A cocompact group action is a group action by homeomorphisms on a topological
space whose quotient space is compact
(Bridson and Haefliger 1999; Löh 2018). For an isometric
action on a boundedly compact space, also
called a proper metric space, this is equivalent
to the existence of a compact subset
whose translates cover the space (Bridson and Haefliger
1999, p. 131). For example, integer translations
on 
have quotient space the torus 
. A geometric
group action is an isometric group
action that is both proper and cocompact.
Cocompact Group Action
See also
Geometric Group Action, Group Action, Proper Group Action, Quotient SpaceExplore with Wolfram|Alpha
References
Bridson, M. R. and Haefliger, A. Metric Spaces of Non-Positive Curvature. Berlin, Germany: Springer-Verlag, 1999. https://doi.org/10.1007/978-3-662-12494-9.Löh, C. Geometric Group Theory: An Introduction. Cham, Switzerland: Springer, 2018. https://doi.org/10.1007/978-3-319-72254-2.Cite this as:
Weisstein, Eric W. "Cocompact Group Action." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CocompactGroupAction.html