On Lipschitz compactifications of trees

On Lipsc hitz compactifications of trees Beno ˆ ıt Klo ec kner August 23, 202 1 Abstract W e study the Lipschitz structures on t he geo desic compactification of a reg ular tree, that are preserv ed b y the automorph i sm group. They are sho wn to b e similar to the compactifications introd uced by William Flo yd, and a complete description is giv en. In [4], we desc rib ed all p ossible differen tiable structures on the geo desic compactification of the hy p erb olic space, for which the action of its isometries is differen tiable. W e consider here the similar problem for regular trees and obtain a description of “differentiable” compactifications, based on an idea of William Floy d [3]. A tree has a geodesic compactification, but it is ob viously not a manifo ld and w e shall in fact replace the differen tiabilit y condition by a Lipsc hitz one. Note that w e only consider regular trees so that we hav e a large gro up of automorphisms, hence the greatest possible rigidity in our problem. A close case is that of the univ ersal co v ering of a finite graph (that is, when the automorphism group is co compact). Our study do es not extend as it is to this case, in pa r t ic ular one can convinc e onese lf by lo oking at the barycen tric division of a regular tree that condition (1) in theorem 2.1 should b e mo dified. Ho w ev er, similar results should hold, up to considering the translates of a fundamen tal domain instead of the edges at some p oin t. This note is made of tw o sections. The first one recalls some facts ab out regular tree s and their automorphisms, Flo yd compactifications, and giv es the definition of a Lipsc hitz compactification. The second one contains the result and its pro of. 1 1 Preliminaries 1.1 Regular trees and their automorphisms W e denote b y T n the regular tree of v alency n ≥ 3 and b y T n is top o logical realization, o btained by replacing each abstract edge by a segmen t. All considered metrics on T n shall be length metrics, since general metrics could ha v e no relation at all with the comb inatoria l structure of T n . Up t o isometry , t w o length metrics on T n that are compatible with the top ology differ only b y the length of the edges. W e shall therefore identify T n equipped with suc h a metric and T n equipped with a lab elling of the edges b y p o s itiv e real n um b ers (the lab el cor r esp onding to the length of the edge). When all edges a re c hosen o f length 1, w e call the resulting metric space the “standard metric realizatio n” of T n , denoted b y T n (1). Its metric shall be denoted b y d ; it coincides on v ertices with the usual com binatorial distance. There is a natura l one-to-one correspo nde nce b et w een automorphism of T n and isometries of T n (1). W e denote b oth groups b y Aut( T n ) and endow them with the compact-op en to pology , so that a basis of neighbor ho o ds of iden tit y is give n b y the sets B K (Id) = { φ ∈ Aut ( T n ); φ ( x ) = x ∀ x ∈ K } where K runs o v er all finite sets of v ertices. Giv en an automorphism φ , one defines the tr ans l a t ion length of φ as the in teger T ( φ ) = min x { d ( x, φ ( x )) } where the minim um is tak en o v er all p oin ts (not only v ertice s) of T n (1). The follow ing alternativ e is classical: 1. if T ( φ ) > 0 then there is a uniq ue in v arian t bi- infinite path ( x i ) i ∈ Z and φ ( x i ) = x i + T ( φ ) for all i , 2. if T ( φ ) = 0 t hen either φ fixes s ome v ertex , o r φ has a unique fixed p oin t in T n (1), whic h is the midp oin t of an edge. In the first case, φ is said to b e a tr anslation (a unitary translation if T ( φ ) = 1). Any translation is a p o w er of a unitary translation. 1.2 Compactification of trees The standard metric tree T n (1) is a CA T(0) complete length space, th us is a Hadamard space (see for example [2]). Therefore, it has a geo desic compactification we no w briefly desc rib e. A b oundary p oin t p is a class of a s ymptotic geo desic ra ys, where t w o geo desic ray s γ 1 = x 0 , x 1 , . . . , x i , . . . and γ 2 = y 0 , y 1 , . . . , y j , . . . are said to b e asymptotic if they are ev en tually identic al: there are indices i 0 and j 0 so that for all k ∈ N , on has x i 0 + k = y j 0 + k . The po int p is said t o b e the endp oin t of an y geodesic ray of the giv en asymptot y class. 