Algebraic and geometric convergence of discrete representations into $\mathrm{PSL}_2\mathbb{C}$

Abstract

Anderson and Canary have shown that if the algebraic limit of a sequence of discrete, faithful representations of a finitely generated group into ${PSL}_2ℂ$ 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.