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Conjugacy problems in higher-dimensional Thompson groups

  • Ville Salo
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Abstract

We prove that topological conjugacy of elements of the Brin--Thompson group nV is undecidable for every n≥2. 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.

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Cite

BibTeXRIS
BibTeX
@misc{salo2026conjugacy,
  title = {{Conjugacy problems in higher-dimensional Thompson groups}},
  author = {Ville Salo},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.11124},
  eprint = {2610.11124},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.11124},
}
RIS
TY  - UNPB
TI  - Conjugacy problems in higher-dimensional Thompson groups
AU  - Ville Salo
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.11124
UR  - https://arxiv.org/abs/2610.11124
AB  - We prove that topological conjugacy of elements of the Brin--Thompson group nV is undecidable for every n≥2. 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.
ER  -