  Benjamini and Papasoglou (2011) showed that planar graphs with uniform
polynomial volume growth admit $1$-dimensional annular separators: The vertices
at graph distance $R$ from any vertex can be separated from those at distance
$2R$ by removing at most $O(R)$ vertices. They asked whether geometric
$d$-dimensional graphs with uniform polynomial volume growth similarly admit
$(d-1)$-dimensional annular separators when $d > 2$. We show that this fails in
a strong sense: For any $d \geq 3$ and every $s \geq 1$, there is a collection
of interior-disjoint spheres in $\mathbb{R}^d$ whose tangency graph $G$ has
uniform polynomial growth, but such that all annular separators in $G$ have
cardinality at least $R^s$.
