The amenability-rigidity antinomy in the study of type II$_1$ factors

Abstract. We explain a general strategy for studying type II$_1$ factors, which consists in ``playing amenability against rigidity'' whenever some (weak) versions of these properties are met. The coexistence of these oposing properties creates enough ``tension'' within the algebra to unfold much of its structure. We will exemplify with two types of situations and results: 1). When the II$_1$ factor $M$ contains Cartan subalgebras $A\subset M$ such that $A\subset M$ satisfies a combination of ``aT-menability'' and ``(T)'' properties. 2). When $M=N \rtimes_\sigma G$ with $\sigma$ a ``malleable action'' (e.g., Bernoulli shift) and $G$ a weakly rigid group. A notable consequence shows that for any countable group $H \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ and standard equivalence relations $\Cal R$ with $\Cal F(M)=H=\Cal F(\Cal R)$.