What does it mean for a group to act cocompactly by isometries on a topological space $X$?

Question

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.

Answer 1

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) $\text{Id} \cdot x = x$

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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.

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