Topological examples of profinite groups 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.
 A: One natural source of examples is matrix groups over either $\mathbb{Z}_p$ or $\mathbb{F}_q[[t]]$ (formal power series over a finie field). So for instance $SL_n(\mathbb{Z}_p)$ or  $SL_n(\mathbb{F}_q[[t]])$. 
Another good example is the Nottingham group: one way to describe it is as the normalized automorphisms (you can require continuous, but it is the same) of $\mathbb{F}_q[[t]]$. Another way is as power series in $\mathbb{F}_q[[t]]$ of the form $t+a_1t^2+a_2t^3+\cdots$ where the product is composition.
A: I may be way off on what you're looking for (learned from Serre's Galois Cohomology):
If $M$ is a torsion abelian group, then its dual $Hom(M,\mathbb{Q}/\mathbb{Z})$ with the topology of pointwise convergence, is a commutative profinite group.
This gives a "Pontryagin duality": $\lbrace$torsion abelian groups$\rbrace\Longleftrightarrow \lbrace$commutative profinite groups$\rbrace$.
A: One idea would be to introduce the profinite completion. If the students have already met the connection of the fundamental group with covering spaces, why not look at finite covering spaces. (This is essentially doing the SGA1 algebraic fundamental group idea without mentioning the links with alg. geom nor with Galois theory.) You are looking at a group $G$ and the category of finite $G$-sets.  As all the permutation groups of the objects concerned are finite, the group acts via its profinite completion. There is a nice master's thesis by Robalo at the IST in Lisbon which looked at various aspects of this situation and may provide a useful reference document for the students as it is written at only slightly above the level at which they will be working.
A: 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. 
