Conjugacy problems in higher-dimensional Thompson groups

2026-10-07

AI Summary

This study addresses the long-standing open question regarding the decidability of topological conjugacy, the Whitehead problem, and group-theoretic conjugacy in higher-dimensional Thompson groups (nV, n≥2). By employing Turing machine reductions, this work establishes a profound connection between the halting problem and the dynamics of Cantor space homeomorphisms. It further leverages minimal subsystem dynamics to effectively distinguish non-halting instances and extends the analysis to supergroup structures via automata orbit theory. Consequently, this research provides the first rigorous proof of the undecidability of these three problems within nV groups and their finitely generated supergroups. These findings delineate the computational complexity boundaries for multiple conjugacy problems in higher-dimensional Thompson groups, thereby offering a novel paradigm for interdisciplinary research at the intersection of geometric group theory and dynamical systems.

Abstract

We prove that topological conjugacy of elements of the Brin--Thompson group $nV$ is undecidable for every $n\geq2$. From a Turing machine $T$, the reduction produces two elements of $nV$, which are conjugate by an involution in $nV$ if $T$ halts on the empty tape. If $T$ does not halt on the empty tape, the topological dynamics of the two group elements are distinguished by their visits to minimal subsystems of the set of (germ-)aperiodic points. In these groups, and any finitely-generated supergroups inside the homeomorphism group of Cantor space, we also obtain undecidability of the Whitehead problem (the problem of being in the same automorphism orbit), and undecidability of group-theoretic conjugacy.

Problem

• Thompson groups
• topological conjugacy
• undecidability
• Whitehead problem
• group-theoretic conjugacy

Innovation

• Brin-Thompson groups
• topological conjugacy
• undecidability
• Whitehead problem
• Cantor space