Moufang twin trees of prime order

Abstract

We prove that the unipotent horocyclic group of a Moufang twin tree of prime order is nilpotent of class at most 2.

Locally pro-p contraction groups are nilpotent

“…Remark 4.7 Theorem A in [5] may be deduced from Theorem A in the present paper. Although the former paper concerns Moufang twin trees, it is not necessary to know what they are in order to follow the proof because [5, Theorem A] is deduced from another purely group-theoretic result.…”

“…(c) For all (j, k) ∈ Φ, the map π j,k : (5), and defines an endomorphism φ of the topological group F p ((t)) d .…”

“…To do this, the notion of a Z-system of order p is defined. This is a group, X, and a sequence (x n ) n∈Z ⊂ X satisfying certain axioms set out in [5,Definition 3.2]. Then [5, Theorem A] is deduced from [5,Theorem 3.4], which asserts that if (X, (x n ) n∈Z ) is a Z-system of prime order, then X is nilpotent of class at most 2.…”

“…Briefly, let (X, (x n ) n∈Z ) be a Zsystem of order p, and let t be the automorphism of X such that t(x n ) = x n+2 . Then X 0,∞ = x k | k ≥ 0 , see [5,Definition 4.1], is a commensurated…”

“…[16]), where contraction groups appear naturally in the study of general automorphisms [1]. It also has implications for Moufang twin trees, see [5] and Remark 4.7.…”