A classification theorem for boundary 2-transitive automorphism groups of trees

TL;DR

This paper classifies certain highly symmetric automorphism groups of large-valency trees, revealing a countable family with notable simple and universal subgroups, and shows the classification does not extend to smaller trees.

Contribution

It provides a classification of boundary 2-transitive automorphism groups of large-valency trees, identifying key subgroup structures and exceptions for smaller trees.

Findings

Classified groups acting 2-transitively on tree ends with local alternating groups.

Identified a countable family of groups with notable simple and universal subgroups.

Showed classification fails for trees with smaller degree.

Abstract

Let $T$ be a locally finite tree all of whose vertices have valency at least $6$. We classify, up to isomorphism, the closed subgroups of $\mathrm{Aut}(T)$ acting $2$-transitively on the set of ends of $T$ and whose local action at each vertex contains the alternating group. The outcome of the classification for a fixed tree $T$ is a countable family of groups, all containing two remarkable subgroups: a simple subgroup of index $\leq 8$ and (the semiregular analog of) the universal locally alternating group of Burger-Mozes (with possibly infinite index). We also provide an explicit example showing that the statement of this classification fails for trees of smaller degree.