HNN extensions of inverse semigroups, where the associated inverse subsemigroups are order ideals of the base, are defined by means of a construction based upon the isomorphism between the categories of inverse semigroups and inductive groupoids. The resulting HNN extension may conveniently be described by an inverse semigroup presentation, and we determine when an HNN extension with finitely g...

A group is called HNN-free if it has no subgroups that are nontrivial HNN-extensions. 23 We prove that finitely generated HNN-free implies virtually polycyclic for a large class of groups. We also consider finitely generated groups with no free subsemigroups of rank 2 25 and show that in many situations such groups are virtually nilpotent. Finally, as an application of our results, we determine...

We give a criterion for an HNN extension of a finite p-group to be residually p. 1. Statement of the Main Results By an HNN pair we mean a pair (G,φ) where G is a group and φ : A → B is an isomorphism between subgroups A and B of G. Given such an HNN pair (G,φ) we consider the corresponding HNN extension G∗ = 〈G, t | t−1at = φ(a), a ∈ A〉 of G, which we denote, by slight abuse of notation, as G∗...

We prove that the property of a free group endomorphism being irreducible is invariant ascending HNN extension it defines. This answers question posed by Dowdall-Kapovich-Leininger. further and atoroidal commensurability invariant. The invariance follows from an algebraic characterization extensions determines exactly when their defining endomorphisms are atoroidal; specifically, we show if onl...