## On the zero-in-the-spectrum conjecture.(English)Zbl 0992.58012

The authors answer a question of J. Lott [Enseign. Math. (2) 42, 341-376 (1996; Zbl 0874.58086)] by showing:
Theorem 1: For any $$n\geq 6$$, there exists a closed $$n$$ dimensional smooth manifold $$M$$ so that for any $$p=1,\dots ,n$$, $$0$$ does not belong to the spectrum of the Laplacian acting on the space of $$L^2$$ forms on the universal covering of $$M$$.
The authors prove Theorem 1 by restating it in an equivalent form using extended $$L^2$$ homology:
Theorem 2: For any $$n\geq 6$$, there exists a closed orientable smooth $$n$$ dimensional manifold $$M$$ so that the extended $$L^2$$ homology $$H_p(M;\ell^2(\pi))$$ vanishes for all $$p$$ where $$\pi$$ is the fundamental group of $$M$$ and $$\ell^2(\pi)$$ is the $$L^2$$ completion of the group ring $$C[\pi]$$.
This yields $$L^2$$ invisible manifolds. Theorem 2 rests in turn on
Theorem 3: There exists a finite $$3$$ dimensional polyhedron $$Y$$ with fundamental group $$\pi=F\times F\times F$$ where $$F$$ is the free group with two generators such that the extended $$L^2$$ homology $$H_p(Y;\ell^2(\pi))$$ vanishes for all $$p=0,1,\dots$$.
This yields as a corollary
Theorem 4: There exists an aspherical $$3$$ dimensional finite polyhedron $$Z$$ and a normal subgroup $$H$$ in $$\pi=\pi_1(Z)$$ such that the extended $$L^2$$ homology $$H_p(Z;\ell^2(\pi/H))$$ vanishes for all $$p=0,1,\dots\;$$.

### MSC:

 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Zbl 0874.58086
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