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 $\widehat T$.