GGeodesics and Visual boundary of Horospherical Products
Tom FerragutSeptember 11, 2020
Abstract
Horospherical products of two hyperbolic spaces unify the construction of metric spaces suchas the Diestel-Leader graphs, the SOL geometry or the treebolic spaces. Given two proper, geodesi-cally complete, Gromov hyperbolic, Busemann spaces H p and H q , we study the geometry of theirhorospherical product H ∶= H p & H q through a description of its geodesics.Specifically we introduce a large family of distances on H p & H q . We show that all these distancesproduce the same large scale geometry. This description allows us to depict the shape of geodesicsegments and geodesic lines. The understanding of the geodesics’ behaviour leads us to the char-acterization of the visual boundary of the horospherical products. Our results are based on metricestimates on paths avoiding horospheres in a Gromov hyperbolic space. Contents H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 H H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 A horospherical product is a metric space constructed from two Gromov hyperbolic spaces H p and H q .It is included in their cartesian product H p × H q and can be seen as a diagonal in it. The definition ofthis horospherical product makes use of the so-called Busemann functions. Let us assume that thereexists a unique geodesic ray k in H p starting at a given base point w p ∈ H p and leading in the directionof a given point on the boundary a p ∈ ∂H p . Then the Busemann function with respect to a p and w p a r X i v : . [ m a t h . M G ] S e p ssociates to a point x p ∈ H p the delay it has in a race towards a p against k . Given a base point anda point on the boundary on H p and H q we have two respective Busemann functions. We define theheight functions of these spaces H p and H q to be the opposite of the Busemann functions. Hencethe level-lines of the Busemann functions, which are called horospheres, are also the level-lines of theheight functions. Then the horospherical product H = H p & H q is built by gluing H p with an upsidedown copy of H q along their respective horospheres. More precisely with a given height h p on H p (Definition 2.1) and a given height h q on H q , the horospherical product H = H p & H q is defined asfollows. H ∶= {( x p , x q ) ∈ H p × H q / h p ( x p ) + h q ( x q ) = } . Since we are considering only couples of points with opposite heights in this set, we define the heighton the horospherical product H as the height on the first component H p . This notion of horosphericalproduct generalizes the description of the Diestel-Leader graphs, the SOL geometry and the Cayley2-complexes of Baumslag-Solitar groups BS ( , n ) . In the second chapter of [14, Woess], the last threeexemples are presented as horocyclic products of metric spaces. We choose the name horosphericalproduct instead of horo cyclic product since in higher dimension, level-lines according to a Busemannfunction are not horocycles but horospheres. As Woess suggested in [14, W], we explore here a gener-alization for horospherical products.The Diestel-Leader graphs are horospherical products of two regular trees. If the two trees’ degreeare equal, their horospherical product is the Cayley graph of a lamplighter group, see [13, Woess] forfurther details. A motivation to study this construction are the results from [4, Eskin, Fisher, Whyte],[5, E,F,W] and [6, E,F,W]. They state that the Diestel-Leader graphs constructed from two regular treeswith no common divisor in their degree are vertex-transitive graphs which are not quasi-isometric toany Cayley Graphs. The existence of such a graph was a long open problem.The SOL geometry, one of the eight Thurston geometries, is presented in [14, Woess] as the horospher-ical product of two hyperbolic planes. In [6, Eskin, Fisher, Whyte], they also prove rigidity resultson lattices of the SOL geometry. A third example is related to the family of Baumslag-Solitar groups BS ( , n ) , their Cayley 2-complex are described in [1, Bendikov, Saloff-Coste, Salvatori, Woess] as thehorospherical product of a hyperbolic plane and a homogeneous tree. Similar rigidity results as in [4,Eskin, Fisher, Whyte] are presented in [7, Farb, Mosher] and [8, F,M] for Baumslag-Solitar groups.For our generalization, we require that our components H p and H q are two proper geodesically com-plete Gromov hyperbolic Busemann spaces. A Busemann space is a metric space where the distancebetween any two geodesics is convex, and metric space X is geodesically complete if and only if ageodesic segment α ∶ I → X can be prolonged into a geodesic line ˆ α ∶ R → X . The Busemann hypoth-esis suits with the definition of horospherical product since we require that the opposite heights areexactly equal. Furthermore, adding the hypothesis that H p and H q are geodesically complete allows usto prove that the horospherical product H = H p & H q is connected.There are many possible choices for the distance on H p & H q . In this paper we work with a familyof length path metrics induced by distances on H p × H q (see precise definition 4.2). We require thatthe distance on H p & H q comes from a norm N on R that is greater than the normalized (cid:96) norm.Such norms are called admissible norms. A description of the distances on horospherical products isgiven by Corollary 4.17. This corollary shows that any distance we described earlier provides the samegeodesic shapes, up to an additive constant depending only on H p , H q and on the norm N . To do sowe introduce a notion of vertical geodesics, which are geodesics heuristically "normal" to horospheres(see precise definition 4.7). The shapes of geodesic segments are described in the following theorem. Theorem 1.1.
