Groups acting on trees with prescribed local action

We extend the Burger–Mozes theory of closed, nondiscrete, 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 Td of degree d∈N≥3. Three applications are given. First, we characterize the automorphism types that the quasicentre of a nondiscrete subgroup of Aut(Td) may feature in terms of the group’s local action. In doing so, we explicitly construct closed, nondiscrete, compactly generated subgroups of Aut(Td) with nontrivial quasicentre, and see that the Burger–Mozes theory does not extend further to the transitive case. We then characterize the (Pk)-closures of locally transitive subgroups of Aut(Td) containing an involutive inversion, and thereby partially answer two questions by Banks et al. [‘Simple groups of automorphisms of trees determined by their actions on finite subtrees’, J. Group Theory 18(2) (2015), 235–261]. Finally, we offer a new view on the Weiss conjecture.