120

数理解析研究所講究録
第2061巻 2018年 120-125

On the low dimensional cohomology groups of
the IA‐automorphism group of the free group ofrank three

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 announcement we consider the structure of the rational cohomology
groups of the IA‐automorphism group \mathrm{I}\mathrm{A}_{3} of the free group of rank three by using
combinatorial group theory and representation theory. In particular, we detect a
non‐trivial irreducible component in the second cohomology group of \mathrm{I}\mathrm{A}_{3} , which
does not contained in the image of the cup product map of the first cohomology
groups. We also show that the image of the triple cup product map of the firstcohomology groups in the third cohomology group is trivial. As a corollary, we
obtain that the fourth term of the lower central series of \mathrm{I}\mathrm{A}_{3} has finite index inthat of the Andreadakis‐Johnson filtration.

1

Introduction

Let F_{n} be a free group of rank n\geq 2 with basis x_{1} , . . . , x_{n} , and Aut F_{n} the automorphismgroup of F_{n} . As far aồ we know, the first contribution to the study of the (co)homology
groups of Aut F_{n} was given by Nielsen [36] in 1924, who showed H_{1} (Aut F_{n}, \mathrm{Z} ) =\mathrm{Z}/2\mathrm{Z}for n\geq 2 by using a presentation for Aut F_{n} . Now we have a broad range of results forthe (co)homology groups of Aut F_{n} due to many authors. In 1984, Gersten [16] showed\mathrm{Z}/2\mathrm{Z} for n \geq 5 In 1980\mathrm{s} , by introducing the Outer space, Culler
H_{2}(\mathrm{A}\mathrm{u}\mathrm{t}F_{n}, \mathrm{Z})and Vogtmann [10] made a breakthrough in the computation of homology groups ofthe outer automorphism groups Out F_{n} of free groups F_{n} . To put it briefly, the Outer
space is an analogue of the Teichmüller space on which the mapping class of a surface
naturally acts. By using the geometry of the Outer space, Hatcher and Vogtmann\mathrm{Q} , and On the other hand, by using sophisticated[17] computed H_{4}(\mathrm{A}\mathrm{u}\mathrm{t}F_{4}, \mathrm{Q})homotopy theory, Galatius [14] showed that the stable integral homology groups of==

Aut F_{n} are isomorphic to those of the symmetric group \mathfrak{S}_{n} of degree n . In particular,
the stable rational homology groups H_{q}(\mathrm{A}\mathrm{u}\mathrm{t}F_{n}, \mathrm{Q}) are trivial for n\geq 2q+1 . This result*\mathrm{e} ‐address:

takao@rs.tus.ac.jp

121

is a quite contrast to the case of the mapping class groups of surfaces. Intuitively, we
can see this from the fact that the free group has no geometric extra structure likesurface groups.
With respect to unstable cohomology groups, Aut F_{n} behave in much different andmysterious way. The unstable cohomology groups of the (outer) automorphism groupsof free groups has also been studied by many authors. For unstable case, the Outer space
is a powerful tool for computation of the cohomology groups. For example, in 1993,

Brady [6] computed the integral cohomology groups of Out F_{3} . Other than Hatcher andVogtmann’s results for the rational cohomology as mentioned above, Gerlitz showed\mathrm{Q} in 2002, and Ohashi [38] computed H_{8} (Aut F_{6}, \mathrm{Q} )H7(Aut F_{5}, \mathrm{Q} )Q. On theother hand, in 1999, Morita [33] constructed a series of unstable homology classes of
Out F_{n} with Kontsevich’s results [22] and [23]. (See also [34].) These homology classes==

