Auxiliary free constructions for explicit embeddings of recursive groups

TL;DR

This paper introduces a new auxiliary free construction based on HNN-extensions and free products, simplifying embedding methods for recursive groups and enabling the creation of specific subgroups within free groups.

Contribution

It generalizes Higman's embedding techniques using a novel construction, improving the process of embedding recursive groups into finitely presented groups.

Findings

Properties of the new construction are established.

Technical results on subgroups in HNN-extensions and free products are proved.

Application to constructing infinitely generated benign subgroups in free groups.

Abstract

An auxiliary free construction $*_{i=1}^{r}(K_i, L_i, t_i)_M$ based on HNN-extensions and on generalized free product of groups with amalgamated subgroups is suggested, and some of its basic properties are displayed. The proposed construction is a generalization of a series of structures used by Higman for embeddings of recursive groups into finitely presented groups, and also for certain structures we recently applied in the research on embeddings of recursive groups. Usage of this technical tool substantially simplifies some embedding methods for recursive groups. A few technical results on specific subgroups in the suggested $*$-constructions, in the HNN-extensions of groups, and in free products of groups with amalgamated subgroup are proved. The obtained properties are applied to build infinitely generated benign subgroups inside free groups of small rank.