  This work is motivated by the problem of finding locally compact group
topologies for piecewise full groups (a.k.a.~ topological full groups). We
determine that any piecewise full group that is locally compact in the
compact-open topology on the group of self-homeomorphisms of the Cantor set
must be uniformly discrete, in a precise sense that we introduce here.
Uniformly discrete groups of self-homeomorphisms of the Cantor set are in
particular countable, locally finite, residually finite and discrete in the
compact-open topology. The resulting piecewise full groups form a subclass of
the ample groups introduced by Krieger. We determine the structure of these
groups by means of their Bratteli diagrams and associated dimension ranges
($K_0$ groups). We show through an example that not all uniformly discrete
piecewise full groups are subgroups of the ``obvious'' ones, namely, piecewise
full groups of finite groups.
