Codimension-$1$ Simplices in Divisible Convex Domains

Abstract

Properly embedded simplices in a convex divisible domain Ω ⊂ RP d behave somewhat like flats in Riemannian manifolds [Sch90], so we call them flats. We show that the set of codimension-1 flats has image which is a finite collection of disjoint virtual (d − 1)-tori in the compact quotient manifold. If this collection of virtual tori is non-empty, then the components of its complement are cusped convex projective manifolds with type d cusps.

Dynamical properties of convex cocompact actions in projective space

“…Maximal properly embedded simplices in Ω can be thought of as analogues of maximal flats in CAT(0) spaces; see, for example, [4,9,29,31]. However, in general, the metric space (Ω, 𝑑 Ω ) is not CAT(0); in fact this occurs if and only if Ω is an ellipsoid [37].…”

“…These nonhyperbolic convex cocompact groups are somewhat more mysterious than their hyperbolic counterparts. Recently, however, there has been significant progress toward a deeper understanding of them, especially in the case where the Γ-action on the entire domain Ω is cocompact: see, for example, [9,28,47]. Of particular relevance to this paper is the description, due to Islam-Zimmer [29,31], of the domains with a convex cocompact action by a relatively hyperbolic group relative to a family of virtually abelian subgroups of rank at least 2.…”

Patterson--Sullivan densities in convex projective geometry

“…Such a simplex S can be interpreted as a flat of Ω since it is isometric to R k endowed with some norm. In many examples of convex projective manifolds M = Ω/Γ, for instance when M is 3-dimensional, compact and irreducible, the maximal (for inclusion) PES's of Ω satisfy good properties (such as being isolated, see [Ben06,Bob,IZ]) which imply that M is rank-one. More precisely, Islam used [IZ] to establish [Isl,Prop.…”

Dynamical properties of convex cocompact actions in projective space

“…Maximal properly embedded simplices in Ω can be thought of as analogues of maximal flats in CAT(0) spaces; see e.g. [Ben06a], [IZ19a], [IZ19b], [Bob20]. However, in general, the metric space (Ω, d Ω ) is not CAT(0); in fact this occurs if and only if Ω is an ellipsoid [KS58].…”

“…Recently, however, there has been significant progress towards a deeper understanding of them, especially in the case where the Γ-action on the entire domain Ω is cocompact: see e.g. [Isl19], [Bob20], [Zim20]. Of particular relevance to this paper is the description, due to Islam-Zimmer [IZ19a], [IZ19b], of the domains with a convex cocompact action by a relatively hyperbolic group relative to a family of virtually abelian subgroups of rank at least two.…”