For a discrete metric space (or more generally a large scale space) X and an action of a group G on X by coarse equivalences, we define a type of coarse quotient space X G , which agrees up to coarse equivalence with the orbit space X G when G is finite. We then restrict our attention to what we call coarsely discontinuous actions and show that for such actions the group G can be recovered as an appropriately defined automorphism group Aut(X X G ) when X satisfies a large scale connectedness condition. We show that for a coarsely discontinuous action of a countable group G on a discrete bounded geometry metric space X there is a relation between the maximal Roe algebras of X and X G , namely that there is a * -isomorphism C * max (X G ) K ≅ C * max (X) K⋊G, where K is the ideal of compact operators. If X has Property A and G is amenable, then we show that X G has Property A, and thus the maximal Roe algebra and full crossed product can be replaced by the usual Roe algebra and reduced crossed product respectively in the above equation.

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If one has a collection of large scale spaces {(X s , LSS s )} s∈S with certain compatibility conditions one may define a large scale space on X = ⋃ s∈S X s in a way where every function on X is large scale continuous if and only if the function restricted to every X s is large scale continuous. This large scale structure is called the asymptotic filtered colimit of {(X s , LSS s )} s∈S . In this paper, we explore a wide variety of coarse invariants that are preserved between {(X s , LSS s )} s∈S and the asymptotic filtered colimit (X, LSS). These invariants include finite asymptotic dimension, exactness, property A, and being coarsely embeddable into a separable Hilbert space. We also put forth some questions and show some examples of filtered colimits that give an insight into how to construct filtered colimits and what may not be preserved as well. Hilbert space, Inventiones mathematicae 139.

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