup

[2]

S.

_Andreadakis;

On the

_{automorphisms}

offree _groups and free

_nilpotent

_groups,

Proc. London Math. Soc.

₍₃₎

15

_(1965),

239‐268.

[3]

H. Cartan and S.

_{Eilenberg; Homological Algebra,}

Princeton

_University

Press.

[4]

M.

_Day;

Extensions ofJohnson’s and Morita’s

_{homomorphisms}

thatmap to

finitely

generated

abelian groups, Journal of

_Topology

and

_Analysis,

Vol.

_5,

No. 1

_(2013),

57‐85.

[5]

N. Enomoto and T.

_Satoh,

On the derivation

_algebra

of the free Lie

_algebra

and

trace maps.

Alg.

and Geom.

_Top.,

11

₍₂₀₁₁₎

2861‐2901.

[6]

R. Fricke and F.

_Klein;

_Vorlesungen

über dieTeorie der

_automorphen

Functionen.

Vol.

_1,

365‐370.

_Leipzig:

B.G. Teuber 1897.

[7]

M.

_Hall;

The

_theory

_{of groups,} second

_edition,

AMS Chelsea

Publishing

1999.

[8]

R.

_Hain;

Infinitesimal

_{presentations}

_{of the Torelli group, Journal of the American}

Mathematical

_Society

10

_(1997),

597‐651.

[9]

E. Hatakenaka and T.

_Satoh;

Onthe

_{graded quotients}

ofthe

_ring

ofFricke char‐

actersofa free_group, Journal of

Algebra,

430

_(2015),

94‐118.

[10]

E. Hatakenaka and T.

_Satoh;

On the

_rings

of Fricke characters of free abelian

groups, Journal of Commutative

Algebra,

to_appear.

[11]

R.

_Horowitz;

Characters of free_groups

_represented

inthe two‐dimensional

_special

linear group, Comm. on Pure and

Applied

Math.,

Vol. XXV

(1972),

635‐649.

[12]

R.

_Horowitz;

Induced

_{automorphisms}

on Fricke characters of free _{groups, Trans.}

ofthe Amer. Math. Soc. 208

_(1975),

41‐50.

[13]

D.

_Johnson;

An abelian

quotient

ofthe

_mapping

_{class group, Mathematshe} An‐

nalen 249

_(1980),

225‐242.

[14]

D.

_Johnson;

Thestructureof the_{Torelli group I:} AFiniteSet ofGeneratorsfor

_\mathcal{I},

Annals of

_Mathematics,

2nd Ser.

_118,

No. 3

_(1983),

423‐442.

[15]

D.

_Johnson;

Thestructureof_{the Torelli group II:} A characterization of_{the group}

generated by

twists on

bounding

_curves,

Topology,

24,

No. 2

(1985),

113‐126.

[16]

D.

_Johnson;

Thestructureofthe Torelli_{group III:} The abelianization of\mathcal{I},

Topol‐

ogy 24

(1985),

127‐144.

[17]

N.

_Kawazumi;

_{Cohomological}

_aspects

of

_{Magnus expansions,}

_preprint,

arXiv: math.

\mathrm{G}\mathrm{T}/0505497.

[18]

W.

_Magnus;

Über

n‐dimensinale

Gittertransformationen,

Acta Math. 64

(1935),

[19]

W.

_{Magnus; Rings}

ofFrickecharacters and

_automorphism

groups of free groups,

Math. Z. 170

_(1980),

91‐103.

[20]

W.

_Magnus,

A. Karras and D.

_Solitar;

Combinatorial group

theory,

Interscience

Publ.,

New York

_(1966).

[21]

S.

_Morita;

Abelian

_quotients

of

_subgroups

ofthe

_mapping

classgroup of

surfaces,

Duke Mathematical Journal 70

_(1993),

699‐726.

[22]

S.

_Morita;

The extension of Johnson’s

_homomorphism

from the _{Torelli group} to

the

_mapping

classgroup, Invent. math. 111

(1993),

197‐224.

[23]

J.

_Nielsen;

Die

_{Isomorphismengruppe}

der freien

_Gruppen,

Math. Ann. 91

_(1924),

169‐209.

[24]

T.

_Satoh;

Twisted first

_homology

_{group of the}

_automorphism

_groupofafree_group,

J. ofPureand

_{Appl. Alg.,}

204

_(2006),

334‐348.

[25]

T.

_Satoh;

New obstructions for the

_surjectivity

of the Johnson

_homomorphism

ofthe

_automorphism

_group ofa free_{group, J.} of theLondon Math.

Soc.,

(2)

74

(2006)

, 341‐360.

[26]

T.

_Satoh;

Thecokernel of theJohnson

_{homomorphisms}

ofthe

_automorphism

_group

ofa free metabelian group, Transactions of American Mathematical

Society,

361

(2009)

, 2085‐2107.

[27]

T.

_Satoh;

First

_cohomologies

and the Johnson

_{homomorphisms}

of the automor‐

phism

group of a free _group. Journal of Pure and

Applied Algebra

217

(2013),

137‐152.

[28]

T.

_Satoh,

Onthe lower central seriesof the

_{IA‐automorphism}

_groupofafree_group.

J. ofPure and

_{Appl. Alg.}

216

_(2012),

709‐717.

[29]

T.

_Satoh;

A _survey of the Johnson

_{homomorphisms}

ofthe

_automorphism

_groups

offreegroups and related

topics,

to_{appear in}Handbook ofTeichmüller

_theory

_(A.

Papadopoulos,

ed.)

Volume

_V,

167‐209.

[30]

T.

_Satoh;

The Johnson‐Morita

_theory

for the

_ring

of Fricke characters of free

groups, Pacific Journal of

Math.,

275

(2015),

443‐461.

[31]

T.

_Satoh;

On the universal SL

_{‐representation rings}

of free _{groups, Proc. Edinb.}

Math.

_Soc.,

_{to appear.}

[32]

T.

_Satoh;

On the

_{\mathrm{S}\mathrm{L}(2, \mathrm{C})}

_{‐representation rings}

of free_{abelian groups,}

_preprint.

[33]

T.

_Satoh;

First

_cohomologies

and the Johnson

_{homomorphisms}

of the automor‐

phism

groupsof freegroups

II, preprint.

[34]

H.

_Vog;

Surles invariants fondamentaux des

_équations

différentielles linéaires du