  We study profinite actions of residually finite groups in terms of weak
containment. We show that two strongly ergodic profinite actions of a group are
weakly equivalent if and only if they are isomorphic. This allows us to
construct continuum many pairwise weakly inequivalent free actions of a large
class of groups, including free groups and linear groups with property (T). We
also prove that for chains of subgroups of finite index, Lubotzky's property
($\tau$) is inherited when taking the intersection with a fixed subgroup of
finite index. That this is not true for families of subgroups in general leads
to answering the question of Lubotzky and Zuk, whether for families of
subgroups, property ($\tau$) is inherited to the lattice of subgroups generated
by the family. On the other hand, we show that for families of normal subgroups
of finite index, the above intersection property does hold. In fact, one can
give explicite estimates on how the spectral gap changes when passing to the
intersection. Our results also have an interesting graph theoretical
consequence that does not use the language of groups. Namely, we show that an
expander covering tower of finite regular graphs is either bipartite or stays
bounded away from being bipartite in the normalized edge distance.
