  We study certain polynomial trace identities in the group $SL(2,\IC)$ and
their application in the theory of discrete groups. We obtain canonical
representations for two generator groups in \S 4 and then in \S 5 we give a new
proof for Gehring and Martin's polynomial trace identities for good words, and
extend that result to a larger class which is also closed under a semigroup
operation inducing polynomial composition. This new approach is through the use
of quaternion algebras over indefinites and an associated group of units. We
obtain structure theorems for these quaternion algebras which appear to be of
independent interest in \S 8. Using these quaternion algebras and their units,
we consider their relation to discrete subgroups of $SL(2,\IC)$ giving
necessary and sufficient criteria for discreteness, and another for
arithmeticity \S 9. We then show that for the groups $\IZ_p*\IZ_2$, the
complement of the closure of roots of the good word polynomials is precisely
the moduli space of geometrically finite discrete and faithful representations
a result we show holds in greater generality in \S12.
