  We exhibit a finitely generated group $G$ and a sequence of finite index
normal subgroups $N_n\trianglelefteq G$ such that for every finite generating
subset $S\subseteq G$, the sequence of finite Cayley graphs $(G/N_n, S)$ does
not coarsely embed into any $L^p$-space for $1\leqslant p<\infty$ (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.
