The Conjugacy Problem and Higman Embeddings (Memoirs of the American Mathematical Society)

For every finitely generated recursively presented group $\mathcal G$ we construct a finitely presented group $\mathcal H$ containing $\mathcal G$ such that $\mathcal G$ is (Frattini) embedded into $\mathcal H$ and the group $\mathcal H$ has solvable conjugacy problem if and only if $\mathcal G$ has solvable conjugacy problem. Moreover $\mathcal G$ and $\mathcal H$ have the same r.e. Turing degrees of the conjugacy problem. This solves a problem by D. Collins.