Outer automorphism groups of hyperbolic groups, bounded extensions, and hierarchical hyperbolicity

Abstract

We prove that the outer automorphism group of a one-ended hyperbolic group is virtually a hierarchically hyperbolic group (HHG), under mild orientability conditions on the associated JSJ decomposition. This is done by proving that a finite-index subgroup is a central extension of a product of orbifold mapping class groups, and the extension has bounded Euler class. Our theorem is sharp: we exhibit a surface amalgam whose fundamental group has full outer automorphism group which is not a HHG. To prove this, the main technical tool is the fact that a top-dimensional Abelian subgroup of a HHG is a standard flat.