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 $\mathrm{A}\mathrm{u}\mathrm{t}(T_d)$ may feature in terms of the group’s local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{A}\mathrm{u}\mathrm{t}(T_d)$ 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 $\mathrm{A}\mathrm{u}\mathrm{t}(T_d)$ containing an involutive inversion, and thereby partially answer two questions by Banks–Elder–Willis. Finally, we offer a new view on the Weiss conjecture.