Limits of limit sets II: Geometrically Infinite Groups

TL;DR

This paper investigates the convergence properties of Cannon-Thurston maps in sequences of Kleinian groups, demonstrating uniform convergence in certain cases and failure of pointwise convergence in others, revealing limits of geometric and algebraic convergence.

Contribution

It establishes conditions under which Cannon-Thurston maps converge uniformly and provides examples where pointwise convergence fails, advancing understanding of Kleinian group limits.

Findings

Uniform convergence of Cannon-Thurston maps for strongly convergent sequences.

Existence of algebraically convergent sequences with non-convergent Cannon-Thurston maps.

Differentiation between geometric and algebraic convergence effects.

Abstract

We show that for a strongly convergent sequence of purely loxodromic finitely generated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as maps from a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically convergent sequences we show that there exist examples where even pointwise convergence of Cannon-Thurston maps fails.