Let $G$ be a finitely generated group with symmetric generating set $S$. Then $S$ induces a distance $d$ on $G$ by letting $d(a,b) = $ the minimum $n$ such that there are generators $s_1,...,s_n$ with $as_1...s_n = b$. In cases where $G$ is of polynomial growth, it is known that the metric space $(G,d)$ satisfies the doubling property, that there is a constant $C$ such that every ball of radius $r$ can be covered by $C$ or fewer balls of radius $r/2$. However, it is also known that examples exists of metric spaces with the doubling property that do not have the Besicovitch Covering Property (BCP), such as the Heisenberg group with the Cygan-Korany metric. And yet, there do exist distances which DO have the BCP on the Heisenberg group. My question is this: does the metric space $(G,d)$ have the BCP whenever $G$ is of polynomial growth and $d$ is the word length metric on some generating set? If not, what additional conditions would be necessary?
Here is one formulation of BCP: Let $B$ be a family of balls in $(G,d)$ with nonempty intersection and such that the centers of the balls are only contained in a single element of $B$. We call such a $B$ a Besicovitch Family of Balls. Then, we say that $(G,d)$ has the BCP if there is a constant $D$ such that the number of balls in any Besicovitch Family of Balls is always less than $D$.