A modified proof for Higman's embedding theorem

Abstract

We suggest a modified and briefer version for the proof of Higman's embedding theorem stating that a finitely generated group can be embedded in a finitely presented group if and only if it is recursively presented. In particular, we shorten the main part of original proof establishing characterization of recursive relations in terms of benign subgroups in free groups. Also, some technical lemmas on homomorphisms in free constructions are replaced by simple combinatorial observations on words of specific type."We do fundamental research, not only to acquire results solely, but because the process is an ennobling one"

Embeddings using universal words in the free group of rank 2

“…Background information on these constructions can be found in [10,3,26]. Also, we refer to our recent note [18] for specific notations that we also share here to write the free product G * ϕ H = 〈G, H | a = a ϕ for all a ∈ A〉 of groups G and H with subgroups A and B amalgamated under the isomorphism ϕ : A → B; and the HNN-extension G * ϕ t = 〈G, t | a t = a ϕ for all a ∈ A〉 of the base group G by the stable letter t with respect to the isomorphism ϕ : A → B of the subgroups A, B ≤ G. We also use HNN-extensions…”

“…Lemma 2.1 is a slight variation of Lemma 3.1 given on p. 465 of [8] without a proof as "obvious from the normal form theorem for free products with an amalgamation". The proof can be found in subsection 2.5 of [18]. (1)…”

The Higman operations and embeddings of recursive groups

“…For background on free constructions, such as, free products, free products with amalgamated subgroups, HNN-extension we refer to textbooks [9,1,18]. See also the recent note [12] from where we adopt the notations related to free constructions without restating the definitions here. Information on varieties of groups can be found in Hanna Neumann's monography [15].…”

“…To get familiar with these operations the reader may check examples and basic lemmas in Section 2 of [6] or in Subsection 2.2 of [12].…”

“…For basic properties and examples of benign subgroups we refer to Section 3 in [6] or to subsections 3.1 and 3.2 in [12].…”