Skip to main content Accessibility help
×
Home

On the Universal SL2-Representation Rings of Free Groups

  • Takao Satoh (a1)

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.

Copyright

References

Hide All
1. Andreadakis, S., On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc. s3-15(1) (1965), 239268.
2. Cohen, F. and Pakianathan, J., On automorphism groups of free groups, and their nilpotent quotients, Preprint.
3. Cohen, F. and Pakianathan, J., On subgroups of the automorphism group of a free group and associated graded Lie algebras, Preprint.
4. Farb, B., Automorphisms of Fn which act trivially on homology, In preparation.
5. Fricke, R. and Klein, R., Vorlesungen über die Teorie der automorphen Funktionen, I (Teuber, Leipzig/Stuttgart, 1897).
6. Hain, R., Infinitesimal presentations of the Torelli group, J. Am. Math. Soc. 10 (1997), 597651.
7. Hall, M., The theory of groups, 2nd edn (American Mathematical Society, Providence, RI, 1999).
8. Hatakenaka, E. and Satoh, T., On the graded quotients of the ring of Fricke characters of a free group, J. Alg. 430 (2015), 94118.
9. Horowitz, R., Characters of free groups represented in the two-dimensional special linear group, Commun. Pure Appl. Math. 25 (1972), 635649.
10. Horowitz, R., Induced automorphisms on Fricke characters of free groups, Trans. Am. Math. Soc. 208 (1975), 4150.
11. Johnson, D., An abelian quotient of the mapping class group, Math. Annalen 249 (1980), 225242.
12. Johnson, D., The structure of the Torelli group I: a finite set of generators for , Annals Math. 118(3) (1983), 423442.
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), 113126.
14. Johnson, D., The structure of the Torelli group III: the abelianization of , Topology 24 (1985), 127144.
15. Kawazumi, N., Cohomological aspects of Magnus expansions, Preprint (arXiv:math/0505497 [math.GT]; 2006).
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).
17. Magnus, W., Über n-dimensinale Gittertransformationen, Acta Math. 64 (1935), 353367.
18. Magnus, W., Rings of Fricke characters and automorphism groups of free groups, Math. Z. 170 (1980), 91103.
19. Magnus, W., Karras, A. and Solitar, D., Combinatorial group theory (Interscience Publications, New York, 1966).
20. Morita, S., Abelian quotients of subgroups of the mapping class group of surfaces, Duke Math. J. 70 (1993), 699726.
21. Morita, S., The extension of Johnson's homomorphism from the Torelli group to the mapping class group, Invent. Math. 111 (1993), 197224.
22. Nielsen, J., Die Isomorphismengruppe der freien Gruppen, Math. Annalen 91 (1924), 169209.
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).
24. Satoh, T., Twisted first homology group of the automorphism group of a free group, J. Pure Appl. Alg. 204 (2006), 334348.
25. Satoh, T., The cokernel of the Johnson homomorphisms of the automorphism group of a free metabelian group, Trans. Am. Math. Soc. 361 (2009), 20852107.
26. Satoh, T., First cohomologies and the Johnson homomorphisms of the automorphism group of a free group, J. Pure Appl. Alg. 217 (2013), 137152.
27. Satoh, T., The Johnson–Morita theory for the ring of Fricke characters of free groups, Pac. J. Math. 275 (2015), 443461.
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. 167209 (European Mathematical Society, Zürich, 2016).
29. Satoh, T., First cohomologies and the Johnson homomorphisms of the automorphism groups of free groups, II, In preparation.

Keywords

MSC classification

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed