Codimension-1 simplices in divisible convex domains

Properly embedded simplices in a convex divisible domain Ω⊂ R Pᵈ behave somewhat like flats in Riemannian manifolds, 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.