On the quasi-isometry invariance of \(L^ 2\)-Betti numbers (Q918017)

In two previous papers [J. Differ. Geom. 27, No.1, 87-113, 115-136 (1988; Zbl 0657.58041)] the author developed an index theory on open manifolds of bounded geometry which are equipped with a regular exhaustion. In this setting he defined ``renormalized \(L^ 2\)-Betti numbers''. These Betti numbers depend on the germ of the spectrum of the Laplacian near zero and not just on the space of harmonic forms. Their alternating sum is related to the ``mean Euler characteristic'' of the manifold, which is defined as a limit of ``normalized'' integrals of the Euler form using the regular exhaustion of the manifold. In the present paper it is proved that these \(L^ 2\)-Betti numbers are invariants of strict quasi-isometry. The definition of strict quasi-isometry is a technical one, but it ``may be considered as the analogue of diffeomorphism in the category of manifolds of bounded geometry''.