Trees, contraction groups, and Moufang sets

TL;DR

This paper classifies certain automorphism groups of locally finite trees, linking their structure to local fields, Moufang sets, and p-adic groups, revealing deep connections between group actions, geometry, and algebra.

Contribution

It establishes a classification of boundary-Moufang groups acting on trees, connecting their properties to local fields and p-adic analytic groups, and analyzes contraction groups in this context.

Findings

If a stabilizer of an end is metabelian, the group embeds into PGL_2 over a local field.

Closed, torsion-free contraction groups correspond to rank one p-adic analytic groups.

Boundary actions are Moufang sets, with a complete classification when root groups are torsion-free.

Abstract

We study closed subgroups $G$ of the automorphism group of a locally finite tree $T$ acting doubly transitively on the boundary. We show that if the stabiliser of some end is metabelian, then there is a local field $k$ such that $\mathrm{PSL}_2(k) \leq G \leq \mathrm{PGL}_2(k)$. We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if $G$ is (virtually) a rank one simple $p$-adic analytic group for some prime $p$. A key point is that if some contraction group is closed, then $G$ is boundary-Moufang, meaning that the boundary $\partial T$ is a Moufang set. We collect basic results on Moufang sets arising at infinity of locally finite trees, and provide a complete classification in case the root groups are torsion-free.