  We study the topological full group of ample groupoids over locally compact
spaces. We extend Matui's definition of the topological full group from the
compact, to the locally compact case. We provide two general classes of
groupoids for which the topological full group, as an abstract group, is a
complete isomorphism invariant. Hereby extending Matui's Isomorphism Theorem.
As an application, we study graph groupoids and their topological full groups,
and obtain sharper results for this class. The machinery developed in this
process is used to prove an embedding theorem for ample groupoids, akin to
Kirchberg's Embedding Theorem for $C^*$-algebras. Consequences for graph
$C^*$-algebras and Leavitt path algebras are also spelled out. In particular,
we improve on a recent embedding theorem of Brownlowe and S{\o}rensen for
Leavitt path algebras.
