  Let G a be subgroup of SL(2,C), the group of 2x2 matrices of determinant 1
with complex entries. Let h map onto h(G) be a homomorphism. We call h a trace
preserving homomorphism if tr(h(g))=tr(g) for all g in G,where tr(g) is the
trace of g. We solve the question of when a trace invariant homomorphism is a
conjugation by some A in SL(2,R). Moreover, if the group G is finitely
presented, this paper determines which traces of the generators and products of
the generators determine the group up to conjugation. Incomplete solutions are
known from the study of Fuchsian groups. Our theorems in this paper will expand
the results in the literature to include Fuchsian Groups with elliptic
elements, which have not been considered before. Moreover, they will be
applicable to any class of subgroups of SL(2,C). The methods used will be
relatively elementary and will indicate how many traces are needed, and the
role that any relator equation plays in the parameterization by traces of a
class of groups.
