The conjugacy problem for groups, and Higman embeddings

Abstract

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