Relative expanders

We exhibit a finitely generated group $G$ and a sequence of finite index normal subgroups $N_n⊴G$ such that for every finite generating subset $S\subseteq G$, the sequence of finite Cayley graphs $(G\mathrm{/}{N}_n,S)$ does not coarsely embed into any $L^p$-space for $1⩽p<\mathrm{\infty }$ (moreover, into any uniformly curved Banach space), and yet admits no weakly embedded expander. The reason why our examples do not coarsely embed is a new phenomenon called relative expansion, which we define in terms of Poincar'e inequalities.