  We introduce a wide class of countable groups, called properly proximal,
which contains all non-amenable bi-exact groups, all non-elementary convergence
groups, and all lattices in non-compact semi-simple Lie groups, but excludes
all inner amenable groups. We show that crossed product II$_1$ factors arising
from free ergodic probability measure preserving actions of groups in this
class have at most one weakly compact Cartan subalgebra, up to unitary
conjugacy. As an application, we obtain the first $W^*$-strong rigidity results
for compact actions of $SL_d(\mathbb Z)$ for $d \geq 3$.
