Strong solidity classification of Coxeter groups
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915646423433216 |
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| author | Blufstein, Martín Goldman, Katherine Oyakawa, Koichi |
| author_facet | Blufstein, Martín Goldman, Katherine Oyakawa, Koichi |
| contents | We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled Coxeter groups by Borst-Caspers. However, our proof is conceptually different, which leads to a significantly streamlined argument. We also provide additional equivalent geometric and group-theoretic characterizations of strong solidity for Coxeter groups that allow us to completely classify those with a strongly solid group von Neumann algebra. In particular, we characterize strong solidity purely in terms of the defining Coxeter-Dynkin diagram. Finally, we obtain the same dichotomy for virtually cocompact special groups. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_20559 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Strong solidity classification of Coxeter groups Blufstein, Martín Goldman, Katherine Oyakawa, Koichi Operator Algebras Group Theory 20F55, 46L10, 20F65, 46L05 We prove the dichotomy that every Coxeter group either has a strongly solid group von Neumann algebra or contains the product of an infinite cyclic group and a free group of rank 2. This generalizes the same dichotomy for right-angled Coxeter groups by Borst-Caspers. However, our proof is conceptually different, which leads to a significantly streamlined argument. We also provide additional equivalent geometric and group-theoretic characterizations of strong solidity for Coxeter groups that allow us to completely classify those with a strongly solid group von Neumann algebra. In particular, we characterize strong solidity purely in terms of the defining Coxeter-Dynkin diagram. Finally, we obtain the same dichotomy for virtually cocompact special groups. |
| title | Strong solidity classification of Coxeter groups |
| topic | Operator Algebras Group Theory 20F55, 46L10, 20F65, 46L05 |
| url | https://arxiv.org/abs/2511.20559 |