Let δ ≥ and let N be an admissible norm. Let H p and H q be two proper, geodesicallycomplete, δ -hyperbolic, Busemann spaces. Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q and let α be a geodesic segment of (H , d H ,N ) linking x to y . There exists a constant κ depending only on δ and N , and there exist two vertical geodesics V = ( V ,p , V ,q ) and V = ( V ,p , V ,q ) such that: V ( V ,p , V ,q ) H p H q h N κ ( V ) x y h ( y ) α Figure 1: Shape of geodesic segments when h ( x ) ≤ h ( y )− κ in H = H p & H q . The neighbourhoods’ shapeare distorted since when going upward, distances are contracted in the "direction" H p and expanded inthe "direction" H q .
1. If h ( x ) ≤ h ( y ) − κ then α is in the κ -neighbourhood of V ∪ ( V ,p , V ,q ) ∪ V
2. If h ( x ) ≥ h ( y ) + κ then α is in the κ -neighbourhood of V ∪ ( V ,p , V ,q ) ∪ V
3. If ∣ h ( x ) − h ( y )∣ ≤ κ then at least one of the conclusions of . or . holds.Specifically, V and V can be chosen such that x is close to V and y is close to V . This behaviour is illustrated on Figure 1 for h ( x ) ≤ h ( y ) − κ . This result is similar to the hyperboliccase, where a geodesic segment is in the constant neighbourhood of two vertical geodesics. The heuris-tic comprehension of Theorem 1.1 is, say in the case h ( x ) ≤ h ( y ) − κ , that a geodesic segment travelsfirst along a copy of the component H q (which is upside down) as a geodesic in it, and last travels alonga copy of the component H p as a geodesic in it.To prove Theorem 1.1 we need to control the lengths of the geodesics’ projections on H p and H q .This work is done in section 3. The relative distance is defined as the distance minus the differenceof height, it can be understood as the distance on horospheres. We first exhibit that in a hyperbolicspace the maximal height of a geodesic segment and the relative distance between the end points ofthat geodesic segment are tightly related. We also have a lower bound on the length of paths avoidinghoroballs as in Proposition . p400 of [2, Bridson, Haefliger]. Then we refine this last result into a con-trol on the length of paths which avoid horoballs and which reach a given point. Since the projectionson H p and H q of geodesics in H are such paths, Theorem 1.1 follows.This result leads us to show the existence of unextendabled geodesics, which are called dead-ends. Thiswas well known for lamplighter groups. This description of geodesic segments also allows us to provethat for any geodesic ray k of H = H p & H q , there exists a vertical geodesic ray at finite Hausdorffdistance. Therefore we classify all possible shapes for geodesic lines and then give a description of thevisual boundary of H . The notion of H p -type and H q -type geodesics at scale κ are described in Defini-tion 5.10 and illustrated on Figure 2. They are essentially geodesics of H in a constant κ -neighbourhoodof geodesics in a copy of H p or in a copy of H q in H . We show that the geodesic lines of H p & H q areeither H p -type, H q -type or both. Theorem 1.2.
