On Lipschitz compactifications of trees.docx

On Lipschitz compactifications of trees We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given. a r X i v :0804.2357v 1 [m a t h .M G ] 15 A p r 2021On Lipschitz compacti?cations of trees Beno??t Kloeckner April 16,2021Abstract We study the Lipschitz structures on the geodesic compacti?cation of a regular tree,that are preserved by the automorphism group.They are shown to be similar to the compacti?cations introduced by William Floyd,and a complete description is given.In [4],we described all possible di?erentiable structures on the geodesic compacti?cation of the hyperbolic space,for which the action of its isometries is di?erentiable.We consider here the similar problem for regular trees and obtain a description of “di?erentiable”compacti?cations,based on an idea of William Floyd [3].A tree has a geodesic compacti?cation,but it is obviously not a manifold and we shall in fact replace the di?erentiability condition by a Lipschitz one.Note that we only consider regular trees so that we have a large group of automorphisms,hence the greatest possible rigidity in our problem.A close case is that of the universal covering of a ?nite graph (that is,when the automorphism group is cocompact).Our study does not extend as it is to this case,in particular one can convince oneself by looking at the barycentric division of a regular tree that condition (1)in theorem 2.1should be modi?ed.However,similar results should hold,up to considering the translates of a fundamental domain instead of the edges at some point. This note is made of two sections.The ?rst one recalls some facts about regular trees and their automorphisms,Floyd compacti?cations,and gives the de?nition of a Lipschitz compacti?cation.The second one contains the result and its proof. 1 We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given. 1Preliminaries 1.1Regular trees and their automorphisms We denote by T n the regular tree of valency n≥3and by T n is topological realization,obtained by replacing each abstract edge by a segment.All considered metrics on T n shall be length metrics,since general metrics could have no relation at all with the combinatorial structure of T n.Up to isometry, two length metrics on T n that are compatible with the topology di?er only by the length of the edges.We shall therefore identify T n equipped with such a metric and T n equipped with a labelling of the edges by positive real numbers(the label corresponding to the length of the edge).When all edges are chosen of length1,we call the resulting metric space the“standard metric realization”of T n,denoted by T n(1).Its metric shall be denoted by d;it coincides on vertices with the usual combinatorial distance. There is a natural one-to-one correspondence between automorphism of T n and isometries of T n(1).We denote both groups by Aut(T n)and endow them with the compact-open topology,so that a basis of neighborhoods of identity is given by the sets B K(Id)={φ∈Aut(T n);φ(x)=x?x∈K} where K runs over all?nite sets of vertices. Given an automorphismφ,one de?nes the translation length ofφas the integer T(φ)=min x{d(x,φ(x))}where the minimum is taken over all points (not only vertices)of T n(1).The following alternative is classical: 1.if T(φ)0then there is a unique invariant bi-in?nite path(x i)i∈Z and φ(x i)=x i+T(φ)for all i, 2.if T(φ)=0then eitherφ?xes some vertex,orφhas a unique?xed point in T n(1),which is the midpoint of an edge. In the?rst case,φis said to be a translation(a unitary translation if T(φ)= 1).Any translation is a power of a unitary translation. 1.2Compacti?cation of trees The standard metric tree T n(1)is a CAT(0)complete length space,thus is a Hadamard space(see for example[2]).Therefore,it has a geodesic compacti?cation we now brie?y describe. A boundary point p is a class of asymptotic geodesic rays,where two geodesic raysγ1=x0,x1,...,x i,...andγ2=y0,y1,...,y j,...are said to be asymptotic if they are eventually identical:there are indices i0and j0so that for all k∈N,on has x i 0+k =y j 0+k .The point p is said to be the endpoint of any geodesic ray of the given asymptoty class. 2 We study the Lipschitz structures on the geodesic compactification of a regular tree, that are preserved by the automorphism group. They are shown to be similar to the compactifications introduced by William Floyd, and a complete description is given. The union T n by homeomorphisms for this topology.Our goal will be to see which additional structure can be added to this topology,。