We prove a representation stability result for the codimension-one cohomology of the level three congruence subgroup of $\mathbf{SL}_n(\mathbb{Z})$. This is a special case of a question of Church-Farb-Putman which we make more precise. Our methods involve proving several finiteness properties of the Steinberg module for the group $\mathbf{SL}_n(K)$ for $K$ a field. This also lets us give a new proof of Ash-Putman-Sam's homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church-Putman's homological vanishing theorem for the Steinberg module for the group $\mathbf{SL}_n(\mathbb{Z})$.

中文翻译：

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同余子群高维上同调的稳定性

我们证明了$\mathbf{SL}_n(\mathbb{Z})$ 的三级同余子群的余维一上同调的表示稳定性结果。这是 Church-Farb-Putman 问题的一个特例，我们对其进行了更精确的描述。我们的方法涉及为 $K$ 一个字段证明组 $\mathbf{SL}_n(K)$ 的 Steinberg 模块的几个有限性属性。这也让我们为 Steinberg 模块提供了 Ash-Putman-Sam 同调消失定理的新证明。我们还证明了群 $\mathbf{SL}_n(\mathbb{Z})$ 的 Steinberg 模的 Church-Putman 同调消失定理的整体改进。