28

On the Andreadakis conjecture of
the automorphism group of a free group
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 II,
Tokyo University of Science

Abstract

In this article, we show that the third subgroup of the Andreadakis‐Johnson
filtration of the automorphism group of a free group coincides with the the thirdgroup of the lower central series of the IA‐automorphism group.

The origin of the study of surface automorphisms goes back to the pioneer works byDehn and Nielsen in early 20th century. In particular, they studied the homeomorphismgroups and the mapping class groups of surfaces. In order to describe mapping classesof surfaces, the most basic and standard way is to consider their actions on the homol‐
ogy groups of the surfaces. In most cases, however, such descriptions kill many deep
and important deta of surface automorphisms in general. To describe surface automor‐
phisms completely, Nielsen considered to study the actions of the mapping class groups
on the fundamental groups of surfaces, and obtained plenty of remarkable results.
In general, the fundamental groups are non‐abelian groups, and to deal with non‐abelian groups and their automorphisms is sometimes quite complicated, compared tothe case of finitely generated abelian groups. In order to investigate the automorphismgroups of non‐abelian groups, Andreadakis [1] introduced descending filtrations of theautomorphism groups of groups. Let G be a group, and Aut G its automorphism group.

G=\Gamma_{G}(1)\supset\Gamma_{G}(2)\supset. . . be the lower central series of G . The action of Aut G
on the (k+1)-st nilpotent quotient group G/\Gamma_{G}(k+1) induces the homomorphism
Aut Garrow Aut (G/\Gamma_{G}(k+1)) . Its kernel is denoted by \mathcal{A}_{G}(k) . Then we have theLetdescending filtration
Aut G\supset \mathcal{A}_{G}(1)\supset \mathcal{A}_{G}(2)\supset. . .

of Aut G . We call this filtration the Andreadakis‐Johnson filtration. (We will explain
why we attach the name “Johnson” in the next paragraph.) Andreadakis [1] showedthat this filtration is central. More precisely, the commutator subgroup of \mathcal{A}_{G}(k) and\mathcal{A}_{G}(l) is contained in \mathcal{A}_{G}(k+l) for any k, l\geq 1 . Hence, each of the graded quotient\mathcal{A}_{G}(k)/\mathcal{A}_{G}(k+1) for k\geq 1 is an abelian group. The graded quotients \mathcal{A}_{G}(k)/\mathcal{A}_{G}(k+1)
are considered to be a sequence of approximations of Aut G by abelian groups, and are*e ‐address:

takao@rs.tus.ac.jp

29
one of powerful tools to study the group structure of Aut G . If G is finitely generated,then so is \mathcal{A}_{G}(k)/\mathcal{A}_{G}(k+1) for any k\geq 1 . However, to determine the structure of it isquite a difficult problem in general.
Here we remark Johnson’s results on mapping class groups of surfaces in the 1980s.
The mapping class group of a compact oriented surface with one boundary component
can be embedded into the automorphism group of a free group by classical works of
Dehn and Nielsen in the 1910s and in early 1920s respectively. Hence we can consider
the descending filtration of the mapping class group by restricting the Andreadakis‐
Johnson filtration to it. This filtration is called the Johnson filtration of the mappingclass group, and it became famous among topologists. The first subgroup of the Johnson
filtration is called the Torelli group. In the 1980s , Johnson studied the group structureof the Torelli group in a series of works [15], [16], [17] and [18]. In particular, he gave afinite set of generators of the Torelli group, and constructed a certain homomorphism\tau to determine the abelianization of it. Today, his homomorphism \tau is called the first
Johnson homomorphism, and it is generalized to Johnson homomorphisms of higher
degrees. Over the last two decades, good progress was made in the study of the Johnson
homomorphisms of mapping class groups through the works of many authors including

