LOCAL FIELDS AND TREES 5
trees of larger degree. For the case
q =
2 the reader may refer to Fig.1, ignoring
for the time being the notations at the margins of the figure.
THE GROUP OF AUTOMORPHISMS OF A TREE
A tree automorphism is simply a bijection carrying vertices into vertices and
preserving edges. The automorphims of a tree form a group under the composition
of two automorphisms. We shall denote this group by
Aut( X). If
a vertex is mapped
into another vertex by an automorphism the two vertices must belong to the same
number of edges. Thus the condition of homogeneity is a necessary condition to
have a transitive action of
Aut(X)
on the set of vertices. It is not difficult to see
that the condition is also sufficient.
The set of vertices X of a tree is, in a natural way, a metric space (X, d). It is
sufficient to define the distance
d(x,
y) between two vertices
x
andy as the number
of edges defined by the unique chain that joins x
toy. It
is also evident that every
automorphism of the tree is an isometry and vice versa, because two vertices belong
to the same edge if and only if their distance is one. Thus
Aut( X)
may be considered
as the group of isometries of
(X, d).
The group
Aut( X)
may be naturally endowed
with a topology (the compact open topology). The basic open sets of this topology
are associated to finite sets of vertices F
=
{Xi :
i
=
1, ... ,
n}
and F'
=
{Yi :
i
=
1, ... ,
n },
and are defined as follows:
U
F,F'
=
{g
E
Aut(X) :
g(xi)
=
Yi,
i
=
1, ...
n }.
TRANSLATIONS ROTATIONS AND INVERSIONS
There is a simple classification of automorphisms of a tree which is due to J. Tits.
If g
E
Aut( X)
is not the identity then one and only one of the following happens:
a) g fixes a vertex; b) g leaves an edge invariant exchanging the two vertices of this
edge; c) There exists a doubly infinite chain { ...
X-n• ...
xo, ... ,
Xn . .. }
(that is a
double sequence of vertices such that the two-elements sets
{Xi,
xi+
1}
are edges and
Xi
-::f
Xi+2)
and an integer k
E
Z, such that
gxi
=
Xi+k·
It
is natural to call rotation an automorphism satisfying a); inversion an auto-
morphism satisfying b); and translation an automorphism satisfying c). The integer
k (or rather its absolute value) appearing in the condition b) is called the step of
the translation. Observe that every translation is the composition of translations
of step one. The group of all rotations about x (a compact group) is denoted by
Kx.
A doubly infinite chain{ ...
X-n ...
xo, ... ,
Xn ... }
defined as above is sometimes
called an (infinite) geodesic and is denoted by
'Y·
We can consider then the group
G'"Y
= {g: Y'Y = 'Y}
which contains all translations along
'Y,
but also some inversions
about the edges contained in
'Y,
and some rotations about vertices of
'Y·
Similarly
we consider the compact group
K{a,b}
consisting of all inversion about the edge
{a,
b} and all rotations which
fix
both a and
b.
Another easy observation is that the group generated by all rotations (about all
possible vertices) is a subgroup of index
2
in
Aut(X),
and that when q 2 every
automorphism is
a
product of translations.
BOUNDARY OF THE TREE
We shall assume in the sequel that
q ::::::
2. An infinite chain in the tree is a
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