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