Morita [24, 25, 26], Hain [12] and others. The definition of the Johnson homomorphismsof the mapping class group can be easily generalized to those of Aut G for any group G.
To put it plainly, the Johnson homomorphisms are powerful tools to study the graded

quotients \mathcal{A}_{G}(k)/\mathcal{A}_{G}(k+1) of the Andreadakis‐Johnson filtration of Aut G. (For details,
see our survey papers [36] and [35].)
Now, the first subgroup \mathcal{A}_{G}(1) is called the IA‐automorphism group of G , and usu‐
ally denoted by IA(G) . The letters I and A stands for “Identity” and “Automorphism”

respectively due to Bachmuth [2]. Let \mathcal{A}_{G}'(1)\supset \mathcal{A}_{G}'(2)\supset. . . be the lower central seriesof IA(G) . Since the Andreadakis‐Johnson filtration is central, \mathcal{A}_{G}(k) contains \mathcal{A}_{G}'(k)for each k\geq 1 . Then, it is a natural question to ask: How much is the difference
between \mathcal{A}_{G}(k) and \mathcal{A}_{G}'(k) ? Andreadakis focused his interests on the case where G isa free group, and studied the above question.

Let F_{n} be the free group of rank n with basis x_{1} , . . . , x_{n} . Nielsen [27] showed that
IA(F_{2}) coincides with the inner automorphism group of F_{2} . In 1935, Magnus [21]showed that IA(F_{n}) is finitely generated. The group structure of IA(F_{n}) is, however,less well‐understood in general at the moment. Krstič and McCool [20] showed that
IA(F_{3}) is not finitely presentable. For n\geq 4 , it is not known whether IA(F_{n}) is finitely
presentable or not. We should remark that Day and Putman [10] obtained an infinite
presentation for IA(F_{n}) One of reasons why the study of IA(F_{n}) has not achieved good
progress so much seems that the combinatorial complexity increases quite rapidly as ntends to large.

Andreadakis [1] showed that each of \mathcal{A}_{F_{n}}(k)/\mathcal{A}_{F_{n}}(k+1) is free abelian group offinite rank, and gave its rank for k=1 by using Magnus’s generators. He also showedthat \mathcal{A}_{F_{2} (k)=\mathcal{A}_{F_{2} '(k) for any k\geq 1 , and \mathcal{A}_{F_{3} (k)=\mathcal{A}_{F_{3} '(k) for 1\leq k\leq 3 . Then heconjectured that \mathcal{A}_{F_{n}}(k)=\mathcal{A}_{F_{n}}'(k) for any n\geq 3 and k\geq 1 . Today, this conjecture

is called the Andreadakis conjecture. So far, in our previous works [32, 37], we proved

30
that the Andreadakis conjectures restricted to some subgroups are true. However, toattack the original problem is quite difficult due to combinatorial complexities. For any
n\geq 2 , Bachmuth [3] showed that\mathcal{A}_{F_{n}}(2)=\mathcal{A}_{F_{n}}'(2) . This fact is also obtained fromthe independent works by Cohen‐Pakianathan [7, 8], Farb [11] and Kawazumi [19] who
determined the abelianization of IA_{n} . Pettet [28] showed \mathcal{A}_{F_{n} '(3) has at most finite index
in \mathcal{A}_{F_{n} (3) by using the representation theory of the general linear group. Recently,
Bartholdi [4, 5] showed that this conjecture is not true in general. In particular, heshowed that\mathcal{A}_{3}(4)/\mathcal{A}_{3}'(4)\cong(Z/2Z)^{\oplus 14}\oplus(Z/3Z) ^{\oplus 3},\mathcal{A}_{3}(5)/\mathcal{A}_{3}'(5)\cong Z^{\oplus 3}\oplus( torsions)by using a computer.

For a general n\geq 4 , the conjecture is still open.

Here weshould remark a remarkable result due to Darné [9]. Recently, he proved that thestable Andreadakis conjecture is true. Namely, the natural map \mathcal{A}_{F_{n}}'(k)/\mathcal{A}_{F_{n}}'(k+1)arrow\mathcal{A}_{F_{n}}(k)/\mathcal{A}_{F_{n}}(k+1) induced from the inclusion is surjective for sufficiently large n . The
purpose of the paper is to consider the unstable case, and to improve the above Pettet’s
result. More precisely, we show the following.
Theorem 1. For any n\geq 3, \mathcal{A}_{F_{n} '(3)=\mathcal{A}_{F_{n} (3) .

