A branch group with unsolvable conjugacy problem

TL;DR

The paper answers whether the conjugacy problem can obstruct the solvability of the word problem within finitely generated branch groups by showing how to embed any finitely generated residually finite group $G$ into a finitely generated branch group $\Gamma$ via a Frattini embedding that preserves conjugacy. The construction relies on a computable residual chain and a careful assembly of groups acting on a rooted tree, producing a sequence $\Gamma_\ell$ with a wreath product structure and a robust notion of conjugacy preservation. Consequently, if $G$ has unsolvable conjugacy (via Miller’s example), the same holds for $\Gamma$, while $\Gamma$ can have a solvable word problem when $G$ is recursively presented and effectively residually finite (EFRF+). This yields finitely generated branch groups with solvable word problem but unsolvable conjugacy problem, addressing a question of Bartholdi, Grigorchuk, and Šunić and informing Boone–Higman-type considerations for branch groups.

Abstract

We prove that every finitely generated residually finite group $G$ can be embedded in a finitely generated branch group $Γ$ such that two elements in $G$ are conjugate in $G$ if and only if they are conjugate in $Γ$. As an application we construct a finitely generated branch group with solvable word problem and unsolvable conjugacy problem and thereby answer a question of Bartholdi, Grigorchuk, and Šunik.

Figures (1)

• Figure 1: The action of $\widetilde{h}^{[0]}$ on $\mathcal{T}_X = \mathcal{T}_X^{[0]}$.

Theorems & Definitions (33)

• Theorem A
• Theorem B
• Corollary 1.2
• Definition 2.1
• Lemma 2.2
• proof
• Corollary 2.3
• proof
• Lemma 4.1
• proof