Highly transitive group actions on trees and normalizing Tits systems (Q1065152)

The theory of Tits systems associates to each group \(G\) with Tits system a simplicial complex, together with a ``numbering'' of its set of vertices, on which the group acts in a highly transitive manner. This numbered simplicial complex is a tree if and only if the Weyl group of the Tits system is an infinite dihedral group, for example when \(G\) is \(\text{PSL}(2,K)\), \(K\) is a local field, with its affine Tits system structure, or when \(G\) is the central quotient of the group associated to a Kac-Moody Lie algebra of rank 1. There are non-algebraic examples of such groups as well, such as the full automorphism group of a numbered tree. In this paper, we investigate the structure of groups acting highly transitively on a tree without preserving a given numbering of the set of vertices. Such groups no longer possess the structure of a Tits system. However, we show that such groups have a pair of subgroups \(B\) and \(N\) which satisfy all the properties of a Tits system, except the requirement that the generators of the Weyl group should not normalize \(B\). We have called a group \(G\) with \(B\) and \(N\) satisfying these properties a normalizing Tits system. We show that these groups have some properties closely related to, but different from, those arising in the theory of Tits systems, such as the structure of the set of `parabolic subgroups' of \(G\). There are very simple examples of such groups, for instance the full group of automorphisms of a tree of \(\text{GL}(2,K)\), \(K\) a local field.