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 2008On Lipschitz compacti?cations of trees

Beno??t Kloeckner April 16,2008Abstract 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.

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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.

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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,that is preserved by Aut(T n).

We have no di?erentiable structure on

T n,whereδis a length metric,and such that the action of Aut(T n)on

T n is the length metric obtained from a vertex x0and a Floyd function h by assigning to each edge e the length h(d), where d∈N is the combinatorial distance between e and x0.

By a Floyd compacti?cation of T n we mean the topological space

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.

h(r)converges ensures that we do get a distance The condition that

on

T n(h).

Proof.We?rst prove that any Lipschitz compacti?cation of T n is a Floyd compacti?cation.

Letδ′be any length metric in the given Lipschitz class,and?x any vertex x0of T n.We de?ne h by h(r)=minδ′(x,y)where the minimum is taken over all edges xy that are at combinatorial distance r from x0.Then h is a Floyd function because x0is at?niteδ′distance from the boundary.Denote byδthe Floyd metric obtained from x0and h,and let us prove that[δ]=[δ′]. It is su?cient to prove that there is a constant C so that for all r,two edges that are at combinatorial distance r from x0have theirδ′lengths that di?er by a factor at most C.

For any R∈N,let B(R)be the closed ball of radius R and center x0in T n(1).It contains a?nite number of edges,so that there is a constant C R that satis?es the above property for all r≤R.

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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.

Since the compacti?cation is assumed to be Lipschitz,for all p∈?T n there are a neighborhood V of p,a neighborhood U of the identity and a constant k so that anyφ∈U is k-Lipschitz on V.Since?T n is compact,we can?nd a?nite number of such quadruples(p i,V i,U i,k i)so that the V i cover ?T n.Moreover we can assume that the V i are the connected components of T n\B(R)for some radius R,and that U=∩U i=B B(R)(Id).Since for all i and r>R,U acts transitively on the set of edges of V i that are at combinatorial distance r from x0,those edges have theirδ′-length that di?er by a factor at most C′=sup k i.Moreover,there is an automorphismφ0that ?xes x0and permutes cyclically the V i.Sinceφ0is locally Lipschitz,there is a R′and a C′′so that for all r≥R′and all couple(i1,i2),there are edges of

V i

1and V i

2

that are at combinatorial distance r from x0and whoseδ′lengths

di?er by a factor at most C′′.The supremum C of C R′and C′′C′2is the needed constant.

Consider now the Floyd compacti?cation obtained from x0and h and denote byδthe associated Floyd metric.By construction,any automorphism φof T n that?xes x0is an isometry forδ,thus is locally bilipschitz for the corresponding Lipschitz structure.

Two translations are close to one another when they di?er by an element close to identity.An element close enough to identity must?x x0,thus is an isometry.Therefore,we only need to prove that a given translation is Lipschitz to get that all automorphisms in a neighborhood are equilipschitz. Checking unitary translations is su?cient since any translation is an iterate of one of those.

Letφbe a unitary translation,andγ=...,y?1,y0,y1,...be its trans-lated geodesic,where we assume that y0realizes the minimal combinatorial distance d0between vertices ofγand x0.By local?niteness,φis locally bilipschitz around any point of T n and we need only check the boundary.

Let us start with the attractive endpoint p ofγ.Assume that our Floyd compacti?cation is Lipschitz.It implies thatφis locally bilipschitz around p,in particular there is a r0>0and a k>1such that for any r≥r0,

kδ(y r+1,y r+2)≥δ(y r,y r+1)

h(r+d0+1)≥k?1h(r+d0)

which gives condition(1).

Conversely,assume that(1)holds.

For any vertex x we have

|d(φ(x),x0)?d(x,x0)|≤1+2d0

since the worst case is when x=x0or x is in a connected component of T n\{x0}other than that ofγ.Therefore,the length of an edge and of its

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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.

image byφdi?er by a factor bounded byη?(1+2d0).Therefore,φis Lipschitz. Sinceφ?1is also a unitary translation,φis bilipschitz.

It would be interesting to consider more general spaces,for example eu-clidean buildings or CAT(-1)buildings like the I pq described by Bourdon in [1].It is not obvious how to de?ne the Floyd compacti?cation:for exam-ple,a mere scaling of the distance in each cell by a factor depending on the combinatorial distance to a?xed cell would create gluing problems(an edge shared by two faces having two di?erent length).This spaces could therefore be less?exible than trees.

References

[1]M.Bourdon.Immeubles hyperboliques,dimension conforme et rigidit´e

de Mostow.Geom.Funct.Anal.,7(2):245–268,1997.

[2]Dmitri Burago,Yuri Burago,and Sergei Ivanov.A course in metric ge-

ometry,volume33of Graduate Studies in Mathematics.American Math-ematical Society,Providence,RI,2001.

[3]William J.Floyd.Group completions and limit sets of Kleinian groups.

Invent.Math.,57(3):205–218,1980.

[4]Beno??t Kloeckner.On di?erentiable compacti?cations of the hyperbolic

space.Transform.Groups,11(2):185–194,2006.

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