In [31, 36], we showed this type of theorems for some quotient groups of the McCool
subgroup of IA(F_{n}) . The above main theorem is the refinement of these previous results.
Here we recall some results about the Johnson filtration of the mapping class groups.
Let \Sigma_{g,1} be the compact oriented surface of genus g\geq 1 with one boundary component,
and \mathcal{M}_{g,1} its mapping class group. The fundamental group of \Sigma_{g,1} is a free group F_{2g} ofrank 2g . Let \mathcal{M}_{g,1}(1)\supset \mathcal{M}_{g,1}(2)\supset. . . be the Johnson filtration of \mathcal{M}_{g,1} . Namely, if we
consider \mathcal{M}_{g,1} as a subgroup of Aut F_{2g} through the Dehn‐Nielsen embedding, we have\mathcal{M}_{g,1}(k)=\mathcal{A}_{F_{2g}}(k)\cap \mathcal{M}_{g,1} . The subgroup \mathcal{M}_{g,1}(1) is the Torelli group. Let \mathcal{M}_{g,1}'(1)\supset
be the lower central series of the Torelli group. So far, it is known that\mathcal{M}_{g,1}'(2)\supset
the Johnson filtration does not coincide with the lower central series of the Torelli group.

In fact, Johnson [18] determined the abelianization of \mathcal{M}_{g,1}(1) , and showed that it has
many direct summands of Z/2Z by using the Birman‐Craggs homomorphism. From
this, it immediately follows that \mathcal{M}_{g,1}(2)\neq \mathcal{M}_{g,1}'(2) . In addition to this, we should

remark Morita’s remarkable result. Morita [23] showed that (\mathcal{M}_{g,1}(3)/\mathcal{M}_{g,1}'(3))\otimes_{Z}Q is

not trivial for g\geq 3 by using the Casson invariants. Thus, the Andreadakis conjecture
restricted the mapping class group never holds for topological reasons.
To the best of our knowledge, it seems that there are very few results on the original
Andreadakis conjecture for general n\geq 3 and k\geq 4 . It seems to be important to givea theoretical proof if the conjecture is true. In addition to this, if the conjecture is nottrue, it also seems to be interesting to describe obstructions in algebraic way with the
combinatorial group theory, the representation theory, and so on.
In this section, we introduce a strategy to study the difference between the An‐
dreadakis Johnson filtration and the lower central series of IA_{n} by using combinatorialgroup theory. Basically, it is the same method as in our previous work [37] for the
lower‐triangular automorphism groups of free groups.

31
31
The strategy to attack the conjecture
Let n\geq 3 and fix it. For k=1 and 2, the conjecture is true. We assume that\mathcal{A}_{n}(k)=\mathcal{A}_{n}'(k) for k\geq 1 . Then we have the homomorphism gr^{k}(\mathcal{A}_{n}')arrow gr^{k}(\mathcal{A}_{n})induced from the natural inclusion map. If the conjecture is true, it suffices to showthat this map is injective. In fact, if this map is injective, we can conclude \mathcal{A}_{n}(k+1)=\mathcal{A}_{n}'(k+1) from the fact that \mathcal{A}_{n}(k)=\mathcal{A}_{n}'(k) and \mathcal{A}_{n}'(k+1)\subset \mathcal{A}_{n}(k+1) . In general,
however, it is quite difficult to study the structure of \mathcal{A}_{n}'(k)/\mathcal{A}_{n}'(k+1) directly. Thus,

