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ISSN 1088-6850(e) ISSN 0002-9947(p)

The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group

Author(s): Takao Satoh
Journal: Trans. Amer. Math. Soc. 361 (2009), 2085-2107.
MSC (2000): Primary 20F28; Secondary 20J06
Posted: November 5, 2008
MathSciNet review: 2465830
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

Abstract: In this paper, we determine the cokernel of the -th Johnson homomorphisms of the automorphism group of a free metabelian group for $k \geq 2$ and $n \geq 4$ . As a corollary, we obtain a lower bound on the rank of the graded quotient of the Johnson filtration of the automorphism group of a free group. Furthermore, by using the second Johnson homomorphism, we determine the image of the cup product map in the rational second cohomology group of the IA-automorphism group of a free metabelian group, and show that it is isomorphic to that of the IA-automorphism group of a free group which is already determined by Pettet. Finally, by considering the kernel of the Magnus representations of the automorphism group of a free group and a free metabelian group, we show that there are non-trivial rational second cohomology classes of the IA-automorphism group of a free metabelian group which are not in the image of the cup product map.

References:

1.: S. Andreadakis; On the automorphisms of free groups and free nilpotent groups, Proc. London Math. Soc.(3) 15 (1965), 239-268. MR 0188307 (32:5746)
2.: S. Bachmuth; Automorphisms of free metabelian groups, Trans. Amer. Math. Soc. 118 (1965), 93-104. MR 0180597 (31:4831)
3.: S. Bachmuth; Induced automorphisms of free groups and free metabelian groups, Trans. Amer. Math. Soc. 122 (1966), 1-17. MR 0190212 (32:7626)
4.: S. Bachmuth and H. Y. Mochizuki; The non-finite generation of $\mathrm{Aut}(G)$ , free metabelian of rank , Trans. Amer. Math. Soc. 270 (1982), 693-700. MR 645339 (83f:20026)
5.: S. Bachmuth and H. Y. Mochizuki; $\mathrm{Aut}(F) \rightarrow \mathrm{Aut}(F/F'')$ is surjective for free group for rank $\geq 4$ , Trans. Amer. Math. Soc. 292, no. 1 (1985), 81-101. MR 805954 (87a:20032)
6.: Y. A. Bakhturin, Identities in Lie algebras, Nauka, Moscow 1985; English translation, Identical relations in Lie Algebras, VNU Science Press, Utrecht (1987). MR 886063 (88f:17032)
7.: J. S. Birman; Braids, Links, and Mapping Class Groups, Annals of Math. Studies 82 (1974). MR 0375281 (51:11477)
8.: K. T. Chen; Integration in free groups, Ann. of Math. 54, no. 1 (1951), 147-162. MR 0042414 (13:105c)
9.: F. Cohen and J. Pakianathan; On Automorphism Groups of Free Groups, and Their Nilpotent Quotients, preprint.
10.: F. Cohen and J. Pakianathan; On subgroups of the automorphism group of a free group and associated graded Lie algebras, preprint.
11.: B. Farb; Automorphisms of which act trivially on homology, in preparation.
12.: W. Fulton; Young Tableaux, London Mathematical Society Student Texts 35, Cambridge University Press (1997). MR 1464693 (99f:05119)
13.: W. Fulton, J. Harris; Representation Theory, Graduate Texts in Mathematics 129, Springer-Verlag (1991). MR 1153249 (93a:20069)
14.: R. Hain; Infinitesimal presentations of the Torelli group, Journal of the American Mathematical Society 10 (1997), 597-651. MR 1431828 (97k:14024)
15.: M. Hall; A basis for free Lie rings and higher commutators in free groups, Proc. Amer. Math. Soc. 1 (1950), 575-581. MR 0038336 (12:388a)
16.: P. J. Hilton and U. Stammbach; A Course in Homological Algebra, Graduate Texts in Mathematics 4, Springer-Verlag, New York (1970). MR 1438546 (97k:18001)
17.: D. Johnson; An abelian quotient of the mapping class group, Math. Ann. 249 (1980), 225-242. MR 579103 (82a:57008)
18.: D. Johnson; The structure of the Torelli group III: The abelianization of $\mathcal{I}_g$ , Topology 24 (1985), 127-144. MR 793179 (87a:57016)
19.: N. Kawazumi; Cohomological aspects of Magnus expansions, preprint, arXiv:math.GT/0505497.
20.: S. Krstić, J. McCool; The non-finite presentability in and $GL_{2}(\mathbf{Z}[t,t^{-1}])$ , Invent. Math. 129 (1997), 595-606. MR 1465336 (98h:20053)
21.: R. C. Lyndon, P. E. Schupp; Combinatorial Group Theory, Springer (1977). MR 0577064 (58:28182)
22.: W. Magnus; $\ddot{\mathrm{U}}$ ber -dimensinale Gittertransformationen, Acta Math. 64 (1935), 353-367. MR 1555401
23.: W. Magnus, A. Karras, D. Solitar; Combinatorial group theory, Interscience Publ., New York (1966). MR 2109550 (2005h:20052)
24.: S. Morita; Abelian quotients of subgroups of the mapping class group of surfaces, Duke Mathematical Journal 70 (1993), 699-726. MR 1224104 (94d:57003)
25.: S. Morita; Structure of the mapping class groups of surfaces: a survey and a prospect, Geometry and Topology Monographs Vol. 2 (1999), 349-406. MR 1734418 (2000j:57039)
26.: S. Morita; Cohomological structure of the mapping class group and beyond, preprint. MR 2264550 (2007j:20079)
27.: J. Nielsen; Die Isomorphismen der allgemeinen unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1918), 385-397. MR 1511907
28.: J. Nielsen; Die Isomorphismengruppe der freien Gruppen, Math. Ann. 91 (1924), 169-209. MR 1512188
29.: J. Nielsen; Untersuchungen zur Topologie der geschlossenen Zweiseitigen Fl $\ddot{\mathrm{a}}$ schen, Acta Math. 50 (1927), 189-358. MR 1555256
30.: I. B. S. Passi; Group rings and their augmentation ideals, Lecture Notes in Mathematics 715, Springer-Verlag (1979). MR 537126 (80k:20009)
31.: A. Pettet; The Johnson homomorphism and the second cohomology of , Algebraic and Geometric Topology 5 (2005), 725-740. MR 2153110 (2006j:20050)
32.: C. Reutenauer; Free Lie Algebras, London Mathematical Society Monographs, New Series, no. 7, Oxford University Press (1993). MR 1231799 (94j:17002)
33.: T. Satoh; New obstructions for the surjectivity of the Johnson homomorphism of the automorphism group of a free group, Journal of the London Mathematical Society, (2) 74 (2006), 341-360. MR 2269583 (2007i:20060)
34.: E. Witt; Treue Darstellung Liescher Ringe, Journal f $\ddot{\mathrm{u}}$ r die Reine und Angewandte Mathematik, 177 (1937), 152-160.
35.: V. M. Zhuravlev; A free Lie algebra as a module over the full linear group, Sbornik Mathematics 187 (1996), 215-236. MR 1392842 (97f:20053)

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