are called the Morita classes. It is known that the first and the second one are non‐trivial, and hence are generators of H_{4}(\mathrm{A}\mathrm{u}\mathrm{t}F_{4}, \mathrm{Q}) and H_{8}(\mathrm{A}\mathrm{u}\mathrm{t}F_{6}, \mathrm{Q}) respectively. (See[34] and [9] respectively.) Today, the non‐triviality of the higher Morita classes is underintense study by many authors.
Let H be the abelianization of F_{n} , and \mathrm{I}\mathrm{A}_{n} the kernel of the natural homomorphism
Aut F_{n}\rightar ow \mathrm{A}\mathrm{u}\mathrm{t}H induced from the abelianization homomorphism F_{n}\rightarrow H . The group\mathrm{I}\mathrm{A}_{n} is called the IA‐automorphism group of F_{n} . By the spectral sequence of the groupextension of \mathrm{I}\mathrm{A}_{n} by Aut H , the cohomology groups of \mathrm{I}\mathrm{A}_{n} are closely related to thoseof Aut F_{n} . However, the structure of the cohomology groups of \mathrm{I}\mathrm{A}_{n} is far from well‐understood in contrast to those of Aut F_{n} . To our best knowledge, in the (co)homology
groups of \mathrm{I}\mathrm{A}_{n} , completely determined and explicitly written down one is only the first

integral homology group H_{1} (IAn, Z), which is obtained by Cohen‐Pakianathan [7, 8],
Farb [13] and Kawazumi [21] independently. Krstič and McCool [24] showed that \mathrm{I}\mathrm{A}_{3} is
not finitely presentable. This shows that the second homology group H_{2} (IA3, Z) is not
finitely generated. This fact also follows by a recent work of Bestvina, Bux and Margalit[4]. By using the Outer space, they showed that the quotient group of \mathrm{I}\mathrm{A}_{n} by the innerautomorphism group Inn F_{n} has a 2n-4 ‐dimensional Eilenberg‐Maclane space, andthat H_{2n-4}(\mathrm{I}\mathrm{A}_{n}/\mathrm{I}\mathrm{n}\mathrm{n}F_{n}, \mathrm{Z}) is not finitely generated. For n\geq 4 , it is not known whether\mathrm{I}\mathrm{A}_{n} is finitely presentable or not. Namely, at the present stage, even H_{2}(\mathrm{I}\mathrm{A}_{n}, \mathrm{Z}) is notdetermined explicitly. Pettet [39] determined the image of the rational cup productof the first cohomologies in H^{2} (IAn, Q), and gave its irreducible GL‐decomposition.
Furthermore, recently Day and Putman [11] obtained an explicit finite set of generatorsfor H_{2} (IAn, Z) as a \mathrm{G}\mathrm{L}(n, \mathrm{Z}) ‐module.

In this announcement, we mainly study the second rational cohomology groupH^{2} (IAn, Q)

3.
for the case where n
In particular, we detect a new \mathrm{G}\mathrm{L}(3, \mathrm{Q})irreducible component of H^{2} (IA3, Q) by using combinatorial group theory and repre‐
sentation theory. By Pettet [39], the \mathrm{G}\mathrm{L}(3, \mathrm{Q}) ‐irreducible decomposition of the imageof the cup product \displaystyle\bigcup_{\mathrm{Q} : $\Lambda$^{2}H^{1} (IA3, \mathrm{Q} ) \rightarrow H^{2} (IA3, Q). We obtain the following.
=

Theorem 1. The quotient module H^{2}(\displaystyle \mathrm{I}\mathrm{A}_{3}, \mathrm{Q})/{\rm Im}(\bigcup_{\mathrm{Q} ) contains the \mathrm{G}\mathrm{L}(3, \mathrm{Q}) ‐irreducible
representation D^{-3}\otimes_{\mathrm{Q} [5 , 1 ].
where D is the one dimensional representation coming from the determinant, and [ $\lambda$]

