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An auxiliary free construction based on HNN-extensions and on free product ofgroups with amalgamated subgroups is suggested, and some of its basicproperties are displayed. This construction is a generalization for many ofconstructions used by Higman in embeddings of recursive groups into finitelypresented groups, as well as for constructions we used in research onembeddings of recursive groups recently. Usage of this auxiliary technicalconstruction simplifies some embedding methods for recursive groups. A fewother results on specific subgroups of HNN-extensions of groups and of freeproduct of groups with amalgamated subgroup are obtained.

Auxiliary free constructions for explicit embeddings of recursive groups