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On the Universal SL2-Representation Rings of Free Groups

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