Abstract. The deformation space C(L) of convex RP2-structures on a closed surface I with #(5)) < 0 is closed in the space Hom(7t, SL(3, R))/SL(3, R) of equivalence classes of representations nx (Z) -» SL(3, R). Using this fact, we prove Hitchin's conjecture that the contractible "Teichmiiller component" (Lie groups and Teichmiiller space, preprint) of Homfw, SL(3, R))/SL(3, R) precisely equals C(2).Let X be a closed orientable surface of genus g > 1 and n = nx(L) its fundamental group. A convex EP2-structure on M is a representation (uniformization) of M as a quotient £l/T where Q c RP2 is a convex domain and T c SL(3, R) is a discrete group of collineations of RP2 acting properly and freely on Q. (See [5] for basic theory of such structures.) The space of projective equivalence classes of convex RP2-structures embeds as an open subset in the space of equivalence classes of representations n -* SL(3, R). The purpose of this note is to show that this subset is also closed.In [7], Hitchin shows that the space of equivalence classes of representations n -> SL(3, R) falls into three connected components: one component C_i consisting of classes of representations for which the associated flat R3-bundle over £ has nonzero second Stiefel-Whitney class; a component Co containing the class of the trivial representation; a component Cx diffeomorphic to a cell of dimension 16(g -1), which he calls the "Teichmiiller component." While C-i can be distinguished from Co and Cx by a topological invariant [3, 4], no characteristic invariant distinguishes representations in the Teichmiiller component from those in Co. The Teichmiiller component is defined as follows. Using the Klein-Beltrami model of hyperbolic geometry, a hyperbolic structure on Z is a special case of a convex RP2-structure Q/T where Q is the region bounded by a conic. In this case I" is conjugate to a cocompact lattice in SO(2,l) c SL(3, R). The space T(S) of hyperbolic structures ("Teichmiiller

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An orbifold is a topological space modeled on quotient spaces of a finite group actions. We can define the universal cover of an orbifold and the fundamental group as the deck transformation group. Let G be a Lie group acting on a space X. We show that the space of isotopy-equivalence classes of (G, X)-structures on a compact orbifold Σ is locally homeomorphic to the space of representations of the orbifold fundamental group of Σ to G following the work of Thurston, Morgan, and Lok. This implies that the deformation space of (G, X)-structures on Σ is locally homeomorphic to the space of representations of the orbifold fundamental group to G when restricted to the region of proper conjugation action by G.

Abstract. We determine that the deformation space of convex real projective structures, that is, projectively flat torsion-free connections with the geodesic convexity property on a compact 2-orbifold of negative Euler characteristic is homeomorphic to a cell of certain dimension. The basic techniques are from Thurston's lecture notes on hyperbolic 2-orbifolds, the previous work of Goldman on convex real projective structures on surfaces, and some classical geometry.

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