  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.
