  The unique irreducible representation of $\SL_2(\R)$ on $\R^n$ induces an
action, called the \textit{linear action}, of $\SL_2(\Z)$ on the torus $\T^n$
for every $n\geq 2$. For $n$ odd, it factors through $\PSL_2(\Z)$, so we denote
by $G_n$ the group $\SL_2(\Z)$ for $n$ even, and $\PSL_2(\Z)$ for $n$ odd. We
prove that the action is free and ergodic for every $n\geq 2$, that if $h\in
\SL_2(\Z)$ is a hyperbolic element and if $n$ is even, then the action of the
subgroup generated by $h$ is still ergodic, but also that, for $n$ odd, no
amenable subgroup of $\PSL_2(\Z)$ acts ergodically on $\T^n$. We deduce also
that every ergodic sub-equivalence relation $\Rr$ of the orbital equivalence
relation $\mathcal{S}_n$ of $G_n$ on $\T^n$ is either amenable or rigid,
extending a result by Ioana for $n=2$. This result has the following
corollaries: firstly, for $n\geq 2$ even, if $H$ is a maximal amenable subgroup
of $\SL_2(\Z)$ containing an hyperbolic matrix, then the associated crossed
product II$_1$ factor $L^\infty(\T^n)\rtimes H$ is a maximal Haagerup
subalgebra of $L^\infty(\T^n)\rtimes \SL_2(\Z)$; secondly , for every $n$, the
fundamental group of $L^\infty(\T^n)\rtimes G_n$ is trivial.
