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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 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 Gl(2,K), K a local
field.
The most important property familiar from the theory of Tits systems which still
holds for the groups we study here is the existence of a Bruhat decomposition.
However, while the Weyl group is still a Coxeter group, with a distinguished set S of
generators, the rule of multiplying double cosets by elements of S is very different
from the familiar situation: there are elements s in S for which for all w in the Weyl
group, s.B.w.B = B.s.w.B.
For the theory of Tits systems and some of its applications, the reader can consult
Tits, Bruhat and Tits, Iwahori and Matsumoto and Garland and the bibliographies
referred to therein. The idea of studying groups with Bruhat decomposition more
general than those with Tits system was first introduced in our work. It was
in that paper that the intimate relation between multiple transitivity and
groups with Bruhat decomposition was first noticed. But while this paper
is thus conceptually closely related to our work, the notion of transitivity
here introduced and the proof of Bruhat decomposition are wholly different
Also, the central notion of this paper—that of a normalizing Tits system—is
new.
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