Invariants of 2 × 2 matrices, irreducible $${\hbox{SL}(2, \mathbb{C})}$$ characters and the Magnus trace map

Abstract

We obtain an explicit characterization of the stable points of the action of G=SL(2,C) on the cartesian product G^n by simultaneous conjugation on each factor, in terms of the corresponding invariant functions, and derive from it a simple criterion for irreducibility of representations of finitely generated groups into G. We also obtain analogous results for the action of SL(2,C) on the vector space of n-tuples of 2 by 2 complex matrices. For a free group F_n of rank n, we show how to generically reconstruct the 2^{n-2} conjugacy classes of representations F_n -> G from their values under the map T_n : G^n = Hom(F_n,G) -> C^{3n-3} considered in [M], defined by certain 3n-3 traces of words of length one and two.