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We study certain polynomial trace identities in the group $SL(2,\IC)$ andtheir application in the theory of discrete groups. We obtain canonicalrepresentations for two generator groups in \S 4 and then in \S 5 we give a newproof for Gehring and Martin's polynomial trace identities for good words, andextend that result to a larger class which is also closed under a semigroupoperation inducing polynomial composition. This new approach is through the useof quaternion algebras over indefinites and an associated group of units. Weobtain structure theorems for these quaternion algebras which appear to be ofindependent interest in \S 8. Using these quaternion algebras and their units,we consider their relation to discrete subgroups of $SL(2,\IC)$ givingnecessary and sufficient criteria for discreteness, and another forarithmeticity \S 9. We then show that for the groups $\IZ_p*\IZ_2$, thecomplement of the closure of roots of the good word polynomials is preciselythe moduli space of geometrically finite discrete and faithful representationsa result we show holds in greater generality in \S12.

Polynomial Trace Identities in $SL(2,{\bf C})$, Quaternion Algebras, and Two-generator Kleinian Groups