Thomas Schick: L2-determinant class and approximation of L2-Betti numbers
A standing conjecture in $L^2$-cohomology is that every finite
$CW$-complex $X$ is of $L^2$-determinant class. In this
paper, we show this whenever the fundamental group belongs to a
large class of groups containing f.e.~all extensions of
residually finite groups with amenable quotients, all
residually amenable groups and free products of these. If, in
addition, $X$ is $L^2$-acyclic, we also
prove that the $L^2$-determinant is a homotopy invariant, and we
give approximation formulas for the $L^2$-Betti numbers.
Also in
the known cases, our proof of homotopy invariance is much shorter and easier than the
previous ones.
The paper generalizes results of L"uck;
Dodziuk, Mathai, Rothenberg; and Clair.
Thomas Schick