Let δ ≥ and let N be an admissible norm. Let H p and H q be two proper, geodesicallycomplete, δ -hyperbolic, Busemann spaces. Let H = H p & H q be the horospherical product of H p and H q .Then there exists κ ≥ depending only on δ and N such that for all geodesic α ∶ R → H of (H , d H ,N ) atleast one of the two following statements holds. H p H qH p − type H q − typeV ertical geodesic Figure 2: Different type of geodesics in
H = H p & H q . α is a H p -type geodesic at scale κ of H α is a H q -type geodesic at scale κ of H If a geodesic is both H p -type and H q -type at scale κ , it is in the κ -neighbourhood of a verticalgeodesic of H p & H q .The notion of visual boundary of H & H is presented in the work of Troyanov in [12, Troyanov]through several definitions of horizons. We expand the definition and the description of the visualboundary in the general case of horospherical products as follows. Two geodesics are called asymptoticif they are at finite Hausdorff distance from each other. Let o ∈ H , the visual boundary of H is thendenoted by ∂ o (H) and stands for the set of families of asymptotic geodesic rays starting at o . We have: Theorem 1.3.
Let δ ≥ and let H p and H q be two proper, geodesically complete, δ -hyperbolic, Busemannspaces. We fix base points and directions on H p and H q as follows, ( w p , a p ) ∈ H p × ∂H p , ( w q , a q ) ∈ H q × ∂H q . Let H = H p & H q be the horospherical product with respect to ( w p , a p ) and ( w q , a q ) . Then thevisual boundary of H with respect to a given point o = ( o p , o q ) is: ∂ o H =(( ∂H p ∖ { a p }) × { a q }) ⋃ ({ a p } × ( ∂H q ∖ { a q }))=(( ∂H p × { a q }) ⋃ ({ a p } × ∂H q )) ∖ {( a p , a q )} This last result is similar to the Proposition 6.4 of [12, Troyanov]. However, unlike Troyanov inhis work, we are focusing on minimal geodesics and not on local ones. One can see that this visualboundary neither depends on the chosen admissible norm N nor the base point o .The figures of this paper depict lemmas and theorems when the two components H p and H q are hy-perbolic planes H , hence when H = H p & H q is the SOL geometry. In the dimensional figures, wepicture the vertical geodesics as getting closer when going upward since the distance contracts in thisdirection. In the dimensional case we picture the vertical geodesics as straight lines in order to matchwith their shapes in the SOL geometry.The paper is organised with first Section 2 which presents Gromov hyperbolic spaces, the notion ofvertical geodesics in them and the impact of the Busemann hypothesis on the vertical geodesics. ThenSection 3 provides us with an estimate on the length of paths avoiding horoballs in hyperbolic spaces,namely Lemma 3.8, which will be central in our control of the distances in H . In Section 4 we definethe horospherical products and give an estimate of their distance through Corollary 4.17. We hencediscuss the fact that an entire family of distances are close to each others in H . Last, in Section 5, weprove our three main results. Theorem 1.1 follows from the estimates of Corollary 4.17 on the lengthof geodesic segments. The description of geodesic lines of Theorem 1.2 follows from Theorem 1.1 andgives us the tools to prove Theorem 1.3. 4 Context
