  We study the natural representation of the topological full group of an ample
Hausdorff groupoid in the groupoid's complex Steinberg algebra and in its full
and reduced C*-algebras. We characterise precisely when this representation is
injective and show that it is rarely surjective. We then restrict our attention
to discrete groupoids, which provide unexpected insight into the behaviour of
the representation of the topological full group in the full and reduced
groupoid C*-algebras. We show that the image of the representation is not dense
in the full groupoid C*-algebra unless the groupoid is a group, and we provide
an example showing that the image of the representation may still be dense in
the reduced groupoid C*-algebra even when the groupoid is not a group.
