Year: 2026
Type: preprint
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.
Source: arXiv (Cornell University)
Author: Salo, Ville
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221106947