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.

中文翻译：

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离散局部有限全群康托集同胚

这项工作的动机是为分段全群（又名拓扑全群）寻找局部紧凑群拓扑的问题。我们确定在康托集的自同胚群上的紧开拓扑中局部紧的任何分段满群必须是一致离散的，在我们这里介绍的精确意义上。康托集的一致离散自同胚群在紧开拓扑中尤其是可数的、局部有限的、残差有限的和离散的。由此产生的分段完整群形成了 Krieger 引入的充足群的一个子类。我们通过他们的 Bratteli 图和相关的维度范围来确定这些组的结构（${钾}_0$组）。我们通过一个例子表明，并非所有均匀离散的分段满群都是“明显”群的子群，即有限群的分段满群。