# 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 $\Cal F(M)=H=\Cal F(\Cal R)$.