Rank-One Hilbert Geometries [article]

Mitul Islam
2020 arXiv   pre-print
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 or
more » ... 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.
arXiv:1912.13013v2 fatcat:u6vmyj6azrddjdruaeqindojqu