Deforming Convex Projective Manifolds

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 openess of their holonomies. Given a subset Ω ⊂ RP 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 is a convex set with non-empty interior and cl(Ω) is contained in the complement of some hyperplane H , 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 strictly convex. 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 [20, 21] 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 generalised 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. A group Γ ⊂ PGL(n + 1,R) is a virtual flag group if it contains a subgroup of finite index that is conjugate into the upper-triangular group. The set of virtual flag groups is written VFG. A generalized cusp is a properly convex manifold C homeomorphic to ∂C × [0,∞) with compact, strictly-convex boundary and with π1C virtually nilpotent. For an n–manifold M , possibly with boundary, define Rep(π1M) = Hom(π1M,PGL(n + 1,R)) and Repce(M) to be the subset of Rep(π1M) consisting of holonomies of properly convex structures on M with ∂M strictly convex and such that each end is a generalized cusp. For instance, all ends of a properly convex surface with negative Euler characteristic and strictly convex boundary are generalized cusps. Theorem 0.1. Suppose N is a compact connected n–manifold and V is the union of some of the boundary components V1, · · · , Vk ⊂ ∂N. Let M = N \V. Assume π1Vi is virtually nilpotent for each i. Then Repce(M) is an open subset of {ρ ∈ Rep(π1N) : ∀i ρ(π1Vi) ∈ VFG}. A similar statement holds for orbifolds since a properly convex orbifold has a finite cover which is a manifold, and the property of being properly convex is unchanged by coverings. This theorem is a consequence of our main theorem (6.27) that a certain map is open. By (6.9) ρ(π1Vi) ∈ VFG iff there is a finite index subgroup Γ < ρ(π1Vi) such that every eigenvalue of Γ is real. The author(s) acknowledge(s) support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS: GEometric structures And Representation varieties” (the GEAR Network). Cooper was partially supported by NSF grants DMS 1065939, 1207068 and 1045292 and thanks IAS and the Ellentuck Fund for partial support. Long was partially supported by grants from the NSF. Tillmann was partially supported by Australian Research Council grant DP140100158