Acylindrical hyperbolicity of automorphism groups of infinitely ended groups
Anthony Genevois, Camille Horbez
Abstract
We prove that the automorphism group of every infinitely ended finitely generated group is acylindrically hyperbolic. In particular Aut ( F n ) is acylindrically hyperbolic for every n ⩾ 2 . More generally, if G is a group which is not virtually cyclic, and hyperbolic relative to a finite collection P of finitely generated proper subgroups, then Aut ( G , P ) is acylindrically hyperbolic. As a consequence, a free-by-cyclic group F n ⋊ φ Z is acylindrically hyperbolic if and only if φ has infinite order in Out ( F n ) .
Topics & Concepts
MathematicsAutomorphismRelatively hyperbolic groupFinitely-generated abelian groupOrder (exchange)Group (periodic table)Automorphism groupPure mathematicsCyclic groupHyperbolic groupStallings theorem about ends of groupsCombinatoricsHyperbolic manifoldHyperbolic functionMathematical analysisAbelian groupOrganic chemistryChemistryFinanceEconomicsGeometric and Algebraic TopologyMathematical Dynamics and Fractalssemigroups and automata theory