Automorphisms of the Fricke characters of groups
Abstract
Description
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.
17 pages, no figures
17 pages, no figures