A **cocompact group action** refers to a specific type of action of a group on a topological space, particularly in the context of topological groups and geometric topology. In broad terms, if a group \( G \) acts on a topological space \( X \), we say that the action is **cocompact** if the quotient space \( X/G \) is compact.
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A group action on a topological space is cocompact when its quotient space is compact. If a compact subset has translates covering , then its quotient image covers , proving cocompactness.
Conversely, for an action by homeomorphisms on a Hausdorff space that is also a locally compact space, compactness of the quotient supplies such a compact . Choose an open neighborhood with compact closure at each point. The quotient map is open, because the inverse image of the image of is the open union . Finitely many quotient images cover the compact quotient. The union of the corresponding finitely many closures is a compact subset whose translates cover .
For example, the integer lattice acts cocompactly on Euclidean space by translations, with . Cocompactness controls coverage of the space. A properly discontinuous group action adds a separate finiteness condition on translates.