2 The union T n = T n ∪ ∂ T n is given the follow ing top ology: for a p oin t that is not on the b oundary , a basis of neigh b orho o ds is given b y its neigh b orho o ds in T n ; for a b oundary p oin t p , a basis of neigh b orho ods is giv en by the connected comp onen ts of T n \ { x } con taining a geo desic ray asymptotic to p , where x runs o v er t he v ertice s It is a general prop ert y of Hadamard spaces t hat Aut( T n ) acts on T n b y homeomorphisms for this top ology . Our goal will be to see whic h additional structure can be a dde d to this top ology , that is preserv ed b y Aut( T n ). W e ha v e no differen tiable s tructure o n T n , but due to the Ra de mac her theorem it is natural to lo ok at Lipsc hitz structures instead. Definition 1.1 L et X b e a metrizable top o lo gic al sp ac e. A Lipschitz struc- tur e [ δ ] on X is the data of a me tric δ t hat is c omp atible with the top olo gy of X , up to lo c al Lipschitz e quivalenc e (two metrics δ 1 , δ 2 ar e said to b e lo c a l ly Lipschitz e quivalent if the identity map ( X , δ 1 ) → ( X , δ 2 ) is lo c al ly bilipschitz). The natural isomorphisms of a space X endo we d with a Lipsc hitz struc- ture are t he lo cally bilipsc hitz maps . Usually , for an action of a Lie group on a ma nif o ld to b e diffe rentiable, one asks the map G × M → M to b e differen tiable. Similarly , w e sa y that an action of a top ological gro up Γ on a metrizable top ological space X is Lipsc hitz if it is a contin uous action b y lo cally bilip sc hitz maps, and if moreov er the Lipsc hitz factor is lo cally uniform. W e can now define our main o b ject of study . Definition 1.2 A Lip schitz c omp actific ation of T n is a Lipschitz structur e [ δ ] on T n , w her e δ is a le n gt h metric, and such that the action of Aut( T n ) on T n is Lipschitz. In [3], Flo yd in tro duced a metho d for compactifying a gra ph. W e giv e definitions that are adapted to the simple r cas e of trees. Definition 1.3 By a Floyd function we me a n a function h : N → ]0 , + ∞ [ such that P r h ( r ) < + ∞ . Two F loyd f unction s h 1 , h 2 ar e said to b e compa- rable i f ther e is a C > 1 such that for al l r ∈ N one has C − 1 h 2 ( r ) ≤ h 1 ( r ) ≤ C h 2 ( r ) . Definition 1.4 A Floyd metric on T n is the length metric ob tain e d fr om a vertex x 0 and a Floyd function h by ass i g ning to e ach e dge e the le n gt h h ( d ) , wher e d ∈ N is the c o mbinatorial distanc e b etwe en e a nd x 0 . By a Flo yd c om p actific ation of T n we me an the top olo gic al sp ac e T n en- dowe d with the Lipschitz structur e c orr esp o n ding to a F loyd metric. 3 The condition that P h ( r ) conv erges ensures that we do get a distance on T n . F or example, the distance betw een tw o bo und ary p oin ts p and p ′ is 2 P r ≥ R h ( r ) where R is the com binatorial distance b et we en x 0 and the o nly geo desic joining p and p ′ . Tw o Floyd metrics obtained from the same point x 0 and F lo yd functions h 1 , h 2 are easily seen to define the same Lipsc hitz structures if and only if h 1 and h 2 are comparable. 2 Descrip t ion of all Lipsc hitz compactifica- tions of re g ular trees Theorem 2.1 A ny Lipsc hitz c om p actific ation of T n is a Floyd c omp actific a- tion. The Floyd c omp actific ation of T n obtaine d fr om a Floyd function h an d a b ase p oint x 0 is a Lipschitz c omp actific ation if and only if ther e is a c onstant 0 < η < 1 so that fo r al l r ∈ N h ( r + 1) ≥ η h ( r ) . (1) R emark 1 Cond i tion ( 1) implies that h de cr e ases at most exp o nential ly f a st . It is inter esting to c omp ar e this with the usual c onf o rmal c omp actific ation of the hyp erb olic sp ac e , o b tain e d by mult iplying the metric by a factor that is exp onential in the di st anc e to a fixe d p oi nt . R emark 2 Cond i tion (1) implies that the c onsider e d Lips c h it z structur e de- p ends only up on h , not x 0 . We c an ther ef or e denote this c om p actific ation by T n ( h ) . Pr o of. W e first prov e that any Lipsc hitz compactification of T n is a Flo yd compactification. Let δ ′ b e any length metric in the giv en L ipschitz class, and fix a n y v ertex x 0 of T n . W e define h by h ( r ) = min δ ′ ( x, y ) where the minimum is tak en o v er all edges xy that are at com binatoria l dis tance r from x 0 . The n h is a Flo yd function because x 0 is at finite δ ′ distance f r o m the b oundary . Denote b y δ the Floyd metric o btained from x 0 and h , and let us pro v e that [ δ ] = [ δ ′ ]. It is sufficien t to pro v e that there is a constant C so that for all r , t w o edges that are at com binator ial dis tance r from x 0 ha v e their δ ′ lengths that differ b y a factor at most C . F or an y R ∈ N , let B ( R ) b e the closed ball of radius R and cen ter x 0 in T n (1). It con tains a finite n um b er of edges, so that there is a constan t C R that satisfies the ab o v e pro pert y for all r ≤ R . 