  We study the isoperimetric profiles of certain families of finitely generated
groups defined via marked Schreier graphs and permutation wreath products. The
groups we study are among the "simplest" examples within a much larger class of
groups, all defined via marked Schreier graphs and/or action on rooted trees,
which includes such examples as the long range group, Grigorchuck group and the
basillica group. The highly non-linear structure of these groups make them both
interesting and difficult to study. Because of the relative simplicity of the
Schreier graphs that define the groups we study here (the key fact is that they
contained very large regions that are "one dimensional"), we are able to obtain
sharp explicit bounds on the $L^1$ and $L^2$ isoperimetric profiles of these
groups. As usual, these sharp isoperimetric profile estimates provide sharp
bounds on the probability of return of simple random walk. Nevertheless, within
each of the families of groups we study there are also many cases for which the
existing techniques appear inadequate and this leads to a variety of open
problems.
