Consider the lamplighter group, which has the following group operation:
$$ (s_1, T_1) (s_2, T_2) = (s_1 + s_2, T_1 \triangle \{s_1 + t \mid t \in T_2\}) $$
where $s \in \mathbb{Z}$ and $T \subset \mathbb{Z}$. Suppose we have the following generators
\begin{align} R &= (1, \{0\}) \\ L &= (-1, \{\}) \end{align}
I'm looking for the smallest word(s) representing an arbitrary group element $(s, T)$:
$$ \underset{w \in \{R, L\}^* : [w]=(s, T)}{\mathrm{argmin}} \lvert w \rvert $$
Let $\#R$ and $\#L$ be the number of occurrences of $R$ and $L$ in $w$. I'm trying to do this by establishing a lower bound on $\lvert w \rvert = \#R + \#L$. We have:
\begin{align} \#R - \#L &= s \\ \#R &\geq \lvert T \rvert \\ \#R &\geq \max \{s\} \cup T \\ \#L &\geq -\min \{s\} \cup T \\ \#R &\geq 2 \lvert [0, s) \setminus T \rvert \end{align}
How can I provably minimize the word norm with respect to these generators?