Around groups in Hilbert geometry

Marquis, Ludovic

doi:10.4171/147-1/7

Cited by 38 publications

(38 citation statements)

References 73 publications

(155 reference statements)

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“…The Hilbert metric on a properly convex subset Ω ⊂ RP n is a Finsler metric given by the Hilbert-Finsler norm on T x Ω, see Papadopoulos and Troyanov [32] and Marquis [31]. For the definition of Hessian-convex hypersurface see the start of section 8.…”

Section: Hessian Metrics and Convexitymentioning

confidence: 99%

Deforming convex projective manifolds

2018

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We study a properly convex real projective manifold with (possibly empty) compact, strictly-convex boundary, and which consists of a compact part plus finitely many convex ends. We extend a theorem of Koszul which asserts that for a compact manifold without boundary the holonomies of properly convex structures form an open subset of the representation variety. We also give a relative version for non-compact (G, X)-manifolds of the openness of their holonomies.Given a subset Ω ⊂ RP n the frontier is Fr(Ω) = cl(Ω)\int(Ω) and the boundary is ∂Ω = Ω∩Fr(Ω). A properly convex projective manifold is M = Ω/Γ, where Ω ⊂ RP n is a convex set with non-empty interior, and cl(Ω) does not contain any RP 1 , and Γ ⊂ PGL(n + 1, R) acts freely and properly discontinuously on Ω. If, in addition, Fr(Ω) contains no line segment then M and Ω are strictlyconvex. The boundary of M is strictly-convex if ∂Ω contains no line segment.If M is a compact (G, X)-manifold then a sufficiently small deformation of the holonomy gives another (G, X)-structure on M . In [25,26] Koszul proved a similar result holds for closed, properly convex, projective manifolds. In particular, nearby holonomies continue to be discrete and faithful representations of the fundamental group.Koszul's theorem cannot be generalized to the case of non-compact manifolds without some qualification-for example, a sequence of hyperbolic surfaces whose completions have cone singularities can converge to a hyperbolic surface with a cusp. The holonomy of a cone surface in general is neither discrete nor faithful. Therefore we must impose conditions on the holonomy of each end.If M is a geometrically finite hyperbolic manifold M with a convex core that has compact boundary, then every end of M is topologically a product, and is foliated by strictly-convex hypersurfaces. These hypersurfaces are either convex towards M so that cutting along one gives a submanifold of M with convex boundary, and the holonomy of the end contains only hyperbolics; or else convex away from M , in which case the end is a cusp and the holonomy of the end contains only parabolics.This paper studies properly convex manifolds whose ends are either convex towards or away from M . An end that is convex towards M may be compactified by adding a convex boundary. Generalized cusps are those that are convex away from M with virtually nilpotent fundamental group. The holonomy of a generalized cusp may contain both hyperbolic and parabolic elements.Definition 0.1. A generalized cusp is a properly convex manifold C homeomorphic to ∂C × [0, ∞) with compact, strictly-convex boundary and with π 1 C virtually nilpotent.For instance, all ends of a finite volume hyperbolic manifold are generalized cusps. For an nmanifold M , possibly with boundary, define Rep(π 1 M ) = Hom(π 1 M, GL(n + 1, R)) and Rep ce (M ) to be the subset of Rep(π 1 M ) consisting of holonomies of properly convex structures on M with ∂M strictly-convex, and such that each end is a generalized cusp. A group Γ ⊂ GL(n + 1, R) is a virtual flag group if ...

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Section: Hessian Metrics and Convexitymentioning

confidence: 99%

Deforming convex projective manifolds

2018

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“…We will use u 0 = (1, 0) and u 1 = (−1/2, √ 3/2) then u 2 = −u 0 − u 1 . The convex hull of the vectors {±u 0 , ±u 1 , ±u 2 } is a regular Hexagon H. For the Hex plane p-area is different to Busemann volume, used for example in [15], or Holmes-Thompson used in [17]. On a normed plane all these measures are multiples of Haar measure, and so they are multiples of each other.…”

Section: Hex Geometrymentioning

confidence: 99%

The area of convex projective surfaces and Fock–Goncharov coordinates

Adeboye

Cooper

2018

J. Topol. Anal.

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“…For this purpose, we use the Hilbert metric on properly convex domains. The reader can find more information about the Hilbert metric in Marquis [14] or Orenstein [16].…”

Section: A Special Norm Associated To Marked Boxesmentioning

confidence: 99%

Pappus Theorem, Schwartz Representations and Anosov Representations

Barbot¹,

Lee

Valério

2018

Annales De l'Institut Fourier

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In the paper Pappus's theorem and the modular group, R. Schwartz constructed a 2-dimensional family of faithful representations ρΘ of the modular group PSL(2, Z) into the group G of projective symmetries of the projective plane via Pappus Theorem. The image of the unique index 2 subgroup PSL(2, Z)o of PSL(2, Z) under each representation ρΘ is in the subgroup PGL(3, R) of G and preserves a topological circle in the flag variety, but ρΘ is not Anosov. In her PhD Thesis, V. P. Valério elucidated the Anosov-like feature of Schwartz representations: For every ρΘ, there exists a 1-dimensional family of Anosov representations ρ ε Θ of PSL(2, Z)o into PGL(3, R) whose limit is the restriction of ρΘ to PSL(2, Z)o. In this paper, we improve her work: For each ρΘ, we build a 2-dimensional family of Anosov representations of PSL(2, Z)o into PGL(3, R) containing ρ ε Θ and a 1-dimensional subfamily of which can extend to representations of PSL(2, Z) into G . Schwartz representations are therefore, in a sense, the limits of Anosov representations of PSL(2, Z) into G .

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Around groups in Hilbert geometry

Cited by 38 publications

References 73 publications

Deforming convex projective manifolds

Deforming convex projective manifolds

The area of convex projective surfaces and Fock–Goncharov coordinates

Pappus Theorem, Schwartz Representations and Anosov Representations

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