Every locally compact second countable group $G$ has a regular left-invariant measure $h$, the Haar measure. On the other hand the Birkhoff–Kakutani Theorem asserts that such groups also admit a compatible left-invariant metric $d$. Let us denote by $B(id_G,r)$ the $d$-open ball around the identity of radius $r$.
I'm interested in the relations between $h$ and $d$. More precisely I'm curious about the function $$Gr_d:\mathbb R\rightarrow \mathbb R,\quad r\mapsto h(B(id_G,r)).$$
Clearly this function is monotone increasing right-continuous but in general (?) not continuous. The function $Gr_d$ is not continuous at $r\in\mathbb R$ if and only if the boundary of the $r$-ball has positive measure. However since it is monotone there are only countably many point of discontinuity.
My first question is: given a l.c.s.c. group $G$ is there a left-invariant metric $d$ such that $Gr_d$ is continuous?
In most of the groups I know, there exists a canonical metric for which the growth $Gr_d$ is continuous. But in general I have no idea how to construct such a metric.
The second question is about how bad things can go: does anyone have an example of a l.c.s.c. group G and a compatible left-invariant metric $d$ for which $Gr_d$ is not continuous? Can $\mathbb R^2$ have such metrics? Given an at most countable subset $C\subset \mathbb R$ can I find a metric on $\mathbb R^2$ (or your favorite l.c.s.c. group) for which the set of discontinuity points of $Gr_d$ is exactly $C$?
Edit: Here it is what I meant but didn't write (thanks for the comments!):
- The group $G$ is non-discrete
- The metric $d$ is proper
And this third point would make me even happier
- The closure of any open ball is the closed ball of the same radius