Balanced groups and graphs of groups with infinite cyclic edge groups

2015/09/18 by J. O. Button, Button, J. O.
Mathematics · #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR) #math.GR

paper · pdf · doi:10.48550/arxiv.1509.05688

arxiv created 2015/09/18 · openalex publication_date 2015/09/18 · arxiv updated 2015/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 xⁿ implies that |m|=|n| for all infinite order elements x.

Citations

• The Free Product of Two Groups with a Malnormal Amalgamated Subgroup

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