  Building upon work of Y. Shalom we give a homological-algebra flavored
definition of an induction map in group homology associated to a topological
coupling. As an application we obtain estimates of the (co)homological
dimension of groups G and H, where G embeds uniformly into H and the
(co)homological dimension of G is finite. Another consequence of our results is
that the Hirsch ranks of quasi-isometric solvable groups coincide. Further, it
is shown that the real cohomology rings of quasi-isometric nilpotent groups are
isomorphic as graded rings. On the analytic side, we apply the induction
technique to Novikov-Shubin invariants of amenable groups, which can be seen as
homological invariants, and show their invariance under quasi-isometry.
