The fundamental group of the orbit space of a discontinuous group

Abstract

A group G of homeomorphisms of a topological space X will be called discontinuous if(1) the stabilizer of each point of X is finite, and(2) each point x ∈ X; has a neighbourhood U such that any element of G not in the stabilizer of x maps U outside itself (i.e. if gx ≠ x then U ∩ gU is empty). The purpose of this note is to prove the following result.