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We extend Burger--Mozes theory of closed, non-discrete, locallyquasiprimitive automorphism groups of locally finite, connected graphs to thesemiprimitive case, and develop a generalization of Burger--Mozes universalgroups acting on the regular tree $T_{d}$ of degree $d\in\mathbb{N}_{\ge 3}$.Three applications are given: First, we characterize the automorphism typeswhich the quasi-center of a non-discrete subgroup of $\mathrm{Aut}(T_{d})$ mayfeature in terms of the group's local~action. In doing so, we explicitlyconstruct closed, non-discrete, compactly generated subgroups of$\mathrm{Aut}(T_{d})$ with non-trivial quasi-center, and see that Burger--Mozestheory 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 questionsby Banks--Elder--Willis. Finally, we offer a new view on the Weiss conjecture.

Groups Acting on Trees With Prescribed Local Action