Limits of limit sets I

TL;DR

This paper investigates the convergence properties of Cannon-Thurston maps for sequences of geometrically finite Kleinian groups, revealing conditions under which these maps converge uniformly or only pointwise.

Contribution

It establishes new results on the uniform and pointwise convergence of limit set maps in Kleinian groups, especially when algebraic and geometric limits differ.

Findings

Uniform convergence occurs when algebraic and geometric limits coincide.

Pointwise but not uniform convergence occurs when algebraic and geometric limits differ.

Geometric finiteness of the limit influences convergence behavior.

Abstract

We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometric limit is geometrically finite, the maps of limit sets converge pointwise but not uniformly.