2010/11/04 by Suhyoung Choi, Choi, Suhyoung · 1 citation
Mathematics · #20C99 #20F65 #53C10 #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR) #Primary 57M50 #Secondary 53A20 #math.GR #math.GT #msc:20C99 #msc:20F65 #msc:53A20 #msc:53C10 #msc:57M50
paper · pdf · doi:10.48550/arxiv.1011.1060
128 pages 10 figures
arxiv created 2014/05/28 · arxiv updated 2014/05/29
A real projective orbifold is an n-dimensional orbifold modeled on \mathbbRPn with the group PGL(n+1, ℝ). We concentrate on an orbifold that contains a compact codimension 0 submanifold whose complement is a union of neighborhoods of ends, diffeomorphic to closed (n-1)-dimensional orbifolds times intervals. A real projective orbifold has a \it radial end if a neighborhood of the end is foliated by projective geodesics that develop into geodesics ending at a common point. It has a \it totally geodesic end if the end can be completed to have the totally geodesic boundary. The orbifold is said to be \it convex if any path can be homotopied to a projective geodesic with endpoints fixed. A real projective structure sometimes admits deformations to parameters of real projective structures. We will prove the local homeomorphism between the deformation space of convex real projective structures on such an orbifold with radial or totally geodesic ends with various conditions with the PGL(n+1, ℝ)-character space of the fundamental group with corresponding conditions. We will use a Hessian argument to show that under a small deformation, a properly (resp. strictly) convex real projective orbifold with generalized admissible ends will remain properly and properly (resp. strictly) convex with generalized admissible ends. Lastly, we will prove the openness and closedness of the properly (resp. strictly) convex real projective structures on a class of orbifold with generalized admissible ends, where we need the theory of Crampon-Marquis and Cooper, Long and Tillmann on the Margulis lemma for convex real projective manifolds. The theory here partly generalizes that of Benoist on closed real projective orbifolds.