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Isometries of Hilbert geometries

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In a letter to Klein Hilbert remarked that any open, bounded, convex set X in Euclidean space can be equipped with a metric by taking the logarithm of the cross-ratio. The resulting metric space is called the Hilbert geometry on X. Hilbert noted that in case of the open disc one obtains the projective model for the hyperbolic plane. This model is usually called Klein’s model. Hilbert geometries embody geometrically diverse structures and can be viewed as a non-Riemannian generalization of hyperbolic space.

This talk is concerned with the isometries group, Isom(X), of the Hilbert geometry. In the nineteen-eighties P. de la Harpe raised a number of questions concerning Isom(X) and its subgroup, Coll(X), consisting of collineations (projectivities) of X. Among others he conjectured that Isom(X) acts transitively on X if and only if Coll(X) does. I will discuss my recent work with Cormac Walsh (INRIA & Ecole Polytechnique Palaiseau) which resolves De la Harpe’s problems in case X is an open polyhedron.

This talk is part of the Analysis seminar series.

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