Groups Acting on Trees With Prescribed Local Action

We extend Burger--Mozes theory of closed, non-discrete, locally quasiprimitive automorphism groups of locally finite, connected graphs to the semiprimitive case, and develop a generalization of Burger--Mozes universal groups acting on the regular tree $T_{d}$ of degree $d\in\mathbb{N}_{\ge 3}$. Three applications are given: First, we characterize the automorphism types which the quasi-center of a non-discrete subgroup of $\mathrm{Aut}(T_{d})$ may feature in terms of the group's local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{Aut}(T_{d})$ with non-trivial quasi-center, and see that Burger--Mozes theory does not extend further to the transitive case. We then characterize the $(P_{k})$-closures of locally transitive subgroups of $\mathrm{Aut}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks--Elder--Willis. Finally, we offer a new view on the Weiss conjecture.