The Higman operations and embeddings of recursive groups

Abstract

In the context of Higman embeddings for recursive groups into finitely presented groups we suggest an algorithm which uses Higman operations to explicitly constructs the specific recursively enumerable sets of integer sequences arising during the embedding. This makes the constructive Higman embedding a doable task for certain wide classes of groups. Specific auxiliary operations are introduced to make the work with Higman operations a simpler and more intuitive procedure. Also, an automated mechanism of constructive embeddings of countable groups into 2-generator groups is mentioned. CONTENTS 10 4.2. Construction of (n) with restrictions on n 10 4.3. Duplication of the last term 11 4.4. Construction of the pairs (n, −n)114.5. Construction of the triples (p, q, p − q) and (p, q, p + q) 13 4.6. An application of the method 16 References 16

Embeddings using universal words in the free group of rank 2

“…In particular, compare Example 3.4 from this note to Example 3.1, Example 3.2 and Example 4.11 with "abacus machine" in [19].…”

“…This allows to study the recursively enumerable sets of relations by means of certain sets of sequences of integers in [8]. As we see in [19], the embeddings of Theorem 1.1 and Theorem 3.2 guarantee some close correlations between the relations of G and those sets of sequences, which allows us to build constructive embeddings of G into a finitely presented group.…”

“…Moreover, we need embeddings preserving certain structure of relations. For the details we refer to [19], and give just rough idea here. Each relation of a 2-generator group T can be coded by means of a certain sequence of integers.…”

Embedding ℚ into a finitely presented group

“…Further finitely generated examples were later supplied by Mikaelian [18]. As for embeddings into finitely presented groups, Mikaelian [19] has described how to explicitly carry out Higman's construction for Q as well as many other groups of interest, such as GL n (Q).…”