Non-existence of annular separators in geometric graphs

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 $d\ge 3$ and every $s\ge 1$, 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 $R^s$.