Chebyshev Polynomials and Inequalities for Kleinian Groups

TL;DR

This paper introduces new inequalities involving Chebyshev polynomials to characterize when certain representations of free groups into PSL(2,C) are discrete Kleinian groups, advancing understanding in low-dimensional topology.

Contribution

It develops novel inequalities using Chebyshev polynomials and trace identities to identify discrete Kleinian groups and their subgroups, improving upon classical criteria like Jørgensen's inequality.

Findings

Derived new families of inequalities for Kleinian groups.

Identified conditions for principal characters of subgroups.

Some inequalities are proven to be optimal.

Abstract

The principal character of a representation of the free group of rank two into PSL(2, C) is a triple of complex numbers that determines an irreducible representation uniquely up to conjugacy. It is a central problem in the geometry of discrete groups and low dimensional topology to determine when such a triple represents a discrete group that is not virtually abelian, that is a Kleinian group. A classical necessary condition is J{\o}rgensen's inequality. Here we use certainly shifted Chebyshev polynomials and trace identities to determine new families of such inequalities, some of which are best possible. The use of these polynomials also shows how we can identify the principal character of some important subgroups from that of the group itself.