4 Since the compactific ation is assumed to b e Lipsc hitz, for all p ∈ ∂ T n there are a ne ighborho o d V of p , a neighborho o d U of the iden tit y and a constan t k so that an y φ ∈ U is k -Lipsc hitz on V . Since ∂ T n is compact, w e can find a finite num b er of suc h quadruples ( p i , V i , U i , k i ) so that the V i co v er ∂ T n . Moreo v er w e can assume that the V i are the conne cted comp onen ts of T n \ B ( R ) for some radius R , and that U = ∩ U i = B B ( R ) (Id). Since f o r all i and r > R , U acts transitiv ely on the set of edges of V i that are at com binatorial distance r from x 0 , those edges ha v e their δ ′ -length that differ b y a factor at most C ′ = sup k i . Moreo ve r, there is an automorphism φ 0 that fixes x 0 and permutes cyc lically the V i . Since φ 0 is locally Lipsc hitz, there is a R ′ and a C ′′ so that for all r ≥ R ′ and all couple ( i 1 , i 2 ), there are edges of V i 1 and V i 2 that are at combinatorial distance r from x 0 and whose δ ′ lengths differ b y a factor at most C ′′ . The suprem um C of C R ′ and C ′′ C ′ 2 is the needed constant. Consider now the Flo yd compactification obtained from x 0 and h and denote b y δ the asso ciated Flo yd metric. By construction, an y a utomorphism φ of T n that fixes x 0 is a n isometry for δ , th us is lo cally bilipsc hitz for the corresp onding Lipsc hitz structure. Tw o translations are close to one another whe n they differ b y an elemen t close to ide ntit y . An elemen t close enough to iden tity m ust fix x 0 , th us is an isometry . Therefore, w e only need to pro ve that a giv en translation is Lipsc hitz to get that all automorphisms in a neigh b orho o d are equilipsc hitz. Chec king unitary translations is sufficien t since any translation is an iterate of one of tho s e. Let φ b e a unitary translation, and γ = . . . , y − 1 , y 0 , y 1 , . . . b e its trans- lated geo de sic, where w e assume that y 0 realizes the minimal com binatoria l distance d 0 b et w een v ertices of γ and x 0 . By local finiteness, φ is lo cally bilipsc hitz around an y p oin t of T n and w e need only c hec k the b oundary . Let us start with the attractiv e endpo in t p of γ . Assume that our Floy d compactification is Lipsc hitz. It implies that φ is lo cally bilipsc hitz around p , in pa rticular there is a r 0 > 0 and a k > 1 suc h that for an y r ≥ r 0 , k δ ( y r +1 , y r +2 ) ≥ δ ( y r , y r +1 ) h ( r + d 0 + 1) ≥ k − 1 h ( r + d 0 ) whic h giv es condition (1). Con v ersely , ass ume that (1) holds. F or an y v ertex x w e ha v e | d ( φ ( x ) , x 0 ) − d ( x , x 0 ) | ≤ 1 + 2 d 0 since the worst case is when x = x 0 or x is in a connected comp onen t of T n \ { x 0 } other than that of γ . Therefore, the length o f an edge and of its 5 image b y φ differ b y a facto r b ounded by η − (1+2 d 0 ) . The refore, φ is L ipschitz . Since φ − 1 is also a unita ry translation, φ is bilipsc hitz.  It would be in teresting to consider more general spaces, for example eu- clidean buildings or CA T(-1) buildings lik e the I pq described b y Bourdon in [1]. It is not obv ious ho w to define the Flo yd compactification: for e xam- ple, a mere scaling of the distance in each cell b y a factor depending on the com binatorial distance to a fixed cell w o uld create gluing pro blems ( a n edge shared by t w o faces ha ving tw o differen t length). This space s could therefore b e less flexible than trees . References [1] M. Bourdon. Immeubles h yp erb olique s, dimension conforme et rigidit ´ e de Mosto w. Ge om. F unct. A nal. , 7(2):24 5 –268, 1 997. [2] Dmitri Burago, Y uri Burago, and Sergei Iv anov . A c ourse in metric ge- ometry , v olume 33 of Gr aduate Studies in Mathematics . American Math- ematical So ciet y , Providen ce, RI, 2001 . [3] William J. Flo yd. G roup completions and limit sets of Kleinian gro ups. Invent. Math. , 57(3):205 – 218, 1980. [4] Beno ˆ ıt Klo ec kner. On differen tiable compactifications of the h yp erb o lic space. T r ansform. Gr o ups , 11(2):185–19 4 , 2 006. 6