$(X, T)^{(2)} \subseteq X^2$ be the set of the pairs which are connected in the coarse structure generated by $T$. Namely, $(X, T)^{(2)}$ is the set $\bigcup_{n = 1}^\infty (T \cup T^{-1})^{\circ n}$. The set $(X, T)^{(2)}$ naturally has a groupoid structure by

-   range map: $\mathrm{Image}_X \colon (x, y) \mapsto x$,

-   source map: $\mathrm{Dom}_X \colon (x, y) \mapsto y$,

-   product: $(x, y)(y, z) = (x, z)$,

-   inverse: $(x, y)^{-1} = (y, x)$.

We often identify the space of units $\Delta_X$ with $X$. The first goal of this section is Theorem \[TheoremGroupoid\]. The theorem states that there exists a ‘good’ groupoid $\Gamma$ which encodes the partial bijections $\phi(i)$.

Translation C$^*$-algebra associated to label {#SubsectionC$^*$-SubalgebraLabel}
---------------------------------------------

For $i = -k, -k+1, \cdots , -1$, define $\phi(i)$ by the inverse $\phi(-i)^{-1}$. Let $I$ be the index set $\{ -k, -k+1, \cdots , 0, \cdots, k \}$. For $i \in I$, we denote by $v(i) : \ell_2 X \rightarrow \ell_2 X$ the partial isometry defined by $$\begin{aligned}
[v(i)](\delta_y)
=
\left\{
\begin{array}{cc}
\delta_{[\phi(i)](y)}, & \quad y \in \mathrm{Dom}_X (\phi(i)),\\
0,                         & \quad y \notin \mathrm{Dom}_X (\phi(i)).
\end{array}
\right.\end{aligned}$$ The operator $v(i)$ is an element of the uniform Roe algebra $C^*_\mathrm{u}(X) \subseteq 