A group $G$ is co-hopfian if every injection $f\colon G \rightarrow G$ is an automorphism, or equivalently if $G$ is not isomorphic to any of its proper subgroups. Miller and Schupp, using small cancellation theory, showed that every countable group that does not contain elements of every finite order can be embedded in a 2-generated co-hopfian group. I can't find a more general result.
Is it known if every countable group embeds into a finitely generated co-hopfian group? At least, does every finitely generated group embed into a finitely generated co-hopfian group? My instinct is to use small cancellation theory, but if a group contains elements of every finite order it doesn't seem to give enough control over the embeddings $f\colon G\rightarrow G$, as this is usually accomplished using torsion elements.