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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