On the two-parabolic subgroups of SL (2; C)

Abstract

We consider homomorphisms Ht from the free group F of rank 2 onto the subgroup of SL(2;C) that is generated by two parabolic matrices. Up to conjugation, Ht depends only on one complex parameter t. We study the possible relators, that is, the words w 2 F with w 6= 1 such that Ht(w) = I for some t 2 C. We and several families of realtors. Of particular interest here are relators connected with 2-bridge knots, which we consider in a purely algebraic setting. We describe an algorithm to determine whether a given word is a possible relator. Key words and phrases. Representation, Parabolic, Wirtinger presentation, Two- generated groups, Homomorphism, Longitude.