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, x1, ... 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 = x0 , x1 , ... , 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,
Previous Page Next Page