The goal of this section is to recall what is a Gromov hyperbolic space and what are vertical geodesicsin such a space. Let H be a proper geodesic metric space, and d be a distance on H . A geodesic line of H is the isometric image of an Euclidean line in H . A geodesic ray of a metric space H is the isometricimage of a half Euclidean line in H . A geodesic segment of a metric space H is the isometric image ofan Euclidean interval in H . By slight abuse, we will call geodesic, geodesic ray or geodesic segment, amap α ∶ I → H which parametrises our given geodesic by arclength.Let δ ≥ be a non-negative number. Let x , y and z be three points of H . The geodesic triangle [ x, y ] ∪ [ y, z ] ∪ [ z, x ] is called δ -slim if any of its sides is included in the δ -neighbourhood of the re-maining two. The metric space H is called δ -hyperbolic if every geodesic triangle is δ -slim. A metricspace H is called Gromov hyperbolic if there exists δ ≥ such that H is a δ -hyperbolic space.An important property of Gromov hyperbolic spaces is that they admit a nice compactification. In-deed the Gromov boundary allows that. We call two geodesic rays of H equivalent if their images areat finite Hausdorff distance. Let o ∈ H be a base point. We define ∂ o H the Gromov boundary of H asthe set of families of equivalent rays starting from o . In fact, the boundary ∂ o H does not depend on thebase point o , hence we will simply denote it by ∂H . For more details, see [10, Ghys, De La Harpe] orchap.III H p.399 of [2, Bridson, Haefliger]. In this section we fix δ ≥ , H a proper geodesic δ -hyperbolic space, w ∈ H a base point and a ∈ ∂H a point on the boundary of H . We recall the definition of Busemann function firstly presented in theintroduction. ∀ x ∈ H, β a ( x, w ) = sup { lim sup t →+∞ ( d ( x, k ( t )) − t ) ∣ k ∈ a, starting from w } . We want a notion of height on our hyperbolic spaces, a number tending to +∞ when following aselected direction. It is the reason why we define the height on H as the opposite of the Busemannfunction. Definition 2.1 (height with respect to a ∈ ∂H and w ∈ H ) . Let a ∈ ∂H be a direction in H and let w ∈ H be a base point. Then we define: ∀ x ∈ H, h ( a,w ) ( x ) = − β a ( x, w ) . Let us write Proposition 2 chap.8 p.136 of [10, Ghys, De La Harpe] with our notations.
Proposition 2.2 ([10], chap.8 p.136) . Let H be a hyperbolic proper geodesic metric space. Let a ∈ ∂H and w ∈ H , then:1. lim x → a h ( a,w ) ( x ) = +∞ lim x → b h ( a,w ) ( x ) = −∞ , ∀ b ∈ ∂H ∖ { a } ∀ x, y, z ∈ H, ∣ β a ( x, y ) + β a ( y, z ) − β a ( x, z )∣ ≤ δ . Furthermore, a geodesic ray is in a ∈ ∂H if and only if its height tends to +∞ . Corollary 2.3.
Let H be a hyperbolic proper geodesic metric space. Let a ∈ ∂H and w ∈ H , and let α ∶ [ , +∞[→ H be a geodesic ray. The two following properties are equivalent: . lim t →+∞ h ( a,w ) ( α ( t )) = +∞ α ([ , +∞[) ∈ a .Proof. As for any geodesic ray α ∶ [ , +∞[→ H there exists b ∈ ∂H such that α ([ , +∞[) ∈ b , thisproposition is a particular case of Proposition 2.2.We will picture our hyperbolic spaces in a way similar to the Log model for the hyperbolic plane.We send a ∈ ∂H upward to infinity and ∂H ∖ { a } downward to infinity. We then call vertical thegeodesic rays that are in the equivalence class a . Definition 2.4 (Vertical geodesics with repsect to a ∈ ∂H ) . A geodesic of H which satisfies one of theproperties of Corollary 2.3 is called a vertical geodesic relatively to the point a . An important property of the height is to be Lipschitz.
Proposition 2.5.
Let a ∈ ∂H and w ∈ H . The height function h a ∶= − β a (⋅ , w ) is Lipschitz: ∀ x, y ∈ H, ∣ h ( a,w ) ( x ) − h ( a,w ) ( y )∣ ≤ d ( x, y ) . Proof.
By using the triangular inequality we have for all x, y ∈ H : − h ( a,w ) ( x ) = β a ( x, w ) = sup { lim sup t →+∞ ( d ( x, k ( t )) − t ) ∣ k vertical rays starting at w }≤ d ( x, y ) + sup { lim sup t →+∞ ( d ( y, k ( t )) − t ) ∣ k vertical rays starting at w }≤ d ( x, y ) + β a ( y, w ) ≤ d ( x, y ) − h ( a,w ) ( y ) . The result follows by exchanging the roles of x and y .From now on, we fix a given a ∈ ∂H and a given w ∈ H . Therefore we simply denote the height by h instead of h ( a,w ) . Proposition 2.6.
