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.

V2: Many results in the literature were only proven for orbifolds without mirrors; hence we now use a different JSJ decomposition, and reduce the questions in the introduction to mapping class groups of non-orientable surfaces. Also, the technical lemma on maximal periodic flats in HHG was known; the original argument can still be found in version 1. Now 26 pages, 3 figures. Comments are welcome!