Embeddings using universal words in the free group of rank 2

TL;DR

The paper presents a straightforward method to embed countable groups into 2-generator groups with explicit generator images and automatically derived relations, demonstrated on specific examples and applicable to recursive groups.

Contribution

It introduces a simple, explicit embedding technique for countable groups into 2-generator groups with automatic relation deduction, advancing group embedding methods.

Findings

Embedding method explicitly constructs images of generators.

Relations of the target group are automatically deduced.

Method applied successfully to specific group examples.

Abstract

For a countable group G = <A | R> presented by its generators A and defining relations R we discuss a simple method to embed G into such a 2-generator group T that the images of generators from A are explicitly given in T, and the defining relations for T can automatically be deduced from the relations R. The obtained method is applied on particular examples of groups, and references of its application for embeddings of recursive groups into finitely presented groups are given.