Automorphisms of the Fricke characters of groups
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.
Keywords
Cite
@article{arxiv.math/0311119,
title = {Automorphisms of the Fricke characters of groups},
author = {Richard Brown},
journal= {arXiv preprint arXiv:math/0311119},
year = {2007}
}
Comments
17 pages, no figures