Let α be a vertical geodesic of H . We have the following control on the height along α : ∀ t , t ∈ R , t − t − δ ≤ h ( α ( t )) − h ( α ( t )) ≤ t − t + δ. Proof.
Let t , t ∈ R , then: h ( α ( t )) − h ( α ( t )) = β ( α ( t ) , w ) − β ( α ( t ) , w )= β ( α ( t ) , α ( t )) − ( β ( α ( t ) , w ) − β ( α ( t ) , w ) + β ( α ( t ) , α ( t ))) . The third point of Proposition 2.2 applied to the last bracket gives: β ( α ( t ) , α ( t )) − δ ≤ h ( α ( t )) − h ( α ( t )) ≤ β ( α ( t ) , α ( t )) + δ. (1)Since t ↦ α ( t + t ) is a vertical geodesic starting at α ( t ) we have: β ( α ( t ) , α ( t )) = sup { lim sup t →+∞ ( d ( α ( t ) , k ( t )) − t )∣ k vertical rays starting at α ( t )}≥ lim sup t →+∞ ( d ( α ( t ) , α ( t + t )) − t )≥ lim sup t →+∞ (∣ t + t − t ∣ − t ) ≥ t − t , for t large enough.Using this last inequality in inequality (1) we get t − t − δ ≤ h ( α ( t )) − h ( α ( t )) . The resultfollows by exchanging the roles of t and t . 6sing Proposition 2.6 with t = and t = t , the next corollary holds. Corollary 2.7.
Let α be a vertical geodesic parametrised by arclength and such that h ( α ( )) = . Wehave: ∀ t ∈ R , ∣ h ( α ( t )) − t ∣ ≤ δ. In the sequel we want to apply the slim triangles property on ideal triangles, hence we need thefollowing result of [3, Coornaert, Delzant, Papadopoulos].
Property 2.8 (Proposition . page of [3]) . Let a, b and c be three points of X ∪ ∂X . Let α, β, γ bethree geodesics of X linking respectively b to c , c to a , and a to b . Then every point of α is at distance lessthan δ from the union β ∪ γ . We recall here some material from Chap.8 and Chap.12 of [11, Papadopoulos] about Busemann spaces.Busemann spaces are metric spaces where the distance between geodesics are convex functions. Tomake it more precise, a metric space X is called Busemann if it is geodesic, and if for every pair ofgeodesics parametrized by arclength γ ∶ [ a, b ] → X and γ ′ ∶ [ a ′ , b ′ ] → X , the following function isconvex: D γ,γ ′ ∶ [ a, b ] × [ a ′ , b ′ ] → X ( t, t ′ ) ↦ d X ( γ ( t ) , γ ′ ( t ′ )) . As an example, all
CAT ( ) spaces are Busemann spaces. However, being CAT ( ) is stronger thanbeing Busemann convex by Theorem . of [9, Foertsch, Lytchak, Schroeder]. As an example, strictlyconvex Banach spaces are all Busemann spaces, but they are CAT ( ) if and only if they are Hilbertspaces. Something interesting in Busemann spaces is that two points are always linked by a uniquegeodesic (see . . p. of Papadopoulos [11, Papadopoulos] for further details). The next propositiongives us informations on the height functions. Property 2.9 (Prop. . . in p.263 of Papadopoulos [11]) . Let δ ≥ be a non negative number. Let H be a proper δ -hyperbolic, Busemann space. For every geodesic α , the function t ↦ − h ( α ( t )) is convex. From now on, H will be a proper, Gromov hyperbolic, Busemann space. The Busemann hypothesisimplies that the height along geodesic behaves nicely. This means that we can drop the constant δ from Corollary 2.7. It is the main reason why we require our spaces to be Busemann spaces. Proposition 2.10.
