  Given an ample groupoid $G$ with compact unit space, we study the canonical
representation of the topological full group $[[G]]$ in the full groupoid
$C^*$-algebra $C^*(G)$. In particular, we show that the image of this
representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial
state. The techniques that we use include the notion of groups covering
groupoids. As an application, we provide sufficient conditions for
$C^*$-simplicity of certain topological full groups, including those associated
with topologically free and minimal actions of non-amenable and countable
groups on the Cantor set.
