[Paper Review] A modified proof for Higman's embedding theorem

Vahagn H. Mikaelian|arXiv (Cornell University)|Aug 27, 2019

Geometric and Algebraic Topology22 references4 citations

TL;DR

This paper presents a streamlined, more accessible proof of Higman's embedding theorem, demonstrating that a finitely generated group embeds into a finitely presented group if and only if it is recursively presented. By simplifying the characterization of recursive relations via benign subgroups in free groups using word-combinatorics and nested HNN-extensions, the proof shortens the original argument and enables explicit constructive embeddings of recursive groups into finitely presented groups.

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 are replaced by simple combinatorial observations on words in free constructions.

Motivation & Objective

• To provide a shorter, more transparent proof of Higman's embedding theorem, reducing the complexity of the original argument.
• To simplify the characterization of recursively enumerable subsets of functions via benign subgroups in free groups.
• To make the embedding construction explicit and adaptable for concrete group embeddings, such as ℚ into finitely presented groups.
• To replace intricate group-theoretic lemmas in Higman’s original proof with elementary word-combinatorics and normal form analysis.
• To preserve Higman’s original conceptual framework while enhancing clarity and constructivity through structured examples and step-by-step normal form reductions.

Proposed method

• Construct a group $\Omega'$ via three nested HNN-extensions over $\langle b,c\rangle$, enabling a unique nested normal form for elements.
• Use the subgroup $W_{\mathcal{B}} = \langle g_f, a, r \mid f \in \mathcal{B} \rangle$ to model the action of Higman’s operation $\omega_m$ on recursive sets.
• Apply word-combinatorics to analyze when elements in $W_{\mathcal{B}}$ reduce to elements in $\langle a,b,c\rangle$, relying on conjugation relations $a^{g_f} = a^{b_f}$.
• Use normal form analysis to determine when a word $w \in W_{\mathcal{B}}$ lies in $\langle a,b,c\rangle$, showing it lies in $A_{\omega_m \mathcal{B}}$ iff the power of $r$ is zero.
• Leverage Lemma 3.2 and Corollary 2.5 to conclude that $A_{\omega_m \mathcal{B}}$ is benign if $A_{\mathcal{B}}$ is benign, via closure under $\omega_m$.
• Replace Higman’s complex amalgamated product constructions with a simpler, combinatorially transparent HNN-based framework.

Experimental results

Research questions

• RQ1 Can the proof of Higman’s embedding theorem be significantly shortened while preserving its logical structure?
• RQ2 Is the closure of benign subgroups under Higman’s operation $\omega_m$ derivable through elementary word-combinatorics rather than complex group amalgamations?
• RQ3 Can the embedding process be made constructive and explicit, enabling concrete embeddings of recursive groups like $\mathbb{Q}$ into finitely presented groups?
• RQ4 To what extent can the original proof’s reliance on advanced lemmas be replaced by direct normal form analysis and simple homomorphism arguments?
• RQ5 Does a nested HNN-extension construction yield a more transparent and verifiable path to proving that $A_{\omega_m \mathcal{B}}$ is benign when $A_{\mathcal{B}}$ is benign?

Key findings

• The paper provides a shorter proof of Theorem 1.2, showing that a subset $\mathcal{B} \subseteq \mathcal{E}$ is recursively enumerable if and only if the subgroup $A_{\mathcal{B}} = \langle a_f \mid f \in \mathcal{B} \rangle$ is benign in the free group $F = \langle a,b,c \rangle$.
• The closure of benign subgroups under Higman’s operation $\omega_m$ is established via a direct, combinatorial analysis of nested normal forms in a HNN-extension, replacing the original complex amalgamated product construction.
• The construction ensures that $A_{\omega_m \mathcal{B}} \leq W_{\mathcal{B}} \cap \langle a,b,c \rangle$, and that $w \in \langle a,b,c \rangle$ only if the power of $r$ in its normal form is zero, which characterizes membership in $A_{\omega_m \mathcal{B}}$.
• The method allows for explicit embeddings of recursive groups into finitely presented groups, as demonstrated by the construction of such embeddings for $\mathbb{Q}$ in a follow-up work.
• The proof replaces Higman’s intricate lemmas (e.g., Lemma 3.8–3.10) with elementary word-combinatorics and direct verification of conjugation relations, significantly simplifying the argument.
• The subgroup $\Omega'$, built via three nested HNN-extensions, provides a full list of defining relations and enables unique normal forms, making the analysis of subgroup membership decidable and transparent.