Some Questions — Question 1

Definitions and mathematical statement

Let $T_2$ be the rooted binary tree with vertices finite binary strings; an edge joins a string to either of its one-digit extensions. The group $\operatorname{Aut}(T_2)$ consists of root-preserving tree automorphisms and carries the profinite topology of agreement on successively deeper finite levels. Identify a level-$m$ vertex with a residue modulo $2^m$, compatibly with truncation toward the root. The odometer $a$ is the automorphism which adds one modulo $2^m$ on every level. Its closure $\overline{\langle a\rangle}$ is isomorphic to the additive group of $2$-adic integers.

A free pro-$2$ group $\widehat F^{(2)}_r$ is the pro-$2$ completion of the rank-$r$ free group: the inverse limit of all its finite $2$-group quotients. Is there a nonabelian free pro-$2$ group and a continuous injective homomorphism \[ \rho:\widehat F^{(2)}_r\longrightarrow\operatorname{Aut}(T_2), \qquad r\geq2, \] whose image contains the specified odometer $a$?

The point is containment in the tree-automorphism group with this particular action. Abstractly embedding an infinite cyclic or procyclic group into a free pro-$2$ group does not resolve the question. Since the source group is compact, a continuous injection has closed image and is a topological embedding. The element mapping to $a$ is not required to be a member of a chosen free pro-$2$ basis.

Short English statement

Can the binary adding-machine action occur inside a closed nonabelian free pro-two subgroup of the same tree-automorphism group?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 1200001 (AMR-011-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Abert - Some questions (pdf) (2010), Question 1. Original source indexed by AMR-011-0001. https://www.renyi.hu/~abert/questions.pdf.