The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916190975164416 |
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| author | Mészáros, András |
| author_facet | Mészáros, András |
| contents | Let $T_n$ be a $2$-dimensional determinantal hypertree on $n$ vertices. Kahle and Newman conjectured that the $p$-torsion of $H_1(T_n,\mathbb{Z})$ asymptotically follows the Cohen-Lenstra distribution. For $p=2$, we disprove this conjecture by showing that given a positive integer $h$, for all large enough $n$, we have \[\mathbb{P}(\dim H_1(T_n,\mathbb{F}_2)\ge h)\ge \frac{e^{-200h}}{(100h)^{5h}}.\] We also show that $T_n$ is a bad cosystolic expander with positive probability. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_02308 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra Mészáros, András Combinatorics Probability Let $T_n$ be a $2$-dimensional determinantal hypertree on $n$ vertices. Kahle and Newman conjectured that the $p$-torsion of $H_1(T_n,\mathbb{Z})$ asymptotically follows the Cohen-Lenstra distribution. For $p=2$, we disprove this conjecture by showing that given a positive integer $h$, for all large enough $n$, we have \[\mathbb{P}(\dim H_1(T_n,\mathbb{F}_2)\ge h)\ge \frac{e^{-200h}}{(100h)^{5h}}.\] We also show that $T_n$ is a bad cosystolic expander with positive probability. |
| title | The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/2404.02308 |