Automorphism-invariant refinements of weakly branch actions via overlap functions

Let a finitely generated group $G$ act weakly branch on a locally finite rooted tree $T$ with boundary $\partial T$. The rooted tree structure is encoded by the \textit{overlap function}, which is our name for the Gromov product on the boundary: \[
c(\xi,\eta)=|\xi\wedge\eta|. \] We axiomatise this function and show that when it is `admissible', one can recover the rooted tree.
By the boundary rigidity theorem of Lavreniuk and Nekrashevych, $\operatorname{Aut}(G)$ acts canonically on $\partial T$. We therefore form the automorphism symmetrisation of the overlap function: \[
\widehat c(\xi,\eta)
=\inf_{\alpha\in\operatorname{Aut}(G)}c(\alpha \cdot\xi,\alpha \cdot\eta). \]
We prove that $ \widehat c $ is again an admissible $G$-invariant overlap function and that its associated tree $ \widehat T $ is locally finite. The action of $G$ on $ \widehat T $ is faithful and weakly branch, and is branch if and only if the original action on $T$ is branch. Moreover, \[ N_{\operatorname{Aut}(\widehat T)}(G) \cong \operatorname{Aut}(G). \] In particular, $ \operatorname{Aut}(G) $ is weakly branch. We also describe the finite weakly branch extensions of $ G $: they are precisely the pullbacks of finite subgroups of $ \operatorname{Out}(G) $. If $ G $ is branch, all these extensions are branch. In both cases, they act on the same tree $ \widehat T $.