122

is the irreducible polynomial representation associated to the Young diagram $\lambda$.
In order to show Theorem 1, we use our previous results about the Andreadakis‐\mathcal{A}_{m}(1) \supset \mathcal{A}_{m}(2) \supset . . . and the Johnson homomorphismsof Aut F_{3} . Historically, the Andreadakis‐Johnson filtration was originally introducedby Andreadakis [1] in the 1960\mathrm{s} . In 1980\mathrm{s} , Johnson used this filtration to study thegroup structure of the mapping class groups of surfaces. Andreadakis conjectured thatJohnson filtration \mathrm{I}\mathrm{A}_{n}=\mathcal{A}_{n}(1) \supset \mathcal{A}_{n}(2) \supset . . . coincides with the lower central series\mathrm{I}\mathrm{A}_{n}=\mathcal{A}_{n}'(1)\supset \mathcal{A}_{n}'(2)\supset\cdots . Andreadakis showed that this conjecture is true for n=2
and any k \geq 2 , and n=3 and k \leq 3 . Bachmuth [2] showed \mathcal{A}_{n}'(2) =\mathcal{A}_{n}(2) for any
the filtration \mathrm{I}\mathrm{A}_{n}=n\geq 2 . This result is also induced from the fact that the first Johnson homomorphismis the abelianization of \mathrm{I}\mathrm{A}_{n} by independent works Cohen‐Pakianathan [7, 8], Farb [13]and Kawazumi [21]. Pettet [39] showed that \mathcal{A}_{n}'(3) has at most finite index in A_{ $\eta$}(3)for any n \geq 4 . Bartholdi [3] showed that the “rational” version of the Andreadakis
conjecture is not true for n=3 . Fhrom our computation in the proof of Theorem 1, asa corollary, we also obtain the following.
Corollary 1. \mathcal{A}_{3}(4)/\mathcal{A}_{3}'(4) is finite.
We remark that this fact is also obtained by Bartholdi’s computation.

Finally, we consider the third rational cohomology group H^{3} (IA3, Q). The re‐sults by work of Bestvina, Bux and Margalit [4] as mentioned above, we see thatH^{3} (IA3, Q) is infinitely generated. The following theorem shows that non‐trivial ele‐ments in H^{3} (IA3, Q) cannot be detected by the triple cup product of the first cohomol‐
ogy group of \mathrm{I}\mathrm{A}_{3}.
Theorem 2. The image of the triple cup product\displaystle\bigcup_{\mathrm{I}\mathrm{A}_{3}^{3 : $\Lambda$^{3}H^{1} (IA3, \mathrm{Q} ) \rightarrow H^{3} (IA3, Q)
is trivial.

We remark that the arguments and techniques which we use in this paper are appli‐
cable to study the cohomology groups of \mathrm{I}\mathrm{A}_{n} for general n\geq 4 . However, the amountof calculation and the complexity vastly increase with the increasing n . In the present
paper, we give the first combinatorial group theoretic approach to the study of the low
dimensional cohomology groups of the IA‐automorphism groups of free groups.

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, Induced automorphisms of free groups and free metabelian groups.
Trans. Amer. Math. Soc. 122 (1966), 1‐17.

123[3] L. Bartholdi, Automorphisms of free groups I, New York Journal of Mathematics
19 (2013), 395‐421.
[4] M. Bestvina, Kai‐Uwe Bux and D. Margalit; Dimension of the Torelli group for
Out (Fn), Inventiones Mathematicae 170 (2007), no. 1, 1‐32.
[5] N. Bourbaki; Lie groups and Lie algebra, Chapters 1‐3, Softcover edition of the
2nd printing, Springer‐Verlag (1989).

[6] T. Brady; The integral cohomology of \mathrm{O}\mathrm{u}\mathrm{t}_{+}(F_{3}) . J. Pure Appl. Algebra 87 (1993),
123‐167.

[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. Conant and K. Vogtmann; Morita classes in the homology of automorphismgroups of free groups. Geom. Topol. 8 (2004), 1471‐1499.
[10] M. Culler and K. Vogtmann; Moduli of graphs and automorphisms of free groups,
Invent. math., 84 (1986), 91‐119.

[11] M. Day and A. Putman; On the second homology group of the Torelli subgroup ofAut(Fn), preprint, arXiv: 1408. 6242.
[12] N. Enomoto and T. Satoh; On the derivation algebra of the free Lie algebra andtrace maps., Alg. and Geom. Top., 11 (2011) 2861‐2901.
[13] B. Farb; Automorphisms of F_{n} which act trivially on homology, in preparation.

[14] S. Galatius; Stable homology of automorphism groups of free groups, Ann. of Math.
173 (2011), 705‐768.
[15] F. Gerlitz; Ph.D. thesis, Cornell University (2002).
[16] S. M. Gersten; A presentation for the special automorphism group of a free group,
J. Pure and Applied Algebra 33 (1984), 269‐279.

