A surface with discontinuous isoperimetric profile and expander manifolds

We construct sequences of `expander manifolds’ and we use them to show that there is a complete connected 2-dimensional Riemannian manifold with discontinuous isoperimetric profile, answering a question of Nardulli and Pansu. Using expander manifolds in dimension 3 we show that for any $\epsilon , M>0$thereisaRiemannian3-sphere$S$ of volume 1, such that any (not necessarily connected) surface separating $S$ in two regions of volume greater than $ϵ$, has area greater than $M$.