Embedding finitely presented self-similar groups into finitely presented simple groups

TL;DR

This work resolves a broad case of the Boone–Higman conjecture by showing that every finitely presented self-similar group embeds in a finitely presented simple group. The core idea is to use Röver–Nekrashevych groups $V_d(G)$ and their commutator subgroups $[V_d(G),V_d(G)]$, together with a fine-tuned even-multiple construction $V_{md}(G)$ to ensure finite abelianization, making the commutator finitely presented and simple. A key technical tool is a wreath-product embedding ensuring that finite groups act inside the commutator, enabling $G$ to sit inside $[V_{md}(G),V_{md}(G)]$; a linear-algebra argument guarantees the finite abelianization for some even $m$. The results yield wide applicability, including that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies Boone–Higman, and the framework extends via virtual endomorphisms to produce many new self-similar, finitely presented examples.

Abstract

We prove that every finitely presented self-similar group embeds in a finitely presented simple group. This establishes that every group embedding in a finitely presented self-similar group satisfies the Boone-Higman conjecture. The simple groups in question are certain commutator subgroups of Röver-Nekrashevych groups, and the difficulty lies in the fact that even if a Röver-Nekrashevych group is finitely presented, its commutator subgroup might not be. We also discuss a general example involving matrix groups over certain rings, which in particular establishes that every finitely generated subgroup of $\mathrm{GL}_n(\mathbb{Q})$ satisfies the Boone-Higman conjecture.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1: Self-similar
• Definition 2.2: Röver--Nekrashevych group
• Proposition 2.5
• proof
• proof : Proof of Theorem \ref{['thrm:main']}
• Definition 4.1: Tree of cosets
• Definition 4.2: $\Gamma$-stable, mutually stable
• Lemma 4.3: Faithful