We study closed subgroups of the automorphism group of a locally finite tree acting doubly transitively on the boundary. We show that if the stabilizer of some end is metabelian, then there is a local field such that . We also show that the contraction group of some hyperbolic element is closed and torsion-free if and only if is (virtually) a rank simple -adic analytic group for some prime . A key point is that if some contraction group is closed, then is boundary-Moufang, meaning that the boundary 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.
"Trees, contraction groups, and Moufang sets." Duke Math. J. 162 (13) 2413 - 2449, 1 October 2013. https://doi.org/10.1215/00127094-2371640