  We study ascending HNN-extensions $G$ of finitely generated free abelian
groups: examples of such $G$ include soluble Baumslag-Solitar groups and
fundamental groups of orientable prime $3$-manifolds modelled on Sol geometry.
In particular, we study the elliptic subgroup $A \leq G$, consisting of all
elements that stabilise a point in the Bass-Serre tree of $G$. We consider the
density of $A$ with respect to ball counting measures corresponding to finite
generating sets of $G$, and we show that $A$ is exponentially negligible in $G$
with respect to such sequences of measures. As a consequence, we show that the
set of tuples $(x_0,\ldots,x_r) \in G^{r+1}$, such that the $(r+1)$-fold simple
commutator $[x_0,\ldots,x_r]$ vanishes, is exponentially negligible in
$G^{r+1}$ with respect to sequences of ball counting measures.
