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 $T_{d}$ of degree $d\in \mathbb {N}_{\ge 3}$. Three applications are given. First, we characterize the automorphism types that the quasicentre of a nondiscrete subgroup of $\operatorname {\mathrm {Aut}}(T_{d})$ may feature in terms of the group’s local action. In doing so, we explicitly construct closed, nondiscrete, compactly generated subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ with nontrivial quasicentre, and see that the Burger–Mozes theory does not extend further to the transitive case. We then characterize the $(P_{k})$-closures of locally transitive subgroups of $\operatorname {\mathrm {Aut}}(T_{d})$ 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.

A weak form of faithfulness, depending on Green’s equivalence $\mathcal{D}$, is introduced for a ring $R$ graded by a semigroup $S$. Suppose that $R$ satisfies this condition. It is shown that if $e$ and $f$ are $\mathcal{D}$-equivalent idempotents of $S$ and $R_e$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_f$ is semiprime (respectively, prime, semiprimitive, right primitive). In addition, it is shown that if $G$ and $H$ are maximal subgroups of $S$ lying in the same $\mathcal{D}$-class and $R_G$ is semiprime (respectively, prime, semiprimitive, right primitive), then $R_H$ is semiprime (respectively, prime, semiprimitive, right primitive).

AMS 2000 Mathematics subject classification: Primary 16W50. Secondary 20M25