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 $R^s$.

翻译：Benjamini 和 Papasoglou (2011年) 显示,具有统一的多元体积增长的平面图允许以美元为单位的碎裂分离器 : 平面距离的顶部和任何顶部的顶部可以与距离的顶部分离, 去除最多为$O( R) 和 Papasoglou (2011年) 。 他们询问具有统一的多元体积增长的几何方位元图是否同样允许以美元(d-1) 美元为单位的碎裂分离器在 $ > 2 美元时使用。 我们显示,这在强烈意义上是失败的: 对于任何$d\ geq 3 和每 $\ geq 1 美元, 都收集了$\ mathb{ R ⁇ d$ 的内分界域, 其相近方形图 $G$ 具有统一的多元力增长, 但是, $G 中的所有废弃的分离器均具有至少为$R $ 美元的主要性 。