I am preparing a course on profinite groups, to be delievered to early graduate students. The first part of the course will discuss the equivalent characterizations of profinite groups. I will first define a profinite group as a Hausdorff, compact topological group such that the open subgroups form a base for the neighbourhoods of the identity.
I would like to give my students a few ‘natural’ examples. I am looking for examples of profinite groups that give the students something they can wrap their heads around without knowing about products of finite groups and inverse limits. Ideally, I am looking for examples which have more of a topological emphasis.
As an example of what I am looking for: The $p$-adic integers $\mathbb{Z}_p$ can be constructed as the completion of $\mathbb{Z}$ with the metric space structure induced from the $p$-adic absolute value.
I would particularly like to see an example of a group acting on some object which induces some topology on the group. I would like to exclude Galois groups: these will be covered in another part of the course. I would also like to exclude the fundamental groups encountered in algebraic geometry: I do not expect my students be familiar with this material.
Best Answer
If you view the Cantor space as a sequence space, then the isometry group is profinite (using the usual sort of metric that words are close if they have a long common prefix). The automorphism group of a locally finite rooted tree is profinite or more generally the stabilizer of a vertex is profinite in the automorphism group of a locally finite graph. Here use the compact-open topology for the path metric.
Added. More generally the isometry group of any compact totally disconnected metric space is profinite.