Embeddings using universal words in the free group of rank 2

Abstract

For a countable group G = 〈 A | R 〉 presented by its generators A and defining relations R we discuss a simple method of embedding for 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.

The Higman operations and embeddings of recursive groups

“…where each a i corresponds to the primitive (p i )'th root ǫ i of unity [8]. As we found in Example 3.6 in [13], this group can be embedded into the 2-generator group:…”

“…Higman's embedding construction [6] starts by some effective embedding of G into an appropriate T . In the recent note [13] we suggested a method of effective embedding of any countable group G into a 2-generator group T such that the defining relations of T are straightforward to deduce from relations of G. In fact, the very first embedding construction [5] (based on free constructions) and some other embedding constructions (based on wreath products, group extensions, etc.) already allow to find the relations of T .…”

“…. , on letters b, c (see Theorem 1.1 in [13]). If R is recursively enumerable, then the set R ′ of all above relations r ′ s (b, c) also is recursively enumerable.…”