What does it mean for a group to act cocompactly by isometries on a topological space $X$? What does it mean for a group to act cocompactly by isometries on a topological space $X$?  I know if $X$ is a topological group, and $A$ a subspace, then $A$ is cocompact iff $X/A$ is compact.  Not sure what it means in this context. Thanks.
 A: Given a group $G$ and a topological space $X$, an action of $G$ on $X$ is, formally, a homomorphism from $G$ to the group $\text{Homeo}(X)$ of all homeomorphisms from $X$ to itself. This can also be expressed with more notation as a function which associates to each $g \in G$ and each $x \in X$ an element $g \cdot x \in X$ subject to various properties:
(1) For each $g \in G$ the function $x \mapsto g \cdot x$ is a homeomorphism from $X$ to itself.
(2) $g \cdot (h \cdot x) = (gh) \cdot x$
(3) $\text{Id} \cdot x = x$

You also ask what it means for $G$ to act by isometries on $X$. By itself that does not make sense, because "isometries" are not defined for a general topological space. But if $X$ is a metric space then this does make sense and (1) is replaced by
(1') For each $g \in G$ the function $x \mapsto g \cdot x$ is a bijective isometry from $X$ to itself.
This concept can be vastly generalized. Whenever a mathematical object has some extra structure, one can speak about an action that "preserves the structure", which is very simple to formalize with the tiniest bit of category theory.

Finally, for the action to be cocompact means that there exists a compact set $K \subset X$ such that $\cup \{g \cdot K \mid g \in G\} = X$.
