6

ALESSANDRO FIGA-TALAMANCA

sequence

x0,

x 1 , ...

Xn, ... of

distinct

vertices such that {

Xi-1, xi}

is an edge. Two

infinite chains are equivalent if they coincide except for a finite number of vertices.

In other words

x0,

x 1, ... Xn, ... and Yo, y1, ... Yn, ... are equivalent if there exist h

and k such that for every

n,

Xh+n

=

Yk+n· An element

w

of the boundary of a tree

n

is simply an equivalence class of infinite chains. From this point of view it easy to

see that the group Aut( X) acts also on the boundary

n.

Indeed every automorphism

maps an infinite chain into an infinite chain and preserves equivalence classes.

We also observe that given a vertex

x

E

X, and an infinite chain there is a unique

chain starting at

x

and equivalent to the given chain. Indeed let

y

be the point of

the chain of minimum distance from

x

(which is

x

itself if it belongs to the chain).

Then the required chain is made up of the points of the finite chain joining

x

to

y (which we may denote by

[x,

y]) and the points of the chain following y in the

order.

Let

wE

f!. We may at this point introduce the notation

[x,w)

to denote the

unique chain which starts at

x

and belongs to the equivalence class

w.

Another way to define the elements of

n

is to fix a vertex o which we may call

the origin, and consider all infinite chains starting at

o.

They all belong to different

equivalence classes and therefore they correspond exactly to the points of

n.

Once the origin o is fixed we can also define a natural metric on

n

which makes

it into a compact

ultrametric space.

This is how: if two chains starting at o have no

other vertex in common then we declare that their distance, as points of

n

is one. If

they have exactly n

+

1 vertices in common (including o itself) then we declare that

their distance is

~q-n.

In other words a closed ball of radius

~q-n,

to wit a set

defined by

{w: d(w,w')

~ ~q-n},

is associated to a finite

chain

o

=

x0,

x

1

, ... ,

Xn

and consists of all infinite chains starting at o which contain it. The infinite chain

w' (the center of the ball) can be chosen to be any infinite chain starting at o and

containing the finite chain o

=

x

0

,

x

1

, ... , Xn-

In other words

every element of the

closed ball

may

be chosen

as

center of the ball.

It is clear from this description that under the distance so defined

n

becomes a

compact metric space satisfying the ultrametric inequality:

d(w,w')

~

max{d(w,w"),d(w",w')}.

This means in particular that the relation

d(w, w')

~

r,

is an equivalence relation,

and its equivalence classes are the closed balls of radius

r

(if

r

is a value attained

by the distance). In particular closed balls of given radius form a partition of

n.

If together with o we fix a boundary point

w

E

n

we can also define a natural

metric satisfying the ultrametric property on the locally compact space

n \ {

w}.

This definition is given later in the special case in which

n \ {

w}

may be identified,

as a metric space, with a local field.

We observe that if g

E

Aut(X) has the property that go

=

o, then the action

of g on

n

is isometric, that is g induces an isometry on the boundary. But the

converse is also true: if

g

is an isometry of

n

then it maps infinite chains starting

at

o

into infinite chains starting at

o.

This induces a map of X onto itself which

clearly preserves the edges and fixes o. In other words

g

induces an automorphism

having o as a fixed point.

If in place of o we choose another vertex o', the metric induced on

n

is different.

It is not difficult to see however that the new metric is equivalent to the old one,

ALESSANDRO FIGA-TALAMANCA

sequence

x0,

x 1 , ...

Xn, ... of

distinct

vertices such that {

Xi-1, xi}

is an edge. Two

infinite chains are equivalent if they coincide except for a finite number of vertices.

In other words

x0,

x 1, ... Xn, ... and Yo, y1, ... Yn, ... are equivalent if there exist h

and k such that for every

n,

Xh+n

=

Yk+n· An element

w

of the boundary of a tree

n

is simply an equivalence class of infinite chains. From this point of view it easy to

see that the group Aut( X) acts also on the boundary

n.

Indeed every automorphism

maps an infinite chain into an infinite chain and preserves equivalence classes.

We also observe that given a vertex

x

E

X, and an infinite chain there is a unique

chain starting at

x

and equivalent to the given chain. Indeed let

y

be the point of

the chain of minimum distance from

x

(which is

x

itself if it belongs to the chain).

Then the required chain is made up of the points of the finite chain joining

x

to

y (which we may denote by

[x,

y]) and the points of the chain following y in the

order.

Let

wE

f!. We may at this point introduce the notation

[x,w)

to denote the

unique chain which starts at

x

and belongs to the equivalence class

w.

Another way to define the elements of

n

is to fix a vertex o which we may call

the origin, and consider all infinite chains starting at

o.

They all belong to different

equivalence classes and therefore they correspond exactly to the points of

n.

Once the origin o is fixed we can also define a natural metric on

n

which makes

it into a compact

ultrametric space.

This is how: if two chains starting at o have no

other vertex in common then we declare that their distance, as points of

n

is one. If

they have exactly n

+

1 vertices in common (including o itself) then we declare that

their distance is

~q-n.

In other words a closed ball of radius

~q-n,

to wit a set

defined by

{w: d(w,w')

~ ~q-n},

is associated to a finite

chain

o

=

x0,

x

1

, ... ,

Xn

and consists of all infinite chains starting at o which contain it. The infinite chain

w' (the center of the ball) can be chosen to be any infinite chain starting at o and

containing the finite chain o

=

x

0

,

x

1

, ... , Xn-

In other words

every element of the

closed ball

may

be chosen

as

center of the ball.

It is clear from this description that under the distance so defined

n

becomes a

compact metric space satisfying the ultrametric inequality:

d(w,w')

~

max{d(w,w"),d(w",w')}.

This means in particular that the relation

d(w, w')

~

r,

is an equivalence relation,

and its equivalence classes are the closed balls of radius

r

(if

r

is a value attained

by the distance). In particular closed balls of given radius form a partition of

n.

If together with o we fix a boundary point

w

E

n

we can also define a natural

metric satisfying the ultrametric property on the locally compact space

n \ {

w}.

This definition is given later in the special case in which

n \ {

w}

may be identified,

as a metric space, with a local field.

We observe that if g

E

Aut(X) has the property that go

=

o, then the action

of g on

n

is isometric, that is g induces an isometry on the boundary. But the

converse is also true: if

g

is an isometry of

n

then it maps infinite chains starting

at

o

into infinite chains starting at

o.

This induces a map of X onto itself which

clearly preserves the edges and fixes o. In other words

g

induces an automorphism

having o as a fixed point.

If in place of o we choose another vertex o', the metric induced on

n

is different.

It is not difficult to see however that the new metric is equivalent to the old one,