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Automorphisms of the Fricke characters of groups

Richard Brown
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Abstract

In this note, we embed the set of all Fricke characters of a free group F -- the set of all characters of representations of F into SL(2,C) -- as an irreducible affine variety V in complex affine space of dimension 2^n-1. Using the Horowitz generating set as the indeterminates, we show that the ideal I of all polynomials in these indeterminates which vanish on V is finitely generated by the Magnus relation for arbitrary octets of elements in SL(2,C). Using this relation, we produce a basis for I, and show that it is prime. We then show that the natural action of automorphisms of F on V extends to polynomial automorphisms on all of the ambient affine space which, up to sign, preserve a complex volume form. This construction provides an algebraic model for the analysis of the dynamics of the measure preserving action of Out(F) on V.

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Richard Brown. 2003-11-07. Automorphisms of the Fricke characters of groups. https://arxiv.org/abs/math/0311119

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