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Proceedings of the Edinburgh Mathematical Society (1)

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On the Universal SL2-Representation Rings of Free Groups
Part of
Takao Satoh
Journal:

Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 4 / November 2017

Published online by Cambridge University Press:

30 January 2017, pp. 973-1001
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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.