I'm working with the book Self-similar Groups by Volodymyr Nekrashevych. In the chapter $1$ he statements the following proposition:
We have an equality $St_{Aut(X^*)}(n) = RiSt_{Aut(X^*)}(n)$. The subgroups $St_{Aut(X^*)}(n)$ form a system of neighborhoods of identity of a profinite topology on $Aut(X^*)$ coinciding with the topology of pointwise convergence on $X^*$.
Here $X^*$ denotes the set of finite sequences of elements of a non-empty set $X$, $St$ denotes the $n$th level stabilizer, $RiSt$ denotes $n$th level rigid stabilizer. We can see $X^*$ as a rooted tree (I don't know if these are standard definitions. I can write here if necessary).
I want to prove that $Aut(X^*)$ is a profinite group using the system of neighborhoods given in the proposition.
Since $St(n) = \bigcap_{v \in X^n}G_v$ where $G_v$ is the stabilizer of $v$, defining $St_{Aut(X^*)}(n)$ as a system of neighborhoods of identity is easy to see that $Aut(X^*)$ is Hausdorff and totally disconnected. The sets $G_v$ will be clopen sets and the general neighborhoods are obtained by translations.
How can I ensure that $Aut(X^*)$ is compact?
Best Answer
Tychonov's theorem says that a product of compact spaces is compact.
An application is the following: let $X$ be a set with a partition $\mathcal{P}$ $(X_i)_{i\in I}$ into finite subset. Then the stabilizer $\mathrm{Aut}(\mathcal{P})$ of this indexed partition, i.e., the set of permutations of $X$ stabilizing $X_i$ for every $i$, is compact. Precisely, it is a compact subgroup of the group of permutations of $X$, the latter being endowed with the usual topology (of pointwise convergence).
In case $X$ is a rooted connected graph of finite valency, this can be applied to prove that $\mathrm{Aut}(X)$ is compact.
Indeed, take $\mathcal{P}$ to be the partition $(X_n)$, where for each $n$ $X_n$ is the $n$-sphere around the root. (It covers because $X$ is connected, and they are finite because valencies are finite.)
Then $\mathrm{Aut}(X)$ being contained and closed in $\mathrm{Aut}(\mathcal{P})$ (with the same topology), it inherits its compactness.