CLASSIFICATION OF ORDERABLE AND DEFORMABLE COMPACT COXETER POLYHEDRA IN HYPERBOLIC SPACE DHRUBAJIT CHOUDHURY, SUHYOUNG CHOI, AND GYE-SEON LEE Abstract. The aim of this work is to investigate properties of 3-dimensional compact hyperbolic Coxeter polytope which are orderable and deformable in real projective space. We give a complete classification, up to isom- etry, of all 3-dimensional compact hyperbolic Coxeter polytope which are orderable and projectively deformable. By Andreev’s and Choi’s theorem, we proved that number of vertices of polyhedra in such family can’t be more than 10. Using plantri program, we get the complete list of polyhedra having not more than 10 vertices. Using computer pro- gram, from these polyhedra, we found all the 3-dimensional compact hyperbolic Coxeter polytope which are orderable and deformable in real projective space. We also explain the algorithms of computer program behind this result. Introduction A n-dimensional orbifold is a Hausdorff space with a structure based on the quotient space of R n by a finite group action. We only deal with an orbifold of which universal cover is a manifold, the so-called a good orbifold. To give a hyperbolic structure on an orbifold, we model it locally by the orbit spaces of finite subgroups of PO(1,n) acting on open subsets of H n . Similarly, utilizing PGL(n +1, R) and RP n we can give a real projective structure on an orbifold. A real projective structure on an orbifold M im- plies that M has a universal cover ˜ M and the deck transformation group π 1 (M ) acting on ˜ M so that ˜ M/π 1 (M ) is homeomorphic to M . Given a real projective structure on an orbifold M , we can define an immersion D from the universal cover ˜ M to RP n and a homomorphism h : π 1 (M ) → PGL(n +1, R), for the orbifold fundamental group π 1 (M ) of M , where D is called a develop- ing map and h a holonomy homomorphism. However, (D,h) is determined only up to the following action (D,h(·)) → (g ◦ D,g ◦ h(·) ◦ g -1 ). for g ∈ PGL(n +1, R). Conversely, the development pair (D,h) determines the real projective structure. Define ˜ D(M ) by the space of equivalence classes of development pairs of real projective structures on ˜ M under the Date : October 07, 2009 and, in revised form, July –, –. Key words and phrases. Coxeter polyhedron, Orderable Coxeter polyhedron. The first author was supported in part by NSF Grant #000000. This work was completed with the support of an BK 21 Scholarship. 1