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