Let H be a δ -hyperbolic and Busemann space and let V be a path of H . Then V is avertical geodesic if and only if ∃ c ∈ R such that ∀ t ∈ R , h ( V ( t )) = t + c .Proof. Let V be a vertical geodesic in H . By Property 2.9 we have that t ↦ − h ( V ( t )) is convex.Furthermore, from Corollary 2.7, we get ∣ h ( V ( t )) − t ∣ ≤ δ . Thereby the bounded convex function t ↦ t − h ( V ( t )) is constant. Then there exists a real number c such that ∀ t ∈ R , h ( V ( t )) = t + c .We now assume that there exists a real number c such that ∀ t ∈ R , h ( V ( t )) = t + c . Therefore, forall real numbers t and t we have d ( V ( t ) , V ( t )) ≥ ∆ h ( V ( t ) , V ( t )) = ∣ t − t ∣ . By definition V is a connected path, hence ∣ t − t ∣ ≥ d ( V ( t ) , V ( t )) which implies with the previous sentence that ∣ t − t ∣ = d ( V ( t ) , V ( t )) , then V is a geodesic. Furthermore lim t →+∞ h ( V ( t )) = +∞ , which implies bydefinition that V is a vertical geodesic.A metric space is called geodesically complete if all its geodesic segments can be prolonged intogeodesic lines. By adding the hypothesis of geodesically completeness on a hyperbolic Busemann space H we get that any point of H is included in a vertical geodesic line. Property 2.11.
Let H be a δ -hyperbolic Busemann geodesically complete space. Then for all x ∈ H thereexists a vertical geodesic V x ∶ R → H such that V x contains x roof. Let us consider in this proof w ∈ H and a ∈ ∂H , from which we constructed the height h of ourspace H . Then by definition we have h ( a,w ) = h . Proposition 12.2.4 of [11, Papadopoulos] ensures theexistence of a geodesic ray R x ∈ a starting at x . Furthermore as H is geodesically complete R x can beprolonged into a geodesic V x ∶ R → H such that V x ([ +∞[) ∈ a . Hence V x is a vertical geodesic fromDefinition 2.4.In this section we defined all the objects we will use in hyperbolic spaces. We will now focus onproving length estimates on specific paths. They will appear in Section 4 as the projection of geodesicsin a horospherical product. This section focuses on length estimates in Gromov hyperbolic Busemann spaces. The central resultis Lemma 3.8, which present a lower bound on the length of a path staying between two horospheres.Before moving to the technical results of this section, let us introduce some notations.
Notation 3.1.
Unless otherwise specified, H will be a Gromov hyperbolic Busemann geodesically completeproper space. Let γ ∶ I → H be a connected path. Let us denote the maximal height and the minimal heightof this path as follows: h + ( γ ) = sup t ∈ I { h ( γ ( t ))} ,h − ( γ ) = inf t ∈ I { h ( γ ( t ))} . Let x and y be two points of H , we denote the height difference between them by: ∆ h ( x, y ) = ∣ h ( x ) − h ( y )∣ . We define the relative distance between two points x and y of H as: d r ( x, y ) = d ( x, y ) − ∆ h ( x, y ) . Let us denote V x a vertical geodesic containing x , we will consider it to be parametrised by arclength.Thanks to Proposition 2.10 we choose a parametrisation by arclength such that ∀ t ∈ R , h ( V x ( t )) = t + . The relative distance between two points quantifies how far a point is from the nearest verticalgeodesic containing the other point. Next lemma tells us that in order to connect two points a geodesicneeds to go sufficiently high. This height is controlled by the relative distance between those twopoints.
Lemma 3.2.
Let H be a δ -hyperbolic and Busemann metric space, let x and y be two elements of H suchthat h ( x ) ≤ h ( y ) , and let α be a geodesic linking x to y . Let us denote z = α ( ∆ h ( x, y ) + d r ( x, y )) , x ∶= V x ( h ( y ) + d r ( x, y )) the point of V x at height h ( y ) + d r ( x, y ) and y ∶= V y ( h ( y ) + d r ( x, y )) the point of V y at the same height h ( y ) + d r ( x, y ) . Then we have:1. h + ( α ) ≥ h ( y ) + d r ( x, y ) − δ d ( z, x ) ≤ δ d ( z, y ) ≤ δ d ( x , y ) ≤ δ . yV x V y H α α ( t ) ≤ δ ≤ δ h ( x ) h ( y ) z x x y y h x y h ( x ) + t h ( y ) + d r ( x, y ) ∆ h ( x, y ) d r ( x, y ) Figure 3: Proof of Lemma 3.2
Proof.
