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

TL;DR

This paper explores polynomial trace identities in SL(2,C), utilizing quaternion algebras to analyze two-generator Kleinian groups, providing new proofs, structure theorems, and criteria for discreteness and arithmeticity.

Contribution

It introduces a novel quaternion algebra approach to polynomial trace identities and extends existing results to broader classes of groups with applications to discreteness and arithmeticity.

Findings

New proof of Gehring and Martin's identities

Structure theorems for quaternion algebras

Characterization of moduli space of discrete groups

Abstract

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.