Research Talk: Every Countable Group is an Outer Automorphism Group of an Acylindrically Hyperbolic Group with Kazhdan's Property (T)

The combination of Kazhdan’s property (T) and negative curvature typically limits the amount of outer automorphisms. Indeed, it is a result of Paulin that every property (T) hyperbolic group has a finite outer automorphism group. Belegradek and Szczepan ́ski extends Paulin’s result to property (T) relatively hyperbolic groups. We prove that for every countable group Q there is an acylindrically hyperbolic group G such that Out(G) = Q. Therefore the combination of property (T) and acylindrical hyperbolicity is much more flexible in terms of outer automorphisms.

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