Chebyshev Polynomials and Inequalities for Kleinian Groups (original) (raw)

An Overview of Complex Kleinian Groups

Nonlinear Systems and Complexity, 2015

Classical Kleinian groups are discrete subgroups of P SL(2, C) acting on the complex projective line P 1 C , which actually coincides with the Riemann sphere, with non-empty region of discontinuity. These can also be regarded as the monodromy groups of certain differential equations. These groups have played a major role in many aspects of mathematics for decades, and also in physics. It is thus natural to study discrete subgroups of the projective group P SL(n, C), n > 2. Surprisingly, this is a branch of mathematics which is in its childhood, and in this article we give an overview of it.

Representations of The miraculous Klein group

The Klein group contains only four elements. Nevertheless this little group contains a number of remarkable entry points to current highways of modern representation theory of groups. In this paper, we shall describe all possible ways in which the Klein group can act on vector spaces over a field of two elements. These are called representations of the Klein group. This description involves some powerful visual methods of representation theory which builds on the work of generations of mathematicians starting roughly with the work of K. Weiestrass. We also discuss some applications to properties of duality and Heller shifts of the representations of the Klein group.

The Geometry and Arithmetic of Kleinian Groups

2013

In this article we survey and describe various aspects of the geometry and arithmetic of Kleinian groups -discrete nonelementary groups of isometries of hyperbolic 3-space. In particular we make a detailed study of two-generator groups and discuss the classification of the arithmetic generalised triangle groups (and their near relatives). This work is mainly based around my collaborations over the last two decades with Fred Gehring and Colin Maclachlan, both of whom passed away in 2012. There are many others involved as well.

Recent Results in the Geometry of Kleinian Groups

Computational Methods and Function Theory, 2003

A spherical point of a Kleinian group Γ is a point of H 3 that is stabilized by a spherical triangle subgroup of Γ. Such points appear as vertices in the singular graph of the quotient hyperbolic 3-orbifold. We announce here sharp lower bounds for the hyperbolic distances between such points in H 3 . These bound from below the edge lengths of the singular graph. An elliptic element of a Kleinian group is simple if the translates of its axis under the group Γ form a disjoint collection of hyperbolic lines. Here we announce that the minimal covolume Kleinian group contains no simple elliptics of order p ≥ 3.

A Trace-Class Rigidity Theorem for Kleinian Groups

2008

Abstract. Suppose that Γ1 and Γ2 are geometrically finite, convex co-compact, discrete groups of isometries of real hyperbolic space H whose domains of discontinuity are diffeomorphic. We show that if the respective scattering matrices S1(s) and S2(s) differ from each other by a traceclass perturbation on the unitary axis Re(s) = 1 , then Γ1 and Γ2 are conjugate in PSL(2, C) . This result reflects the rigidity of hyperbolic three-manifolds.

Polynomial Trace Identities in SL(2,C), Quaternion Algebras, and Two-generator Kleinian Groups

Handbook of Complex Analysis, 2022

We study certain polynomial trace identities in the group SL(2, C) and their application in the theory of discrete groups. We obtain canonical representations for two generator groups in §4 and then in §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 §8. Using these quaternion algebras and their units, we consider their relation to discrete subgroups of SL(2, C) giving necessary and sufficient criteria for discreteness, and another for arithmeticity §9. We then show that for the groups Z p * Z 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 §12. * Research supported in part by grants from the N.Z. Marsden Fund.

Iteration theory and inequalities for Kleinian groups

Bulletin of the American Mathematical Society, 1989

Introduction. An important problem in the theory of discrete groups is to decide when two Möbius transformations ƒ, g acting on the Riemann sphere C generate a Kleinian group, that is, a discrete group whose limit set contains more than two points. (See [Be and Ml] for further information on such groups.) Solutions to the above problem have quite general applications, for example, to deformation theory, discreteness of limits [Jl] and universal constraints for Kleinian groups [Be], and lower bounds for the volume of hyperbolic manifolds [Me, W].