This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick-thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space.Except in dimension 3, there are only finitely many topological types of strictly convex manifolds with bounded volume. In dimension 4 and higher, the diameter of a closed strictly convex manifold is at most 9 times the diameter of the thick part. There is an algebraic characterization of strict convexity in terms of relative hyperbolicity.

The notion of a layered triangulation of a lens space was defined by Jaco and Rubinstein, and unless the lens space is L(3, 1), a layered triangulation with the minimal number of tetrahedra was shown to be unique and termed its minimal layered triangulation. This paper proves that for each n 2, the minimal layered triangulation of the lens space L(2n, 1) is its unique minimal triangulation. More generally, the minimal triangulations (and hence the complexity) are determined for an infinite family of lens spaces containing the lens space of the form L(2n, 1).Theorem 2. The minimal layered triangulation of L((s + 2)(t + 1) + 1, t + 1), where t > s > 1, s is odd and t is even, is its unique minimal triangulation. The complexity of L((s + 2)(t + 1) + 1, t + 1) is therefore s + t.Theorem 3. The minimal layered triangulation of L((s + 1)(t + 2) + 1, t + 2), where t > s > 1, s is even and t is odd, is its unique minimal triangulation. The complexity of L((s + 1)(t + 2) + 1, t + 2) is therefore s + t.

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 ...

Let M be a cusped 3-manifold, and let T be an ideal triangulation of M. The deformation variety D(T ), a subset of which parameterises (incomplete) hyperbolic structures obtained on M using T , is defined and compactified by adding certain projective classes of transversely measured singular codimension-one foliations of M. This leads to a combinatorial and geometric variant of well-known constructions by Culler, Morgan and Shalen concerning the character variety of a 3-manifold.

Agol recently introduced the concept of a veering taut triangulation of a 3-manifold, which is a taut ideal triangulation with some extra combinatorial structure. We define the weaker notion of a "veering triangulation" and use it to show that all veering triangulations admit strict angle structures. We also answer a question of Agol, giving an example of a veering taut triangulation that is not layered. 57M50