  Let $BS(1,n)= \langle a,b : a b a ^{-1} = b ^n\rangle$ be the solvable
Baumslag-Solitar group for $n \geq 2$. We study representations of $BS(1, n)$
on the plane by orientation preserving homeomorphisms, assuming that $a$ acts
as a linear map and $b$ as a map with bounded displacement. We find that the
possibilities for a faithful action depend greatly on the Jordan canonical form
of the map $h$ defined by the action of $a$. In case $h$ is diagonalizable over
$\mathbb R$, we shall give examples or prove rigidity theorems depending on the
eigenvalues. We also show some rigidity in the cases where $h$ is elliptic or
parabolic. Then we give applications to the actions of $BS(1, n)$ by
homeomorphisms of the torus.