The lemma and its proof are illustrated on Figure 3. Following Property 2.8, the triple of geodesics α , V x and V y is a δ -slim triangle. Since the sets { t ∈ [ , d ( x, y )]∣ d ( α ( t ) , V x ) ≤ δ } and { t ∈[ , d ( x, y )]∣ d ( α ( t ) , V y ) ≤ δ } are closed sets covering [ , d ( x, y )] , their intersection is non empty.Hence there exists t ∈ [ , d ( x, y )] , x ∈ V x and y ∈ V y such that d ( α ( t ) , x ) ≤ δ and d ( α ( t ) , y ) ≤ δ . Let us first prove that t is close to ∆ h ( x, y ) + d r ( x, y ) . By the triangular inequality we havethat: ∣ t − d ( x, x )∣ = ∣ d ( x, α ( t )) − d ( x, x )∣ ≤ d ( x , α ( t )) ≤ δ. Let us denote x ∶= V x ( h ( x ) + t ) the point of V x at height h ( x ) + t , and y = V y ( h ( y ) + d ( x, y ) − t ) the point of V y at height h ( y ) + d ( x, y ) − t . Then by the triangular inequality: d ( α ( t ) , x ) ≤ d ( α ( t ) , x ) + d ( x , x ) = d ( α ( t ) , x ) + ∣ d ( x, x ) − d ( x, x )∣≤ d ( α ( t ) , x ) + ∣ d ( x, x ) − t ∣ ≤ δ. (2)In the last inequality we used that d ( x, x ) = t , which holds by the definition of x . We show in thesame way that d ( α ( t ) , y ) ≤ δ . By the triangular inequality we have d ( x , y ) ≤ δ . As the heightfunction is Lipschitz we have ∆ h ( x , y ) ≤ d ( x , y ) ≤ δ , which provides us with: ∣ d r ( x, y ) + ∆ h ( x, y ) − t ∣ = ∣ d r ( x, y ) + ∆ h ( x, y ) + h ( y ) − h ( x ) − t ∣= ∣ h ( y ) + d ( x, y ) − t − ( h ( x ) + t )∣ =
12 ∆ h ( x , y ) ≤ δ ≤ δ. (3)In particular it gives us that d ( z, α ( t )) ≤ δ . We are now ready to prove the first point using inequal-ities (2) and (3): h + ( α ) ≥ h ( α ( t )) ≥ h ( x ) − ∆ h ( α ( t ) , x ) ≥ h ( x ) + t − δ ≥ h ( x ) + d r ( x, y ) + ∆ h ( x, y ) − δ ≥ h ( y ) + d r ( x, y ) − δ, as we have h ( x ) ≤ h ( y ) . The second point of our lemma is proved by the sequel: d ( z, x ) ≤ d ( z, α ( t )) + d ( α ( t ) , x ) ≤ δ + d ( α ( t ) , x ) + d ( x , x )≤ δ + ∣ t + h ( x ) − ( d r ( x, y ) + h ( y ))∣ = δ + ∣ t − ( ∆ h ( x, y ) + d r ( x, y ))∣ ≤ δ. The proof of . is similar, and . is obtained from . and . by the triangular inequality.9he next lemma shows that in the case where h ( x ) ≤ h ( y ) a geodesic linking x to y is almostvertical until it reaches the height h ( y ) . Lemma 3.3.
Let H be a δ -hyperbolic and Busemann space. Let x and y be two points of H such that h ( x ) ≤ h ( y ) . We define x ′ ∶= V x ( h ( y )) to be the point of the vertical geodesic V x at the same height as y .Then: ∣ d r ( x, y ) − d ( x ′ , y )∣ ≤ δ. (4) Proof.
