# Are there nontrivial generating sets for $\operatorname{Homeo}(\text{Cantor Set})$?

A set $C\subset\mathbb{R}$ is called a Cantor set if it is compact, perfect, and totally disconnected. We may form the group $\operatorname{Homeo}(C):=\{f:C\to C~|~ f$ is a homeomorphism$\}$ under function composition.

Are there any known nontrivial generating sets for this group? This paper gives a rather nice description of a topological generating set, but I have been unable to find any literature for on-the-nose generation.

• I would be very surprised if any natural generating set (besides trivial examples like the whole group) exists. This seems rather like asking for a set of generators of $\mathbb{R}$ as a group under addition. – Eric Wofsey Jun 29 '17 at 23:37
• I think (not 100% sure) that this group has something called Bergman's property, and that implies all Cayley graphs have finite diameter, which feels like that basically means any generating set isn't much better than taking the whole group, at least from some perspectives. – Paul Plummer Jun 30 '17 at 0:05
• This is a simple group so any normal subset not contained in $\{1\}$ is generating. For instance, the set elements of order 2 works. Also, the set of elements of order 2 whose set of fixed points is open and non-empty (this is a single conjugacy class). Or, the set of elements of order 2 without fixed points (this is also a single conjugacy class). Etc. – YCor May 2 at 15:14