Surgery on ${\rm Aut}(F_2)$

Abstract

We provide a new construction, starting from a flat torus, of a locally CAT(0) space whose fundamental group is a finite-index subgroup of ${\rm Aut}(F_2)$, the group of automorphisms of the free group on two generators. We prove that the universal cover of this space is isometrically isomorphic to the standard CAT(0) space $X_0$ on which ${\rm Aut}(F_2)$ acts geometrically (known as the Brady complex). We prove a local rigidity theorem, according to which every CAT(0) space ``locally isomorphic'' to $X_0$ must be isometric to $X_0$. Finally, we explain the relation between these constructions and known surgery constructions for groups acting on CAT(0) spaces, from which they derive.