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We introduce and study the notion of rank-one Hilbert geometries, or rank-one properly convex domains, in ℙ(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 Ω and characterize them precisely as the contracting elements in the automorphism group Aut(Ω) of the Hilbert geometry. We prove that if a discrete subgroup of Aut(Ω) contains a rank-one isometry, then the subgroup is either virtually ℤ orarXiv:1912.13013v2 fatcat:u6vmyj6azrddjdruaeqindojqu