Homological vanishing for the Steinberg representation

Abstract

For a field $k$, we prove that the $i$th homology of the groups $G{L}_n(k)$, $S{L}_n(k)$, $S{p}_{2n}(k)$, $S{O}_{n,n}(k)$, and $S{O}_{n,n+1}(k)$ with coefficients in their Steinberg representations vanish for $n\ge 2i+2$.

21 pages, 1 figure; major revision; to appear in Compositio Mathematica

Cited by in corpus (4)

• Stability in the high-dimensional cohomology of congruence subgroups
• Homological stability for classical groups
• Steinberg homology, modular forms, and real quadratic fields
• Cohomology of congruence subgroups of SL_3(Z), Steinberg modules, and real quadratic fields