Endomorphisms of certain algebras with polynomial identities

Horowitz [2] defined a mapping from a free group Gamma of rank n into a quo tient ring R of a ring of polynomials on Z with 2(n) - 1 variables in such a way that the value of this mapping (a Fricke character) at a point w in G amma allows to compute the value at w of the character of any representatio n of Gamma in SL(2, C). He also showed [3] that an endomorphism of Gamma in duces an endomorphism of R. A way of computing these Fricke characters is t o consider a PI-algebra A universal for the situation Gamma --> SL(2, C) an d perform computations in it. Any endomorphism of Gamma induces an endomorp hism of A. In this work, we show that, in the case of rank 2, some properti es of endomorphisms of A coming from endomorphisms of Gamma extend to all e ndomorphisms of A. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.