Acylindrical hyperbolicity, non simplicity and SQ-universality of groups splitting over Z

2016/03/18 by J. O. Button, Button, J. O.
Computer Science · Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #math.GR #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1603.05909

Much shorter version of 1509.05688 with strengthening of main result

arxiv created 2016/03/18 · arxiv updated 2016/03/21

Abstract

We show, using acylindrical hyperbolicity, that a finitely generated group splitting over \Z cannot be simple. We also obtain SQ-universality in most cases, for instance a balanced group (one where if two powers of an infinite order element are conjugate then they are equal or inverse) which is finitely generated and splits over \Z must either be SQ-universal or it is one of exactly seven virtually abelian exceptions.

Related

• Balanced groups and graphs of groups with infinite cyclic edge groups
• Entropy and finiteness of groups with acylindrical splittings
• The semi-simple theory of higher rank acylindricity
• Complexity volumes of splittable groups
• Acylindricity in Higher Rank, Part I : Fundamentals