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This monograph extends this approach to the more general
investigation of X-lattices, and these "tree lattices" are the main
object of study. The authors present a coherent survey of the
results on uniform tree lattices, and a (previously unpublished)
development of the theory of non-uniform tree lattices, including
some fundamental and recently proved existence theorems. Tree
Lattices should be a helpful resource to researchers in the field,
and may also be used for a graduate course on geometric methods in
group theory.
These notes are a slightly revised and extended version of mim-
graphed notes written on the occasion of a seminar on buildings and
BN-pairs held at Oberwolfach in April 1968. Their main purpose is
to present the solution of the following two problems: (A)
Determination of the buildings of rank >; and irreducible,
spherical type, other than ~ and H ("of spherical type" means "with
finite Weyl 4 group", about the excluded types H, cf. the addenda
on p. 274). Roughly speaking, those buildings all turn out to be
associated to simple algebraic or classical groups (cf. 6. ;, 6.
1;, 8. 4. ;, 8. 22, 9. 1, 10. 2). An easy application provides the
enumeration of all finite groups with BN-pairs of irreducible type
and rank >;, up to normal subgroups contained in B (cf. 11. 7).
(B) Determination of all isomorphisms between buildings of rank
> 2 and spherical type associated to algebraic or classical
simple groups and, in parti cular, description of the full
automorphism groups of such buildings (cf. 5. 8, 5. 9, 5. 10, 6. 6,
6. 1;, 8. 6, 9. ;, 10. 4). Except for the appendices, the notes are
rather strictly oriented - ward these goals.
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