Non proper HNN extensions and uniform uniform exponential growth

TL;DR

This paper investigates how certain growth properties of finitely generated torsion-free groups are preserved under non-proper HNN extensions, focusing on uniform exponential growth conditions.

Contribution

It establishes that non-proper HNN extensions of groups with specific growth properties also maintain those properties, extending understanding of growth behavior in group extensions.

Findings

HNN extensions preserve uniform exponential growth properties

Small subgroups are characterized as cyclic or free abelian of bounded rank

Growth constants are uniformly bounded away from 1 in the extensions

Abstract

If a finitely generated torsion free group K has the property that all finitely generated subgroups S of K are either small or have growth constant bounded uniformly away from 1 then a non proper HNN extension G of K, that is a semidirect product of K by the integers, has the same property. Here small means cyclic or, if the automorphism has no periodic conjugacy classes, free abelian of bounded rank.