I have copied this question from Math.StackExchange, in the hope that some experts here can provide some relevant insight.

Let $M = \oplus_{i\in \mathbb Z} V^{(i)}$ where each $ V^{(i)} \cong \mathbb Z ^5 $. Let $e_1, e_2, e_3, e_4, e_5$ be the standard basis of $\mathbb Z^5$. Let $e_j^{(i)}$ be the element in $M$ whose $i$-th coordinate is $e_j$ and other coordinates are $0$.

Consider the following subgroup $ N = \left\{\begin{array}{c|c} e_1^{(i)}-e_1^{(i+1)} & \\ e_2^{(i)}-e_4^{(i+1)} & i \in \mathbb{Z}\\ e_3^{(i)}-e_5^{(i+1)} & \\ e_3^{(i)}+e_4^{(i)}+e_5^{(i)}+e_1^{(i)}-e_2^{(i+1)} & \end{array}\right\}. $

Let $K = M/N$ and $G = K\rtimes_\phi \mathbb{Z}$ where $\phi(1)$ shifts coordinate one place to the right in $K$. Let $F = \langle e_1^{(i)} \mid i \in \mathbb{Z} \rangle$ be the set of fixed points of $\phi$, this is a normal subgroup of $G$.

Let $T = \{ e_i^{(0)} \mid 1 \leq i \leq 5 \} $ be a finite subset of $K$ and $S = T \cup \{ (0,1) \}$ be the natural generating set of $G$.

**Question**: Can we show that $ \lim _{r \rightarrow \infty} |S^r \cap F| / |S^r| = 0$? (It is clear that $|S^r|$ grows exponentially)

This is an attempt to answer this question. I find it a bit tricky to determine the growth of $|S^r \cap F|$ because $F$ is not a direct summand of $K$. Any ideas would be much appreciated! Thank you for reading.