Locally split and locally finite twin buildings of 2-spherical type

Abstract

We prove a fixed point theorem for twin buildings of arbitrary rank. This theorem is then used to construct certain twin buildings whose existence was conjectured in [12]. As a consequence we obtain a classification of twin buildings whose rank 2 residues correspond to split algebraic groups over a field of cardinality at least 4. A similar result follows for twin buildings whose rank 2 residues are finite.

Groups with a root group datum

“…However, the residual nilpotency of U + is delicate to establish. The way it is done in [63] is by realizing the inductive system of rank two groups of D in a certain large group which is known to possess a root group datum (mostly the latter group is a split Kac-Moody group). This allows to embed U + in some unipotent radical of this larger group.…”

“…Given any field k, the group G B (k) is naturally endowed with a family of subgroups {U α } α∈Φ(B) , all isomorphic to the additive group of k, which is a root group datum for a subgroup The analogy with the theory of reductive algebraic groups can be pushed one step further: Kac-Moody groups admit non-split forms which also possess naturally root group data. The non-split forms may be obtained by an algebraic process of Galois descent, which is defined and studied in [72,, or by using other twisting methods which do not fit into the context of Galois descent: see [45] for Steinberg-Ree type constructions and [63], [64] for some others. In all cases, one obtains groups endowed with root group data; the Weyl group is generally infinite, and the underlying root basis might be of infinite rank as well.…”

Final group topologies, Kac-Moody groups and Pontryagin duality

“…We will show later that, if the Dynkin diagram Δ of K(A) is a tree, then there essentially exists a unique Phan amalgam associated to Δ. On the other hand, if Δ admits cycles, this cannot be expected in view of [86, Section 6.5] and [68].…”

Embeddings of root systems I: Root systems over commutative rings

“…The group G (actually, the isomorphic group of invertible R-linear maps of M acting on each finite rank submodule in the same way as some element of W ) is implicit in [14]. Also, in [17] a combinatorial analogue of the completion G was used to construct buildings.…”