Q:

Asymptotics of orbits on graphs

Let $X$ be a connected, locally finite graph with vertex set $V(X)$ and $G$ a group acting freely on $X$ such that $X/G$ is a finite graph. Fix a vertex $x$ and for $k\in\mathbb N$ set
$$
N(k)=\#\{ g\in G: d(gx,x)\le k\},
$$
where $d$ is the vertex distance in the graph $X$.
Further set
$$
A(k)=\#\{y\in V(X):d(x,y)\le k\}.
$$
Is it true that, as $k\to\infty$, the number $N(k)/A(k)$ tends to $\#V(X/G)^{-1}$? If so, what  error term estimates are known?

A:

It is possible that the limit does not exist at all: Consider the free group on two generators acting on the $(4,2)$-biregular tree in the obvious way. This action is free and has 3 orbits (one containing all vertices of degree 4, and the other two containing "half" of the vertices of degree 2).
Let $x$ be a vertex of degree $4$. Then $N(k)$ is the number of vertices of degree 4 in $B_x(k)$, and $A(k)$ is the total number of vertices in $B_x(k)$. If we write $a_k$ and $b_k$ for the number of vertices at distance exactly $k$ from $x$ which have degree 4 or 2 respectively, we get $a_0 = 1$, and $b_{2l+1} = a_{2l+2} = 4\cdot3^l$  and $b_{2l} = a_{2l+1} = 0$ for $l \geq 0$. Note that
$$\frac{N(k)}{A(k)} = \frac{\sum_{i \leq k} a_i}{\sum_{i \leq k} a_i + b_i}$$
and if I'm not mistaken, plugging in the above values gives a limit of $\frac 12$ for the subsequence of even $k$, and $\frac 14$ for the subsequence of odd $k$.

