Algebraic and geometric convergence of discrete representations into PSL(2,C)

TL;DR

This paper investigates the convergence properties of sequences of discrete representations into PSL(2,C), highlighting limitations of existing theorems and proposing new conditions for convergence in unfaithful cases.

Contribution

The authors construct counterexamples to a known convergence theorem and propose a revised criterion applicable to unfaithful representations.

Findings

Counterexamples to Anderson and Canary's theorem for unfaithful representations

A new criterion for algebraic and geometric convergence in unfaithful cases

Insights into the limits of discrete representations into PSL(2,C)

Abstract

Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into PSL(2,C) does not contain parabolics, then it is also the sequence's geometric limit. We construct examples that demonstrate the failure of this theorem for certain sequences of unfaithful representations, and offer a suitable replacement.