Discrete primitive-stable representations with large rank surplus

TL;DR

This paper constructs a sequence of primitive-stable representations of free groups into PSL(2,C) with increasing rank, where the geometric limits are knot complements, showing no direct correlation between domain rank and image geometry.

Contribution

It introduces a method to generate primitive-stable representations with unbounded rank whose images converge geometrically to knot complements, revealing new insights into the relationship between domain rank and image geometry.

Findings

Constructed sequences of primitive-stable representations with unbounded rank

Demonstrated geometric convergence to knot complements

Showed no constraints on domain rank from image geometry

Abstract

We construct a sequence of primitive-stable representations of free groups into PSL(2,C) whose ranks go to infinity, but whose images are discrete with quotient manifolds that converge geometrically to a knot complement. In particular this implies that the rank and geometry of the image of a primitive-stable representation imposes no constraint on the rank of the domain.