Convergence and divergence of Kleinian surface groups

TL;DR

This paper characterizes the convergence behavior of Kleinian surface groups by analyzing the asymptotic properties of their associated hyperbolic 3-manifolds' end invariants, linking algebraic and geometric limits.

Contribution

It provides a new characterization of convergent sequences of Kleinian surface groups based on end invariants, connecting algebraic and geometric limits.

Findings

Sequences with convergent subsequences are characterized by their end invariants.

Asymptotic end invariants determine the parabolic locus of the algebraic limit.

The wrapping of the algebraic limit within the geometric limit is described by end invariants.

Abstract

We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent sequence predicts the parabolic locus of the algebraic limit as well as how the algebraic limit wraps within the geometric limit under the natural locally isometric covering map.