Balanced groups and graphs of groups with infinite cyclic edge groups

TL;DR

This paper characterizes when the fundamental group of a finite graph of groups with infinite cyclic edges is acylindrically hyperbolic or balanced, providing conditions that clarify the structure and properties of such groups.

Contribution

It offers necessary and sufficient conditions for acylindrical hyperbolicity and balancedness in these groups, advancing understanding of their algebraic and geometric properties.

Findings

Characterization of acylindrical hyperbolicity based on graph of groups

Conditions for the fundamental group to be balanced when vertex groups are torsion free

Implication that finitely generated groups splitting over Z are not simple

Abstract

We give a necessary and sufficient condition for the fundamental group of a finite graph of groups with infinite cyclic edge groups to be acylindrically hyperbolic, from which it follows that a finitely generated group splitting over Z cannot be simple. We also give a necessary and sufficient condition (when the vertex groups are torsion free) for the fundamental group to be balanced, where a group is said to be balanced if $x^m$ conjugate to $x^n$ implies that $|m|=|n|$ for all infinite order elements $x$.