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Non-sofic Tarski monsters

  • Kıvanç Ersoy
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Abstract

We prove the existence of non-sofic Tarski monsters of every sufficiently large prime exponent and of torsion-free non-sofic Tarski monsters. The proof combines a theorem of Sandhu--Thom with quotient theorems of Semenov. These groups are minimal non-sofic, since all their proper subgroups are cyclic. A theorem of Minasyan--Osin--Witzel gives continuum many pairwise non-quasi-isometric torsion-free simple examples. Kulikova's quotient theorem also yields non-simple minimal non-sofic groups of exponent pm for every m≥2 and every sufficiently large prime p. We then study descending chains of non-sofic subgroups in other constructions. We give proper self-embeddings of the binary Leavitt unit group, higher Brin--Thompson groups and the hyperbolic group studied by Douglas and Mghirbi. The explicit Kun--Thom wreath products and doubles are locally graded. These examples are ω-non-sofic, and the locally graded ones contain no minimal non-sofic subgroup.

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Cite

BibTeXRIS
BibTeX
@misc{ersoy2026nonsofic,
  title = {{Non-sofic Tarski monsters}},
  author = {Kıvanç Ersoy},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.42140},
  eprint = {2610.42140},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.42140},
}
RIS
TY  - UNPB
TI  - Non-sofic Tarski monsters
AU  - Kıvanç Ersoy
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2610.42140
UR  - https://arxiv.org/abs/2610.42140
AB  - We prove the existence of non-sofic Tarski monsters of every sufficiently large prime exponent and of torsion-free non-sofic Tarski monsters. The proof combines a theorem of Sandhu--Thom with quotient theorems of Semenov. These groups are minimal non-sofic, since all their proper subgroups are cyclic. A theorem of Minasyan--Osin--Witzel gives continuum many pairwise non-quasi-isometric torsion-free simple examples. Kulikova's quotient theorem also yields non-simple minimal non-sofic groups of exponent p^m for every m≥2 and every sufficiently large prime p. We then study descending chains of non-sofic subgroups in other constructions. We give proper self-embeddings of the binary Leavitt unit group, higher Brin--Thompson groups and the hyperbolic group studied by Douglas and Mghirbi. The explicit Kun--Thom wreath products and doubles are locally graded. These examples are ω-non-sofic, and the locally graded ones contain no minimal non-sofic subgroup.
ER  -