Cannon-Thurston maps do not always exist

TL;DR

This paper provides a counterexample demonstrating that Cannon-Thurston maps, which relate boundaries of hyperbolic groups and subgroups, do not always exist, challenging previous assumptions in geometric group theory.

Contribution

The authors construct the first explicit example of a hyperbolic group with a hyperbolic subgroup lacking a Cannon-Thurston map, showing such maps are not guaranteed.

Findings

Counterexample of non-existent Cannon-Thurston map

Challenges previous beliefs about boundary maps in hyperbolic groups

Highlights limitations in boundary correspondence theory

Abstract

We construct an example of a hyperbolic group with a hyperbolic subgroup for which the Cannon-Thurston map does not exist. That is, inclusion does not induce a map of the boundaries.