I am looking for a reference on properties of the group $G$ of self-homeomorphisms of the Cantor set, or, the group of automorphisms of the countable atomless Boolean algebra $B$. I am aware:
- $G$ is simple. (Anderson)
- $G$ has the small index property. (Truss)
- $G$ as a subgroup of the group of permutations $\Bbb N$ is not locally compact. (easy)
- $G$ has a generic element. (Kechris & Rosendall)
etc.
In particular, I would like to learn easy properties of $G$ that could be used to differentiate $G$ from the automorphism group of a homogeneous structure like $B$ whether as as an abstract group of as a topological group like above.
The properties I have found are scattered in many different papers, surveys, and books, so I would ideally like something that compiles the properties of $B$ in one place.