Groups Acting on Trees With Prescribed Local Action

TL;DR

This paper generalizes Burger--Mozes theory to semiprimitive groups acting on trees, characterizes automorphism types with non-trivial quasi-centers, and explores implications for the Weiss conjecture.

Contribution

It extends existing theory to semiprimitive cases, constructs new examples of groups with specific properties, and addresses open questions about closures and the Weiss conjecture.

Findings

Characterized automorphism types with non-trivial quasi-centers.

Constructed non-discrete, compactly generated subgroups with non-trivial quasi-center.

Partially answered questions on $(P_{k})$-closures and provided new insights on the Weiss conjecture.

Abstract

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 $\mathrm{Aut}(T_{d})$ may feature in terms of the group's local~action. In doing so, we explicitly construct closed, non-discrete, compactly generated subgroups of $\mathrm{Aut}(T_{d})$ 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 $\mathrm{Aut}(T_{d})$ containing an involutive inversion, and thereby partially answer two questions by Banks--Elder--Willis. Finally, we offer a new view on the Weiss conjecture.