Geometric group action

A geometric group action is a specific type of action by a group on a geometric space, which can often be thought of in terms of symmetries or transformations of that space. More formally, if we have a group \( G \) and a geometric object (often a topological space or manifold) \( X \), a geometric group action is defined when \( G \) acts on \( X \) in a way that respects the structure of \( X \).