Rank-One Hilbert Geometries.

We introduce and study the notion of rank-one Hilbert geometries and open properly convex domains in $\mathbb{P}(\mathbb{R}^{d+1})$. This is in the spirit of rank-one non-positively curved Riemannian manifolds. We define rank-one isometries of a Hilbert geometry $\Omega$ and characterize them precisely as the contracting elements in the automorphism group ${\rm Aut}(\Omega)$ of the Hilbert geometry. We prove that if a discrete subgroup of ${\rm Aut}(\Omega)$ contains a rank-one isometry, then the subgroup is either virtually $\mathbb{Z}$ or acylindrically hyperbolic. This leads to some applications like computation of the space of quasimorphisms, genericity results for rank-one isometries and counting results for closed geodesics.