Let G be a group generated by a finite symmetric set S. An isoperimeric inequality on the Cayley graph (G,S) is an inequality stating that, for some function F and all finite subsets A of G of cardinality |A|, F(|A|) is bounded above by the |dA| where dA is the boundary of the finite set A, that is the sets of all edges with one end in A and the other in the complement of A.
After reviewing how isoperimetry relates to volume growth, I will discuss examples of solvable groups admitting isoperimetric function F that are surprisingly close to the non-amenable case F(t)=ct. The results are obtained through the study of simple random walk on these groups.