  Let $K$ be a field, and let $\Aut \,K^2$ be the group of polynomial
automorphisms of $K^2$. If $K$ is infinite, this group is nonlinear. Moreover
it contains nonlinear FG subgroups when $\ch\,K=0$. On the opposite, it
contains some linear "finite codimension" subgroups. This phenomenon is
specific to dimension two: it is also proved that "finite codimension"
subgroups of $\Aut\,K^3$ are nonlinear, even for a finite field $K$.
