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 $\Sigma$ is locally homeomorphic to the space of representations of the orbifold fundamental group of $\Sigma$ to $G$ following the work of Thurston, Morgan, and Lok. This implies that the deformation space of $(G, X)$-structures on $\Sigma$ 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 $G$.