Beschreibung:
<jats:p>We prove a representation stability result for the codimension-one cohomology of the level-three congruence subgroup of <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007046_inline1.png" /><jats:tex-math>$\mathbf{SL}_{n}(\mathbb{Z})$</jats:tex-math></jats:alternatives></jats:inline-formula>. This is a special case of a question of Church, Farb, and Putman which we make more precise. Our methods involve proving finiteness properties of the Steinberg module for the group <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007046_inline2.png" /><jats:tex-math>$\mathbf{SL}_{n}(K)$</jats:tex-math></jats:alternatives></jats:inline-formula> for <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007046_inline3.png" /><jats:tex-math>$K$</jats:tex-math></jats:alternatives></jats:inline-formula> a field. This also lets us give a new proof of Ash, Putman, and Sam’s homological vanishing theorem for the Steinberg module. We also prove an integral refinement of Church and Putman’s homological vanishing theorem for the Steinberg module for the group <jats:inline-formula><jats:alternatives><jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="png" mimetype="image" xlink:href="S0010437X20007046_inline4.png" /><jats:tex-math>$\mathbf{SL}_{n}(\mathbb{Z})$</jats:tex-math></jats:alternatives></jats:inline-formula>.</jats:p>