1 results
Stability in the high-dimensional cohomology of congruence subgroups
- Part of
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- Journal:
- Compositio Mathematica / Volume 156 / Issue 4 / April 2020
- Published online by Cambridge University Press:
- 24 March 2020, pp. 822-861
- Print publication:
- April 2020
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- Article
- Export citation
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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, and Putman which we make more precise. Our methods involve proving 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, 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 
$\mathbf{SL}_{n}(\mathbb{Z})$.
