Limits of limit sets in rank-one symmetric spaces

TL;DR

This paper investigates the continuity properties of limit sets in rank-one symmetric spaces, establishing conditions under which these sets and associated maps vary continuously, extending classical Kleinian results to a broader geometric context.

Contribution

It extends Kleinian theory to rank-one symmetric spaces by proving continuity of limit sets and Cannon-Thurston maps under strong convergence and weak type-preserving conditions.

Findings

Limit sets vary continuously under strong convergence.

Cannon-Thurston maps converge uniformly under weak type-preserving sequences.

The approach uses extended geometrically finite representations.

Abstract

We consider the question of continuity of limit sets for sequences of geometrically finite subgroups of isometry groups of rank-one symmetric spaces, and prove analogues of classical (Kleinian) theorems in this context. In particular we show that, assuming strong convergence of the sequence of subgroups, the limit sets vary continuously with respect to Hausdorff distance, and if the sequence is weakly type-preserving, the sequence of Cannon-Thurston maps also converges uniformly to a limiting Cannon-Thurston map. Our approach uses the theory of extended geometrically finite representations, developed recently by the second author.