A question about group action on topological space

Question

Let $G$ be group and let $X$ be a topological space on which $G$ acts continuously. Now let us consider the following two properties relative to the group action:

(a) For every compact subset $K\subseteq X$ we have that $|\lbrace g\in G:gK\cap K\neq\emptyset\rbrace|<\infty$

(b) For all sequence $\{g_n\}_{n\geq 1}$ of pairwise distinct elements of $G$ and every $x\in X$ the sequence $\{g_n x\}_{n\geq 1}$ has no limit point in $X$.

It is easy to see that $(a)\Rightarrow (b)$. What about the converse?

Under the following assumptions one may show that $(b)\Rightarrow (a)$:

(*) Assume that $X$ is a locally compact metric space where the distance function is denoted by $d$. Assume that there is an absolute constant $C>0$ such that for all $x\in X$ there exists a neighborhood $U_x$ of $x$ such that for all $g\in G$ and all $u,v\in U_x$ one has that $d(gu,gv) < C\cdot d(u,v)$.

For example if $G$ acts through isometries on $X$ on a locally compact space then $(a)$ is equivalent to $(b)$.

1. Is it possible to weaken Assumption $(*)$ ?

Answer 1

Let's assume that $X$ is locally compact and instead of assuming that $G$ is a discrete group, assume more generally that it is locally compact and the action of $G$ on $X$ is continuous (i.e., the map $(g,x) \mapsto gx$ is continuous). Then the natural generalization of (a) is that if $K$ is compact, then the set $((K,K)) = \{ g\in G \mid gK \cap K \ne \emptyset\}$ is relatively compact. If this condition holds then $X$ is called a proper $G$-space, and it is known that this is equivalent to the condition that the map $(g,x) \mapsto (gx,x)$ of $G\times X \to X \times X$ is a proper map (in the sense that the inverse image of a compact set is compact), which is probably what you should assume instead of (b). The theory of proper group actions is a very old and rich one and I think if you do some googling and looking in Wikipedia you will find a lot along the lines that your questions suggest that you are interested in. (In particular, you might want to look at an old Annals paper of mine called "On the Existence of Slices for Actions of Non-Compact Lie Groups".)