This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this question.

Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$. Does there always exist a set $Y\subseteq X$ with the following properties?

• $Y$ is uniformly discrete, i.e. there is an $\varepsilon > 0$ such that for any distinct $x,y\in Y$, $d(x,y) > \varepsilon$.
• $Y$ uniformly covers $X$, i.e. there is a $\delta > 0$ such that for any $x\in X$ there is a $y\in Y$ such that $d(x,y) < \delta$.
• $Y$ is vertex-transitive, i.e. for any $x,y\in Y$ there is a isometry $f:X \rightarrow X$ such that $y=f(x)$ and $f(Y)=Y$.

This is a followup to this easier version of this question on MSE, which Lee Mosher answered in the positive in the special case that $X$ is a hyperbolic space. It's also vaguely related to this question.

Suppose that $(X,d)$ is a locally compact connected homogeneous metric space, where by homogeneous I mean that for any $x_0,x_1 \in X$ there exists an isometry $f:X\rightarrow X$ such that $f(x_0)=x_1$. Does there always exist a set $Y\subseteq X$ with the following properties?

• $Y$ is uniformly discrete, i.e. there is an $\varepsilon > 0$ such that for any distinct $x,y\in Y$, $d(x,y) > \varepsilon$.
• $Y$ uniformly compactly covers $X$, i.e. for some $x \in Y$ there is a compact set $K \ni x$ such that translates of $K$ under $\{f \in \mathrm{Aut}(X) : f(Y) = Y\}$ cover all of $X$.
• $Y$ is vertex-transitive, i.e. for any $x,y\in Y$ there is a isometry $f:X \rightarrow X$ such that $y=f(x)$ and $f(Y)=Y$.

EDIT: I realized I hadn't captured what I wanted with the second bullet point.