Let $G$ be a semisimple Lie group with associated symmetric space $D$, and let $\unicode[STIX]{x1D6E4}\subset G$ be a cocompact arithmetic group. Let $\mathscr{L}$ be a lattice inside a $\mathbb{Z}\unicode[STIX]{x1D6E4}$-module arising from a rational finite-dimensional complex representation of $G$. Bergeron and Venkatesh recently gave a precise conjecture about the growth of the order of the torsion subgroup $H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$ as $\unicode[STIX]{x1D6E4}_{k}$ ranges over a tower of congruence subgroups of $\unicode[STIX]{x1D6E4}$. In particular, they conjectured that the ratio $\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode[STIX]{x1D6E4}_{k}]$ should tend to a nonzero limit if and only if $i=(\dim (D)-1)/2$ and $G$ is a group of deficiency $1$. Furthermore, they gave a precise expression for the limit. In this paper, we investigate computationally the cohomology of several (non-cocompact) arithmetic groups, including $\operatorname{GL}_{n}(\mathbb{Z})$ for $n=3,4,5$ and $\operatorname{GL}_{2}(\mathscr{O})$ for various rings of integers, and observe its growth as a function of level. In all cases where our dataset is sufficiently large, we observe excellent agreement with the same limit as in the predictions of Bergeron–Venkatesh. Our data also prompts us to make two new conjectures on the growth of torsion not covered by the Bergeron–Venkatesh conjecture.

中文翻译：

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论算术群上同调中扭的增长

让$G$是具有关联对称空间的半单李群$D$， 然后让$\unicode[STIX]{x1D6E4}\subset G$是一个协紧算术群。让$\mathscr{L}$成为 a 内的格子$\mathbb{Z}\unicode[STIX]{x1D6E4}$-模由有理有限维复数表示$G$. Bergeron 和 Venkatesh 最近给出了关于扭转子群阶数增长的精确猜想$H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}$作为$\unicode[STIX]{x1D6E4}_{k}$范围在一个全等子群的塔上$\unicode[STIX]{x1D6E4}$. 特别是，他们推测该比率$\log |H_{i}(\unicode[STIX]{x1D6E4}_{k};\mathscr{L})_{\operatorname{tors}}|/[\unicode[STIX]{x1D6E4}:\unicode [STIX]{x1D6E4}_{k}]$当且仅当$i=(\dim (D)-1)/2$和$G$是一组缺陷$1$. 此外，他们对极限给出了精确的表达。在本文中，我们在计算上研究了几个（非协紧）算术群的上同调，包括$\operatorname{GL}_{n}(\mathbb{Z})$为了$n=3,4,5$和$\operatorname{GL}_{2}(\mathscr{O})$对于各种整数环，并观察其作为水平函数的增长。在我们的数据集足够大的所有情况下，我们观察到与 Bergeron-Venkatesh 的预测相同的极限一致性。我们的数据还促使我们对 Bergeron-Venkatesh 猜想未涵盖的扭转增长提出两个新猜想。