Cannon-Thurston maps, subgroup distortion, and hyperbolic hydra

TL;DR

This paper demonstrates the existence of Cannon--Thurston maps for heavily distorted free subgroups within hyperbolic hydra groups, showing such maps can exist despite extreme distortion levels.

Contribution

It proves the existence of Cannon--Thurston maps for free subgroups in hyperbolic hydra groups, even with arbitrarily heavy primitive recursive distortion.

Findings

Cannon--Thurston maps exist despite heavy subgroup distortion

Hyperbolic hydra groups contain heavily distorted free subgroups

Boundary maps can exist with non-recursive distortion levels

Abstract

There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon--Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free subgroups. It is known that Cannon--Thurston maps between hyperbolic space boundaries can exist even in the presence of arbitrarily heavy (even non-recursive) distortion. The hyperbolic hydra examples show that Cannon--Thurston maps can exist even between hyperbolic group boundaries in the presence of arbitrarily heavy primitive recursive distortion.