Negligibity of elliptic elements in ascending HNN-extensions of $\mathbb{Z}^m$

TL;DR

This paper proves that the elliptic subgroup in ascending HNN-extensions of free abelian groups is exponentially negligible, and consequently, the set of tuples with trivial higher-order commutators is also exponentially negligible.

Contribution

It demonstrates the exponential negligibility of elliptic elements and trivial commutators in ascending HNN-extensions of free abelian groups, extending understanding of their algebraic and geometric properties.

Findings

Elliptic subgroup is exponentially negligible in the group.

Tuples with vanishing simple commutators are exponentially negligible.

Results apply to groups like Baumslag-Solitar and Sol 3-manifold groups.

Abstract

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.