  For a field $k$, we prove that the $i$th homology of the groups $GL_n(k)$,
$SL_n(k)$, $Sp_{2n}(k)$, $SO_{n,n}(k)$, and $SO_{n,n+1}(k)$ with coefficients
in their Steinberg representations vanish for $n \geq 2i+2$.
