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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2603.17869 |
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| _version_ | 1866912972985597952 |
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| author | Pikhurko, Oleg Sakamoto, Kohki |
| author_facet | Pikhurko, Oleg Sakamoto, Kohki |
| contents | For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for $n \ge 3$, but obtained only a partial result for $n=2$. In this paper, we prove that the zero-one law also holds for $n=2$. We also remark that a Baire categorical analogue of this result holds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_17869 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the spectral gap conjecture for pairs in SU(2) Pikhurko, Oleg Sakamoto, Kohki Group Theory Dynamical Systems For $n \ge 2$, Gamburd, Jakobson, and Sarnak [J. Eur. Math. Soc. 1, 51-85 (1999)] conjectured that almost every $n$-tuple in $\mathrm{SU}(2)$ has a spectral gap. Toward this conjecture, Fisher [Int. Math. Res. Not. (2006)] established a zero-one law for $n \ge 3$, but obtained only a partial result for $n=2$. In this paper, we prove that the zero-one law also holds for $n=2$. We also remark that a Baire categorical analogue of this result holds. |
| title | On the spectral gap conjecture for pairs in SU(2) |
| topic | Group Theory Dynamical Systems |
| url | https://arxiv.org/abs/2603.17869 |