Discrete locally finite full groups of Cantor set homeomorphisms

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