[17] A. Hatcher and K. Vogtmann; Rational homology of Aut(Fn), Math. Res. Lett. 5(1998), 759‐780.
[18] M. Hall; A basis for free Lie rings and higher commutators in free groups, Proc.
Amer. Math. Soc., 1 (1950), 575‐581.

[19] M. Hall; The theory of groups, second edition, AMS Chelsea Publishing 1999.
[20] P. J. Hilton and U. Stammbach; A Course in Homological Algebra, Graduate Texts
in Mathematics 4, Springer‐Verlag (1997).

124[21] N.

Kawazumi;

Cohomological aspects of Magnus expansions,preprint,

arXiv: math. \mathrm{G}\mathrm{T}/0505497.

[22] M. Kontsevich; Formal (non)commutative symplectic geometry, The Gelfand
Mathematical Seminars, 1990‐1992, 173‐187, Birkhäuser Boston, Boston, MA,
1993.

[23] M. Kontsevich; Feynman diagrams and low‐dimensional topology, First European
Congress of Mathematics, Vol. II (Paris, 1992), 97‐121, Progress in Mathematics,
120, Birkhäuser, Basel, 1994.

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

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

[26] W. Magnus, A. Karras and D. Solitar; Combinatorial group theory, Interscience
Publ., New York (1966).
[27] J. McCool; Some remarks on IA automorphisms of free groups, Can. J. Math. Vol.
XL, no. 5 (1998), 1144‐1155.
[28] S. Morita; On the Homology Groups of the Mapping Class Groups of Orientable
Surfaces with Twisted Coefficients, Proc. Japan Acad., 62, Ser. \mathrm{A} (1986), 148‐151.

[29] S. Morita; Families of Jacobian manifolds and characteristic classes of surface bun‐dles I, Ann. Inst. Fourier 39 (1989), 777‐810.
[30] S. Morita; Families of Jacobian manifolds and characteristic classes of surface bun‐dles, II, Math. Proc. Camb. Phil. Soc. 105 (1989), 79‐101.

[31] S. Morita; Abelian quotients of subgroups of the mapping class group of surfaces,
Duke Mathematical Journal 70 (1993), 699‐726.
[32] S. Morita; The extension of Johnson’s homomorphism from the Torelli group to
the mapping class group, Invent. math. 111 (1993), 197‐224.
[33] S. Morita; Structure of the mapping class groups of surfaces: a survey and aprospect, Geometry and Topology Monographs Vol. 2 (1999), 349‐406.

[34] S. Morita; Cohomological structure of the mapping class group and beyond, Proc.
of Symp. in Pure Math. 74 (2006), 329‐354.
[35] J. Nielsen; Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei
Erzeugenden, Math. Ann. 78 (1918), 385‐397.

[36] J. Nielsen; Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924),
169‐209.

125[37] J. Nielsen; Untersuchungen zur Topologie der geschlossenen Zweiseitigen Fläschen,
Acta Math. 50 (1927), 189‐358.

[38] R. Ohashi; The rational homology group of Out(Fn) for n\leq 6 . Experiment. Math.
17 (2008), no. 2, 167‐179.
[39] A. Pettet; The Johnson homomorphism and the second cohomology of IA_{n} , Alge‐
braic and Geometric Topology 5 (2005) 725‐740.

[40] C. Reutenauer; Free Lie Algebras, London Mathematical Society monographs, newseries, no. 7, Oxford University Press (1993).
[41] 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.

[42] T. Satoh; The cokernel of the Johnson homomorphisms of the automorphism groupof a free metabelian group, Trans. of Amer. Math. Soc. 361 (2009), 2085‐2107.
[43] T. Satoh; A survey of the Johnson homomorphisms of the automorphism groups of
free groups and related topics, Handbook of Teichmüller theory, volume V. (editor:
A. Papadopoulos), 167‐209.
[44] T. Satoh; On the Johnson homomorphisms of the mapping class groups of surfaces,

Handbook of Group actions, Vol. I, ALM31, 373‐405, Higher Education Press andInternational Press Beijing‐Boston.

[45] E. Witt; Treue Darstellung Liescher Ringe, Journal für die Reine und Angewandte
Mathematik, 177 (1937), 152‐160.