Volume 22, issue 1 (2018)

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Gauge-reversing maps on cones, and Hilbert and Thompson isometries

Cormac Walsh

Geometry & Topology 22 (2018) 55–104

We show that a cone admits a gauge-reversing map if and only if it is a symmetric cone. We use this to prove that every isometry of a Hilbert geometry is a projectivity unless the Hilbert geometry is the projective space of a non-Lorentzian symmetric cone, in which case the projectivity group is of index two in the isometry group. We also determine the isometry group of the Thompson geometry on a cone.

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Hilbert metric, Thompson metric, horofunction boundary, isometry group, symmetric cones, antitone map
Mathematical Subject Classification 2010
Primary: 52A99
Received: 23 November 2014
Revised: 19 December 2016
Accepted: 12 May 2017
Published: 31 October 2017
Proposed: Bruce Kleiner
Seconded: Anna Wienhard, Tobias H Colding
Cormac Walsh
INRIA and CMAP, École Polytechnique