  Associated to every group with a weak spherical Tits system of rank n+1 with
an appropriate rank n subgroup, we construct a relative spectral sequence
involving group homology of Levi subgroups of both groups. Using the fact that
such Levi subgroups frequently split as semidirect products of smaller groups,
we prove homological stability results for unitary groups over division rings
with infinite centre as well as for special linear and special orthogonal
groups over infinite fields.
