Geometric Structures on Orbifolds and Holonomy Representations

Abstract

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 character variety of representations of the orbifold fundamental group to G when restricted to the region of proper conjugation action by G.