Galois cohomology and the sign of the Euler characteristic

$L^2$-Betti numbers capture free homology growth of a group along certain sequences of finite index normal subgroups. The question thus arises naturally if $L^2$-Betti can also be determined from the corresponding finite quotient groups.  It turns out that the answer is "yes" in degree one and "no" in higher degrees as exhibited by $S$-arithmetic groups with CSP.  We will explain, however, that Poitou-Tate duality in Galois cohomology shows that just enough information about the $L^2$-cohomology is contained in the finite quotient groups to determine the sign of the Euler characteristic. Joint work with G. Serafini.