GROUPS ACTING ON TREES WITH PRESCRIBED LOCAL ACTION

Abstract We extend the Burger–Mozes theory of closed, nondiscrete, locally quasiprimitive automorphism groups finite, connected graphs to semiprimitive case, and develop a generalization universal acting on regular tree $T_{d}$ degree $d\in \mathbb {N}_{\ge 3}$ . Three applications are given. First, we characterize types that quasicentre nondiscrete subgroup $\operatorname {\mathrm {Aut}}(T_{d})$ may feature in terms group’s local action. In doing so, explicitly construct compactly generated subgroups with nontrivial quasicentre, see does not further transitive case. then $(P_{k})$ -closures containing an involutive inversion, thereby partially answer two questions by Banks et al. [‘Simple automorphisms trees determined their actions finite subtrees’, J. Group Theory 18 (2) (2015), 235–261]. Finally, offer new view Weiss conjecture.