Discrete locally finite full groups of Cantor set homeomorphisms

TL;DR

This paper characterizes locally compact piecewise full groups of Cantor set homeomorphisms, showing they are uniformly discrete, countable, and structurally analyzable via Bratteli diagrams, with some not fitting into finite group subgroups.

Contribution

It proves that locally compact piecewise full groups must be uniformly discrete and provides a detailed structural analysis using Bratteli diagrams and $K_0$ groups.

Findings

Locally compact piecewise full groups are uniformly discrete.

Such groups are countable, locally finite, and residually finite.

Not all uniformly discrete groups are subgroups of finite group piecewise full groups.

Abstract

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.