A Combinatorial Construction For Twin Trees

AbstractThe notions of abuilding [6]and, later, atwin building[7] were introduced to give an axiomatical geometric treatment of groups with BN-pairs and with twin BN-pairs. For this reason, buildings and twin buildings have always been studied from a point of view relevant to (and originated from) BN-pairs.The notion of atwin tree,an affine twin building of rank 2, was introduced in [2] by a self-contained, very elementary and geometric definition. The study of twin trees in [2–4], and the explicit constructions given there, also use notions and groups inspired by BN-pairs.In this paper we shall present an elementary combinatorial construction for twin trees which, in a sense, is orthogonal to the BN-pairs approach. This construction gives all twin trees; in particular, it provides an easy proof that twin trees of any given valency are uncountably many (a result proved independently by M. Ronan [1]).Our approach turns out to be especially fruitful for twin trees of valency 3 (and probably of valency 4; that part of the work is still in progress). In particular, it highlights a new geometric object (a pairing)within a twin tree—and hence a new class of subgroups of its automorphism group. When the twin tree in question has a ‘rich’ automorphism group, study of these objects (and corresponding subgroups) gives a lot of new and unexpected information on combinatorial and group-theoretic structure of the twin tree