Explicit construction of benign subgroup for Higman's reversing operation

Abstract

For the Higman reversing operation $\rho$ and for a set of integer-valued functions $\mathcal{X}$ the following has been proved. Let the subgroup $A_{\mathcal{X}}$ be benign in the free group $F$, let the respective finitely presented overgroup $K_{\mathcal{X}}$ with its finitely generated subgroup $L_{\mathcal{X}}$ be given for $A_{\mathcal{X}}$ explicitly, and let the set $\mathcal{Y}=\rho \mathcal{X}$ be obtained from $\mathcal{X}$ by the reversing operation $\rho$. Then $A_{\mathcal{Y}}$ also is benign in $F$ and, moreover, the finitely presented overgroup $K_{\mathcal{Y}}$ with its finitely generated subgroup $L_{\mathcal{Y}}$ can also be given explicitly for it. The current work is a step in development of a general algorithm for construction of explicit Higman embeddings of recursive groups into finitely presented groups.