Ilya Kapovich, (joint work with Dani Wise)
ABSTRACT: A group is said to be co-Hopf if it is not isomorphic to a proper subgroup of itself.
Z.Sela showed that a torsion-free word-hyperbolic group is co-Hopf if it does not split over cyclic groups. He later used the theory of JSJ decomposition to prove that a non-elementary torsion-free word-hyperbolic group $G$ is co-Hopf if and only if $G$ is freely indecomposable. L.Potiagailo obtained similar results for Kleinian groups.
We show that none of these statements holds for finitely generated subgroups of word-hyperbolic groups, by constructing explicit counter-examples.