Trees, contraction groups, and Moufang sets

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 stabilizer of some end is metabelian, then there is a local field $k$ such that ${PSL}_2\left(k\right)\le G\le {PGL}_2\left(k\right)$. 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 $1$ 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 we provide a complete classification in case the root groups are torsion-free.