we use the Johnson homomorphisms. By using our previous result obtained in [30], we
determined the cokernel of the composition map\tau_{k}':gr^{k}(\mathcal{A}_{n}')arrow gr^{k}(\mathcal{A}_{n})arrow H^{*} \otimes_{Z}\mathcal{L}_{n}^{(}k+1)
for m\geq k+2 . More precisely, let C_{n}(k) be the quotient module of H^{\otimes k} by the actionof cyclic group of order k on the components:\mathcal{C}_{n}(k) :=H^{\otimes k}/\langle\alpha_{1}\otimes a_{2} \otimes\cdots\otimes a_{k}-a_{2}\otimes a_{3}\otimes\cdots\otimes a_{k}\otimes a_{1}|a_{i}\in H\rangle.
In [33], we showed that Coker(\tau_{k}')=C_{n}(k) for any k\geq 2 and n\geq k+2 . Furthermore,
recently, Darné showed that the natural map gr^{k}(\mathcal{A}_{n}')arrow gr^{k}(\mathcal{A}_{n}) is surjective for n\geq
k+2 . This induces that

Coker(\tau_{k})=C_{n}(k)for n\geq k+2 . This means that we can give a lower bound on the number of generators ofgr^{k}(\mathcal{A}_{n}') by considering rank_{Z}({\rm Im}(\tau_{k})) . Thus, if we want to give an affirmative answer tothe conjecture, it suffices to show that gr^{k}(\mathcal{A}_{n}') is generated by rank_{Z}({\rm Im}(\tau_{k})) elements.

Let us consider the case wherek=2 .

Pettet [28] determined the GL(n, Q) ‐module

structure of {\rm Im}(\tau_{2,Q}') , and gaverank({\rm Im}(\tau_{2}))=\dim_{Q}({\rm Im}(\tau_{2,Q}'))= \frac{1}{3}n^{2}(n^{2}-4)+\frac{1}{2}n(n-1)=\frac{1}{6}n(n+1)(2n^{2}-2n-3)
In this article, for

k=2 , we show that.

gr^{2}(\mathcal{A}_{n}') is generated by the above number of

elements for n\geq 3.

Acknowledgments
The author would like to express his sincere gratitude to Professor Shigeyuki Morita
for valuable discussions about the Andreadakis conjecture, and for strong and warm
encouragements.

The part of this work was done when the author stayed at the Mathematical In‐
stitute of the University of Bonn as a visitor in 2017. The author would like to thankthe University of Bonn for its hospitality, {\rm Max} Planck Institute for Mathematics forarranging his office, and Tokyo University of Science for giving him the chance to takea sabbatical. This work is supported by a JSPS KAKENHI Grant Number 24740051.

32
References[1] S. Andreadakis; On the automorphisms of free groups and free nilpotent groups,
Proc. London Math. Soc. (3) 15 (1965), 239‐268.
[2] S. Bachmuth; Automorphisms of free metabelian groups. Trans. Amer. Math. Soc.
118 (1965), 93‐104.

[3] S. Bachmuth; Induced automorphisms of free groups and free metabelian groups.
Trans. Amer. Math. Soc. 122 (1966), 1‐17.
[4] L. Bartholdi; Automorphisms of free groups I, New York Journal of Mathematics
19 (2013), 395‐421.

[5] L. Bartholdi; Erratum; Automorphisms of free groups I, New York Journal of
Mathematics 22 (2016), 1135‐1137.
[6] T. Church, M. Ershov and A. Putman; On finite generation of the Johnson filtra‐
tions, arXiv:1711.04779.

[7] F. Cohen and J. Pakianathan; On Automorphism Groups of Free Groups, andTheir Nilpotent Quotients, preprint.

[8] F. Cohen and J. Pakianathan; On subgroups of the automorphism group of a freegroup and associated graded Lie algebras, preprint.

[9] J. Darné; On the stable Andreadakis Problem, preprint,arXiv: 1711.05991.

[10] M. Day and A. Putman; On the second homology group of the Torelli subgroup ofAut (F_{n}) , arXiv: 1408.6242.
[11] B. Farb; Automorphisms of F_{n} which act trivially on homology, in preparation.

[12] R. Hain; Infinitesimal presentations of the Torelli group, Journal of the American
Mathematical Society 10 (1997), 597‐651.
[13] M. Hall; A basis for free Lie rings and higher commutators in free groups, Proc.
Amer. Math. Soc., 1 (1950), 575‐581.

