## The co-Hopf property and subgroups of hyperbolic groups

**Ilya Kapovich**, (joint work with Dani Wise)

(Rutgers University)

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.