Classical locally finite tits chamber systems of rank 3

Abstract

A chamber system V = ($9, ( j,zi)is,) over some index set Z is a set %’ of “chambers” together with partitions +$, in Z of V. If .ZE Z and c E V then bJ is the join of all partitions fijzi, ~E.Z and d,(c) is the element of fiJ containing c. d,(c) is again a chamber system over J with fij restricted to .Z. We say Q? is connected if d,(c) = $??. Furthermore 111 is the rank of V. The most important examples of chamber systems are those obtained from groups. If G is a group, B a subgroup of G, and Xi, iE Z are subgroups of G containing B then 5?? = %‘(G; B; (Xi)icl) has as chambers the cosets Bg, g E G, two such cosets Bg, Bh being i-adjacent if and only if Xi g = Xih. Then G acts (chamber)-transitively on %’ with kernel B, = n Bg and %? is connected if and only if G = (X, 1 i E Z). Furthermore, each chamber system with transitive automorphism group G can be obtained in this way by setting B = G, and Xi = Gd,Cr). A rank 2 chamber system %? = (U, hi, fi,) is called a classical generalized ma-gon (mq-> 3) if %’ is isomorphic to %?(G; B; Pi, P,), where G is an (essentially) simple rank 2 group of Lie type (in the sense of Cl]), B is a Bore1 subgroup, and P,, Pj are the two maximal parabolic subgroups of G containing B. Here md is given by the relation VU: = wj = (wiw,)“g = 1 defined by the Weyl group of G. A rank 2 chamber system is a generalized digon if %? is isomorphic to V(G, Pin P,, Pi, P,), where ,Pi, P, are different proper subgroups of G satisfying G = PiPj= PjPi. A connected chamber system %’ over Z is called a classical Tits chamber system if for each CE %’ and {i, j} c Z, di, j(c) is either a generalized digon or a classical generalized m,-gon (for some mi, E N). %’ is focally finite if all d,](c) are finite. For motivation to consider such classical locally finite Tits chamber systems see [15] and [16]. If %? is a classical Tits chamber system then the diagram d(Z) is defined in 9 0021-8693/89 $3.00