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The horoboundary and isometry group of Thurston’s Lipschitz metric
, 2011
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Isometries of polyhedral Hilbert geometries
, 2009
"... Abstract. We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an nsimplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the nsimplex for all n ≥ 2, and f ..."
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Abstract. We show that the isometry group of a polyhedral Hilbert geometry coincides with its group of collineations (projectivities) if and only if the polyhedron is not an nsimplex with n ≥ 2. Moreover, we determine the isometry group of the Hilbert geometry on the nsimplex for all n ≥ 2, and find that it has the collineation group as an indextwo subgroup. These results confirm, for the class of polyhedral Hilbert geometries, several conjectures posed by P. de la Harpe. AMS Classification (2000): 53C60, 22F50 Keywords. Hilbert metric, horofunction boundary, detour metric, isometry group, collineations, Busemann points
Busemann points of Artin groups of dihedral type
, 2007
"... Abstract. We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In th ..."
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Cited by 7 (1 self)
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Abstract. We study the horofunction boundary of an Artin group of dihedral type with its word metric coming from either the usual Artin generators or the dual generators. In both cases, we determine the horoboundary and say which points are Busemann points, that is the limits of geodesic rays. In the case of the dual generators, it turns out that all boundary points are Busemann points, but this is not true for the Artin generators. We also characterise the geodesics with respect to the dual generators, which allows us to calculate the associated geodesic growth series. 1.
The horofunction boundary of finitedimensional normed spaces, in "Math
 Proc. Cambridge Philos. Soc
"... Abstract. We determine the set of Busemann points of an arbitrary finite– dimensional normed space. These are the points of the horofunction boundary that are the limits of “almostgeodesics”. We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme ..."
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Abstract. We determine the set of Busemann points of an arbitrary finite– dimensional normed space. These are the points of the horofunction boundary that are the limits of “almostgeodesics”. We prove that all points in the horofunction boundary are Busemann points if and only if the set of extreme sets of the dual unit ball is closed in the Painlevé–Kuratowski topology. 1.
The action of a nilpotent group on its horofunction boundary has finite orbits
 in "Groups, Geometry, and Dynamics", 2009, http://arxiv.org/abs/0806.0966, To appear. Invited Conferences
"... Abstract. We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of the abelianisation of G. We also prove that these ar ..."
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Abstract. We study the action of a nilpotent group G with finite generating set S on its horofunction boundary. We show that there is one finite orbit associated to each facet of the polytope obtained by projecting S into the infinite component of the abelianisation of G. We also prove that these are the only finite orbits of Busemann points. To finish off, we examine in detail the Heisenberg group with its usual generators. 1.
The asymptotic geometry of the Teichmüller metric
 2012, http://fr.arxiv.org/abs/1210.5565. References in notes
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The horofunction boundary and isometry group of the Hilbert geometry. Handbook of Hilbert Geometry
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