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On the Universal SL2-Representation Rings of Free Groups

Part of: Special aspects of infinite or finite groups Linear algebraic groups and related topics General commutative ring theory Theory of modules and ideals

Published online by Cambridge University Press:  30 January 2017

Takao Satoh
Takao Satoh*
Affiliation:
Department of Mathematics, Faculty of Science Division II, Tokyo University of Science, 1–3 Kagurazaka, Shinjuku, Tokyo 162-8601, Japan (takao@rs.tus.ac.jp)
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Abstract

In this paper, we give an explicit realization of the universal SL2-representation rings of free groups by using ‘the ring of component functions’ of SL(2, ℂ)-representations of free groups. We introduce a descending filtration of the ring, and determine the structure of its graded quotients. Then we study the natural action of the automorphism group of a free group on the graded quotients, and introduce a generalized Johnson homomorphism. In the latter part of this paper, we investigate some properties of these homomorphisms from a viewpoint of twisted cohomologies of the automorphism group of a free group.

Keywords

universal SL2-representationFricke characterscharacter varietyJohnson homomorphism

MSC classification

Secondary: 20F28: Automorphism groups of groups 20G05: Representation theory 13A15: Ideals; multiplicative ideal theory 13C05: Structure, classification theorems
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 4 , November 2017 , pp. 973 - 1001
Copyright
Copyright © Edinburgh Mathematical Society 2017 

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References

1. Andreadakis, S., On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc. s3-15(1) (1965), 239–268.Google Scholar
2. Cohen, F. and Pakianathan, J., On automorphism groups of free groups, and their nilpotent quotients, Preprint.Google Scholar
3. Cohen, F. and Pakianathan, J., On subgroups of the automorphism group of a free group and associated graded Lie algebras, Preprint.Google Scholar
4. Farb, B., Automorphisms of Fn which act trivially on homology, In preparation.Google Scholar
5. Fricke, R. and Klein, R., Vorlesungen über die Teorie der automorphen Funktionen, I (Teuber, Leipzig/Stuttgart, 1897).Google Scholar
6. Hain, R., Infinitesimal presentations of the Torelli group, J. Am. Math. Soc. 10 (1997), 597–651.CrossRefGoogle Scholar
7. Hall, M., The theory of groups, 2nd edn (American Mathematical Society, Providence, RI, 1999).Google Scholar
8. Hatakenaka, E. and Satoh, T., On the graded quotients of the ring of Fricke characters of a free group, J. Alg. 430 (2015), 94–118.Google Scholar
9. Horowitz, R., Characters of free groups represented in the two-dimensional special linear group, Commun. Pure Appl. Math. 25 (1972), 635–649.Google Scholar
10. Horowitz, R., Induced automorphisms on Fricke characters of free groups, Trans. Am. Math. Soc. 208 (1975), 41–50.Google Scholar
11. Johnson, D., An abelian quotient of the mapping class group, Math. Annalen 249 (1980), 225–242.CrossRefGoogle Scholar
12. Johnson, D., The structure of the Torelli group I: a finite set of generators for , Annals Math. 118(3) (1983), 423–442.Google Scholar
13. Johnson, D., The structure of the Torelli group II: a characterization of the group generated by twists on bounding curves, Topology 24(2) (1985), 113–126.CrossRefGoogle Scholar
14. Johnson, D., The structure of the Torelli group III: the abelianization of , Topology 24 (1985), 127–144.Google Scholar
15. Kawazumi, N., Cohomological aspects of Magnus expansions, Preprint (arXiv:math/0505497 [math.GT]; 2006).Google Scholar
16. Lubotzky, A. and Magid, A., Varieties of representations of finitely generated groups, Memoirs of the American Mathematical Society, Volume 336 (American Mathematical Society, Providence, RI, 1985).Google Scholar
17. Magnus, W., Über n-dimensinale Gittertransformationen, Acta Math. 64 (1935), 353–367.CrossRefGoogle Scholar
18. Magnus, W., Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), 91–103.CrossRefGoogle Scholar
19. Magnus, W., Karras, A. and Solitar, D., Combinatorial group theory (Interscience Publications, New York, 1966).Google Scholar
20. Morita, S., Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699–726.CrossRefGoogle Scholar
21. Morita, S., The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993), 197–224.Google Scholar
22. Nielsen, J., Die Isomorphismengruppe der freien Gruppen, Math. Annalen 91 (1924), 169–209.CrossRefGoogle Scholar
23. Saito, K., Representation variety of a finitely generated group into SL2 or GL2, Preprint RIMS-958, Research Institute for Mathematical Sciences, Kyoto University (1993).Google Scholar
24. Satoh, T., Twisted first homology group of the automorphism group of a free group, J. Pure Appl. Alg. 204 (2006), 334–348.Google Scholar
25. Satoh, T., The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group, Trans. Am. Math. Soc. 361 (2009), 2085–2107.Google Scholar
26. Satoh, T., First cohomologies and the Johnson homomorphisms of the automorphism group of a free group, J. Pure Appl. Alg. 217 (2013), 137–152.CrossRefGoogle Scholar
27. Satoh, T., The Johnson–Morita theory for the ring of Fricke characters of free groups, Pac. J. Math. 275 (2015), 443–461.Google Scholar
28. Satoh, T., A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics, in Handbook of Teichmüller theory (ed. Papadopoulos, A.), Volume V, pp. 167–209 (European Mathematical Society, Zürich, 2016).CrossRefGoogle Scholar
29. Satoh, T., First cohomologies and the Johnson homomorphisms of the automorphism groups of free groups, II, In preparation.Google Scholar