Explicit construction of benign subgroup for Higman's reversing operation

TL;DR

This paper demonstrates how to explicitly construct benign subgroups for Higman's reversing operation, advancing the development of algorithms for embedding recursive groups into finitely presented groups.

Contribution

It provides a method to explicitly construct benign subgroups after applying Higman's reversing operation, facilitating Higman embeddings of recursive groups.

Findings

Benign subgroups are preserved under Higman's reversing operation.

Explicit constructions of overgroups and subgroups are possible after the operation.

Supports the development of algorithms for Higman embeddings.

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.