The Double Limit Theorem and its Legacy

TL;DR

This paper reviews the development and impact of the Double Limit Theorem in Kleinian group theory, highlighting its role in understanding the convergence of Kleinian representations and related theorems.

Contribution

It provides a comprehensive survey of recent and older results related to the Double Limit Theorem and its influence on the compactness of certain hyperbolic 3-manifold moduli spaces.

Findings

Convergence criteria for Kleinian representations

Implications of the Double Limit Theorem on hyperbolic 3-manifolds

Connections to the AH(acylindrical) compactness theorem

Abstract

This chapter surveys recent and less recent results on convergence of Kleinian representations, following Thurston's Double Limit and "AH(acylindrical) is compact" Theorems.