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

Walsh, Cormac

doi:10.2140/gt.2018.22.55

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

Cormac Walsh

Geometry & Topology 22 (2018) 55–104

DOI: 10.2140/gt.2018.22.55

Abstract

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.

Keywords

Hilbert metric, Thompson metric, horofunction boundary, isometry group, symmetric cones, antitone map

Mathematical Subject Classification 2010

Primary: 52A99

References

Publication

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

Authors

Cormac Walsh
INRIA and CMAP, École Polytechnique Palaiseau France

Volume 22, issue 1 (2018)