Complete embeddings of groups (Q6666500)

\textit{C. F. Miller} and \textit{P. E. Schupp}, in [J. Algebra 17, 171--176 (1971; Zbl 0215.11002)], used small cancellation theory to prove that every countable group \(G\) can be embedded in a finitely generated group \(G^{\ast}\) that is hopfian and complete (asymmetric). Their proof depends on theory of quotients of free products by \textit{C. L. Lyndon} [Math. Ann. 166, 208--228 (1966; Zbl 0138.25702)].\N\NIn the paper under review, the authors present an alternative construction and prove Theorem A: Every countable group \(G\) can be embedded in a finitely generated group \(G^{\ast}\) such that: (1) \(G^{\ast}\) is hopfian and complete; (2) every finite subgroup of \(G^{\ast}\) is conjugate to a finite subgroup of \(G\); (3) if \(G\) has a finite presentation (respectively, a finite classifying space of dimension \(d \geq 3\)), then so does \(G^{\ast}\).\N\NThe proof relies on the existence of asymmetric hyperbolic groups with some additional properties.