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The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra
András Mészáros · 2024-04-03 · via math.PR updates on arXiv.org

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.

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