Explicit construction of benign subgroup for Higman’s reversing operation

For the Higman reversing operation [Formula: see text] and for a set of integer-valued functions [Formula: see text], the following has been proved. Let the subgroup [Formula: see text] be benign in the free group [Formula: see text], let the respective finitely presented overgroup [Formula: see text] with its finitely generated subgroup [Formula: see text] be given for [Formula: see text] explicitly, and let the set [Formula: see text] be obtained from [Formula: see text] by the reversing operation [Formula: see text]. Then [Formula: see text] also is benign in [Formula: see text] and, moreover, the finitely presented overgroup [Formula: see text] with its finitely generated subgroup [Formula: see text] can also be given explicitly for it. This work is a step in development of a general algorithm for construction of explicit Higman embeddings of recursive groups into finitely presented groups.