The $2$-torsion of determinantal hypertrees is not Cohen-Lenstra

TL;DR

This paper disproves a conjecture about the distribution of 2-torsion in the homology of determinantal hypertrees, showing it does not follow the Cohen-Lenstra distribution and establishing new properties like cosystolic expansion.

Contribution

It provides the first counterexample to the Cohen-Lenstra conjecture for 2-torsion in determinantal hypertrees and demonstrates their cosystolic expansion properties.

Findings

Disproves Cohen-Lenstra distribution for 2-torsion in $H_1(T_n,\mathbb{Z})$

Shows $T_n$ is a bad cosystolic expander with positive probability

Provides probability bounds for large 2-torsion in homology

Abstract

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.