[14] M. Hall; The theory of groups, second edition, AMS Chelsea Publishing 1999.
[15] D. Johnson; An abelian quotient of the mapping class group, Math. Ann. 249(1980), 225‐242.

[16] D. Johnson; The structure of the Torelli group I: A Finite Set of Generators for \mathcal{I},Ann. of Math., 2nd Ser. 118, No. 3 (1983), 423‐442.
[17] D. Johnson; The structure of the Torelli group II: A characterization of the group
generated by twists on bounding curves, Topology, 24, No. 2 (1985), 113‐126.

33[18] D. Johnson; The structure of the Torelli group III: The abelianization of \mathcal{I} , Topol‐ogy 24 (1985), 127‐144.
[19] N.

Kawazumi;

arXiv: math.

Cohomologicalaspectsof Magnus expansions,preprint,

GT/0505497.

[20] S. Krstič, J. McCool; The non‐finite presentability in IA(F_{3}) and GL_{2}(Z[t, t^{-1}]) ,Invent. Math. 129 (1997), 595‐606.

[21] W. Magnus; Über ‐dimensinale Gittertransformationen, Acta Math. 64 (1935),n353‐367.

[22] W. Magnus, A. Karras, and D. Solitar; Combinatorial group theory, Interscience
Publ., New York (1966).
[23] S. Morita; Casson’s invariant for Homology 3‐spheres and characteristic classes ofsuface bundle I, Topology 28 (1989), 305‐323.
[24] S. Morita; Abelian quotients of subgroups of the mapping class group of surfaces,
Duke Mathematical Journal 70 (1993), 699‐726.

[25] S. Morita; Structure of the mapping class groups of surfaces: a survey and aprospect, Geometry and Topology Monographs Vol. 2 (1999), 349‐406.
[26] S. Morita; Cohomological structure of the mapping class group and beyond, Proc.
of Symposia in Pure Math. 74 (2006), 329‐354.

[27] J. Nielsen; Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei
Erzeugenden, Math. Ann. 78 (1918), 385‐397.
[28] A. Pettet; The Johnson homomorphism and the second cohomology of IA_{n} , Alge‐
braic and Geometric Topology 5 (2005) 725‐740.

[29] C. Reutenauer; Free Lie Algebras, London Mathematical Society monographs, newseries, no. 7, Oxford University Press (1993).
[30] T. Satoh; New obstructions for the surjectivity of the Johnson homomorphism of
the automorphism group of a free group, Journal of the London MathematicalSociety, (2) 74 (2006) 341‐360.
[31] T. Satoh; On the image of the Burau representation of the IA‐automorphism groupof a free group, Journal of Pure and Applied Algebra, 214 (2010), 584‐595.
[32] T. Satoh; The Johnson filtration of the McCool stabilizer subgroup of the auto‐
morphism group of a free group, Proc. of Amer. Math. Soc. 139 (2011), 1237‐1245.
[33] T. Satoh; On the lower central series of the IA‐automorphism group of a free group,
J. of Pure and Appl. Alg., 216 (2012), 709‐717.

34[34] T. Satoh; A survey of the Johnson homomorphisms of the automorphism groupsof free groups and related topics, Handbook of Teichmueller theory, volume V,
167‐209.

[35] T. Satoh; On the Johnson homomorphisms of the mapping class groups of surfaces,
Handbook of group actions. Vol. I, Adv. Lect. Math. (ALM), 31, Int. Press (2015),
373‐407.

[36] T Satoh; On the basis‐conjugating automorphism groups of free groups and freemetabelian groups, Mathematical Proceedings Cambridge Philosophical Society

158 (2015), 83‐109.

[37] T. Satoh; On the Andreadakis conjecture restricted to the “lower‐triangular” IA‐automorphism groups of free groups, Journal of Algebra and Its Applications, Vol.

16, No. 5 (2017), 1750099, 1‐31.
[38] E. Witt, Treue Darstellung Liescher Ringe. Journal für die Reine und Angewandte
Mathematik, 177 (1937), 152‐160.