Hyperbolic Groups That Are Not Commensurably Co-Hopfian

Sela proved that every torsion-free one-ended hyperbolic group is co-Hopfian. We prove that there exist torsion-free one-ended hyperbolic groups that are not commensurably co-Hopfian. In particular, we show that the fundamental group of every simple surface amalgam is not commensurably co-Hopfian.