Acylindrical hyperbolicity of automorphism groups of infinitely-ended groups

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Publication date: 4 February 2020

Abstract: We prove that the automorphism group of every infinitely-ended finitely generated group is acylindrically hyperbolic. In particular

m a t h r m A u t (m a t h b b F_{n})

is acylindrically hyperbolic for every

n g e 2

. More generally, if

G

is a group which is not virtually cyclic, and hyperbolic relative to a finite collection

m a t h c a l P

of finitely generated proper subgroups, then

m a t h r m A u t (G, m a t h c a l P)

is acylindrically hyperbolic. As a consequence, a free-by-cyclic group

m a t h b b F_{n} t i m e s_{v a r p h i} m a t h b b Z

is acylindrically hyperbolic if and only if

v a r p h i

has infinite order in

m a t h r m O u t (m a t h b b F_{n})

.

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