Complete Embeddings of Groups

Every countable group $G$ can be embedded in a finitely generated group $G^{\ast }$ that is hopfian and complete, i.e. $G^{\ast }$ has trivial centre and every epimorphism $G^{\ast }\to G^{\ast }$ is an inner automorphism. Every finite subgroup of $G^{\ast }$ is conjugate to a finite subgroup of $G$. If $G$ has a finite presentation (respectively, a finite classifying space), then so does $G^{\ast }$. Our construction of $G^{\ast }$ relies on the existence of closed hyperbolic 3-manifolds that are asymmetric and non-Haken.