Moufang Buildings and Twin Buildings

The “Moufang Condition” for spherical buildings was introduced by J. Tits in the appendix of [9], as a tool to give more structure to the classification of spherical buildings of rank at least three (which are automatically “Moufang”). More recently, also the spherical buildings of rank 2 satisfying the Moufang Condition are classified [13]. Hence one could say that, on the geometric level, spherical buildings axiomatize the situation of a simple group of Lie type, while on the group-theoretic level, the Moufang Condition characterizes the groups themselves as automorphism groups. A few years ago, a similar phenomenon occured after the discovery of Kac-Moody algebras and Kac-Moody groups. In [11] Tits gave a group theoretical definition of a “Moufang Condition” intrinsically generalizing the notion of “Moufang spherical building”. This definition was formally translated into geometrical language by Ronan in his book [7]. The main motivation was an attempt to characterize the KacMoody groups as automorphism groups of certain buildings (namely, the Moufang buildings). However, this equivalence is not yet established. On the geometric level, Ronan and Tits introduced the so-called “twin buildings” (see [12]), and these axiomatize the situation of a Kac-Moody group geometrically. Again, the equivalence of (simple) Kac-Moody groups and twin buildings (possibly under some additional hypotheses) has not been established yet. However, the work of Tits (see [11], [12], [9]), Mühlherr (see [4], [5], [6]) and Ronan points in the direction that a classification of 2-spherical twin buildings (i.e., twin buildings with a diagram containing no edges labelled ∞) is feasible. Moreover, Mühlherr and Ronan show in [3] that, under some mild restrictions, both combinatorial buildings of a 2-spherical twin building satisfy the Moufang Condition. Hence, in view of the analog for the spherical case, and in view of a possible classification of Moufang buildings (still under some additional,