Small cancellation and outer automorphisms of Kazhdan groups acting on hyperbolic spaces

Abstract

We show that every finite group realizes as the outer automorphism group of an ICC hyperbolic group with Kazhdan property (T). This result complements the well-known theorem of Paulin stating that the outer automorphism group of every hyperbolic group with property (T) is finite. We also show that, for every countable group $Q$, there exists an acylindrically hyperbolic group $G$ with property (T) such that $Out(G)\cong Q$. The proofs employ strengthened versions of some previously known results in small cancellation theory.

v1: Some results of this paper were originally included in the first version of the paper arXiv:2111.04708 by the same authors; these results have been removed from the latest version (v3) of arXiv:2111.04708. v2: results of Section 3.3 generalized, some typos corrected. v3: The version accepted for publication. Introduction rewritten, some minor corrections throughout the text