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

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 ≤ 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 (x0, . . . , xr) ∈ G r+1 , such that the (r + 1)-fold simple commutator [x0, . . . , xr] vanishes, is exponentially negligible in G r+1 with respect to sequences of ball counting measures.