I think the set $X$ of 2-adic numbers that terminate in $\overline{01}_2$ is super-close to homeomorphic to the free group on two elements, but not quite. Can you help me characterise "not quite"?
Here's the following argument:
The labelled, infinite, binary rooted (at $-\frac13$) tree in which $\{2x,2x+2^{\nu_2(x)}\}$ are the children of $x$ is a cover of the set $X$.
$x\mapsto 2x$ is a homeomorphism.
$x\mapsto 2x+2^{\nu_2(x)}$ is a homeomorphism.
I think the binary rooted tree can be thought of as homeomorphic to the free monoid on two elements with concatenation as the operation. Each of the above functions represents one of the generators.
The composition of the two is therefore a homeomorphism.
I think each homeomorphism constitutes an idempotent operation which orientates the the graph to either left-handed or right-handed - correct?
I think the composition of the two homeomorphisms, maps $-\frac13\to-\frac13$
Since $-\frac13$ is a parent of itself, therefore the infinite, binary unrooted tree also represents $X$ (and hence the two functions above being homeomorphisms rather than just semihomeomorphisms).
Now the tree is unrooted, every binary string has an inverse. Except, I don't quite think it does if I consider handedness of the graph. How do I characterise this?