  We prove that the sign of the Euler characteristic of arithmetic groups with
CSP is determined by the profinite completion. In contrast, we construct
examples showing that this is not true for the Euler characteristic itself and
that the sign of the Euler characteristic is not profinite among general
residually finite groups of type $F$. Our methods imply similar results for
$\ell^2$-torsion as well as a strong profiniteness statement for Novikov-Shubin
invariants.
