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PreprintCC BY 4.0arXiv

On ultraproduct approximations and property (T) factors

  • Jesse Peterson
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Abstract

We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II1 factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras L(SL3(ℤ)) and L 𝔽2 are not elementarily equivalent, and we show that the group von Neumann algebra L𝔽2 is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II1 factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which can be taken to be group von Neumann algebras.

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Cite

BibTeXRIS
BibTeX
@misc{peterson2026ultraproduct,
  title = {{On ultraproduct approximations and property (T) factors}},
  author = {Jesse Peterson},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.16669},
  eprint = {2605.16669},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16669},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - On ultraproduct approximations and property (T) factors
AU  - Jesse Peterson
PY  - 2026
DA  - 2026/10/01
PB  - arXiv
DO  - 10.48550/arxiv.2605.16669
UR  - https://arxiv.org/abs/2605.16669
AB  - We introduce a framework allowing for key aspects of deformation/rigidity theory to be used in the study of continuous model theory of II_1 factors. Using this framework, we solve several well-known open problems in the area. For example, we show that the group von Neumann algebras L(SL_3(ℤ)) and L 𝔽_2 are not elementarily equivalent, and we show that the group von Neumann algebra L𝔽_2 is not pseudomatricial. We also show a Bass-Serre type strong rigidity result in the setting of ultraproducts to provide an infinite family of pairwise non-elementarily equivalent full factors, each of which embeds into an ultraproduct of the hyperfinite II_1 factor. Building on previous work of Boutonnet, Chifan and Ioana, we also provide a continuum of pairwise non-elementarily equivalent full factors, which can be taken to be group von Neumann algebras.
ER  -