On Discrete Chamber-transitive Automorphism Groups of Affine Buildings

1. Introduction. Let A be the affine building of a simple adjoint algebraic group Q of relative rank > 2 over a locally compact local field K. Let Aut A (resp. E Aut A) denote the group of type-preserving (resp. of all) auto-morphisms of A. Note that E Aut A contains the group $(K) of ÜT-rational points of §. We will be interested in discrete subgroups of Aut A which are chamber-transitive on A. It is extremely rare that such groups exist and, as can therefore be expected, exceptions are interesting phenomena; our purpose is to list them all (see the theorem below). In order to describe them we must first introduce some notation. Let ƒ be a quadratic form in n variables over Q p with coefficients in Z. We let Pfi(/, Z[l/p]) denote the intersection PSO(/, Q p)'nPGL(n, Z[l/p]) within PGL(n,Q p), and similarly PGO(/, Z[l/p]) = PGO(/,Q p) nPGL(n,Z[l/p]). In the following list, T will always be a chamber-transitive subgroup of Aut A. The fundamental quadratic form (over Z) of the root lattice of type A n , B n , E n , normalized so that the long roots have squared length 2, will be denoted by a n ,6 n ,e n , respectively; note that b n is Yl" x l-(i) Let ƒ = eg, &7,ci6,66*^6* or as, and let A be the affine building of PSO(/,Q 2). Here T can be any group between r min = Pfi(/,Z[l/2]) and r ma x = PGO(/,Z[l/2]) fi Aut A. The quotient r max /r min is elementary abelian of order 1, 1, 1, 4, 2, or 2, respectively, and r max is generated by Train and reflections. (ii) Let ƒ = &5,e6, or b' e = Xa x ? + ^ x h an(l let A be the building of PSO(/,Q 3). The group r max (/) = PGO(/, Z[l/3]) n Aut A has 3, 5, or 9 conjugacy classes of chamber-transitive subgroups T. Passage mod 2 maps r m ax(b5) onto the symmetric group S5, and the preimages in r max (&5) of S5, A5, or a group of order 20 form the 3 desired conjugacy classes of groups T. The forms e& and b' e are rationally equivalent, and hence the buildings they define over Q3 are the "same" ; with suitable identifications of buildings and groups, T b = r max (ee) …