Negligibity of elliptic elements in ascending HNN-extensions of

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 modeled on Sol geometry. In particular, we study the elliptic subgroup consisting of all elements that stabilize 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 such that the -fold simple commutator vanishes, is exponentially negligible in with respect to sequences of ball counting measures.