  The conjugacy class of a generic unimodular 2 by 2 complex matrix is
determined by its trace, which may be an arbitrary complex number. In the
nineteenth century, it was known that a generic pair (X,Y) of such pairs is
determined up to conjugacy by the triple of traces (tr(X),tr(Y),tr(XY), which
may be an arbitary element of C^3. This paper gives an elementary and detailed
proof of this fact, which was published by Vogt in 1889. The folk theorem
describing the extension of a representation to a representation of the
index-two supergroup which is a free product of three groups of order two, is
described in detail, and related to hyperbolic geometry. When n > 2, the
classification of conjugacy-classes of n-tuples in SL(2,C) is more complicated.
We describe it in detail when n= 3. The deformation spaces of hyperbolic
structures on some simple surfaces S whose fundamental group is free of rank
two or three are computed in trace coordinates. (We only consider the two
orientable surfaces whose fundamental group has rank 3.)

Geometric Topology;Commutative Algebra