Cannon–Thurston maps, subgroup distortion, and hyperbolic hydra

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.