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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 Martins 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.
Given a 2-manifold, a fundamental question to ask is which groups can be realized as the isometry group of a Riemannan metric of constant curvature on the manifold. In this paper, we give a nearly complete classification of such groups for infinite-g
Let $A$ and $B$ be finite-dimensional simple algebras with arbitrary signature over an algebraically closed field. Suppose $A$ and $B$ are graded by a semigroup $S$ so that the graded identitical relations of $A$ are the same as those of $B$. Then $A$ is isomorphic to $B$ as an $S$-graded algebra.
We prove that every finitely generated Kleinian group that contains a finite, non-cyclic subgroup either is finite or virtually free or contains a surface subgroup. Hence, every arithmetic Kleinian group contains a surface subgroup.
Given a Heegaard splitting of a three-manifold Y, we consider the SL(2,C) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible represent
Given a Coxeter system (W,S), there is an associated CW-complex, Sigma, on which W acts properly and cocompactly. We prove that when the nerve L of (W,S) is a flag triangulation of the 3-sphere, then the reduced $ell^2$-homology of Sigma vanishes in all but the middle dimension.