Since H is δ -hyperbolic, the geodesic triangle [ x, y ] ∪ [ y, x ′ ] ∪ [ x ′ , x ] is δ -slim. Then thereexists p ∈ [ x, x ′ ] , p ∈ [ x ′ , y ] and m ∈ [ x, y ] such that d ( p , m ) ≤ δ and d ( p , m ) ≤ δ . Hence, h − ([ x ′ , y ]) − δ ≤ h ( m ) ≤ h + ([ x, x ′ ]) + δ . Let R x ′ and R y be two vertical geodesic rays respectivelycontained in V x and V y and respectively starting at x ′ and y . Then Property 2.8 used on the idealtriangle R x ∪ R y ∪ [ x ′ , y ] implies that h − ([ x ′ , y ]) ≥ h ( y ) − δ , therefore we have h + ([ x, x ′ ]) = h ( y ) .Then h ( y ) − δ ≤ h ( m ) ≤ h ( y ) + δ holds. It follows that m and x ′ are close to each other: d ( m, x ′ ) ≤ d ( m, p ) + d ( p , x ′ ) ≤ δ + ∆ h ( p , x ′ ) ≤ δ + ∆ h ( p , m ) + ∆ h ( m, y ) + ∆ h ( y, x ′ )≤ δ + d ( p , m ) + δ + ≤ δ. (5)Then we give an estimate on the distance between x and m : ∣ d ( x, m ) − ∆ h ( x, y )∣ = ∣ d ( x, m ) − d ( x, x ′ )∣ ≤ d ( m, x ′ ) ≤ δ. (6)However d r ( x, y ) = d ( x, y ) − ∆ h ( x, y ) and d ( x, y ) = d ( x, m ) + d ( m, y ) , therefore: d r ( x, y ) = d ( x, m ) + d ( m, y ) − ∆ h ( x, y ) . (7)Combining inequalities (6) and (7) we have ∣ d r ( x, y ) − d ( m, y )∣ ≤ δ . Then: ∣ d r ( x, y ) − d ( x ′ , y )∣ ≤ δ + d ( x ′ , m ) ≤ δ. The lemmas of this last section allow us to prove the estimate lemmas of the next one.
Consider a path γ and a geodesic α that links the two same points of a proper, Gromov hyperbolic,Busemann space. We prove in this section that if the height of γ does not reach the maximal height ofthe geodesic α , then γ is much longer than α . Furthermore, its length increases exponentially on thedifference of maximal height between γ and α . To do so we need Proposition . p400 of [2, Bridson,Haefliger]. We denote by l ( c ) the length of a path c . Proposition 3.4 ([2]) . Let X be a δ -hyperbolic geodesic space. Let c be a continuous path in X. If [ p, q ] is a geodesic segment connecting the endpoints of c , then for every x ∈ [ p, q ] : d ( x, im ( c )) ≤ δ ∣ log l ( c )∣ + . This result implies that a path of H between x and y which avoids the ball centred in the middle ofa geodesic [ x, y ] has length greater than an exponential in the distance d ( x, y ) . From now on we willadd as convention that δ ≥ . For all δ ≤ δ a δ -slim triangle is also δ -slim, hence all δ -hyperbolicspaces are δ -hyperbolic spaces. That is why we can assume that all Gromov hyperbolic spaces are δ -hyperbolic with δ ≥ . It allows us to consider δ as a well defined term, we hence avoid different casesin the proof of the following lemma. We also use this assumption to simplify constants appearing inthe proof. The next result is a similar control on the length of path as Proposition 3.4, but we considerthat the path is avoiding a horosphere instead of avoiding a ball in H .10 yx y V x V y γ B ( y, ∆ h ( y , y )) B ( x, ∆ h ( x , x )) γ ( t x ) γ ( t y ) h ( x ) H x y V γ ( t x ) V γ ( t y ) Figure 4: Proof of Lemma 3.5
Lemma 3.5.
Let δ ≥ and H be a proper, δ -hyperbolic, Busemann space. Let x and y ∈ H and let V x (respt. V y ) be a vertical geodesic containing x (respt. y ). Let us consider t ≥ max ( h ( x ) , h ( y )) and let usdenote x ∶= V x ( t ) and y ∶= V y ( t ) . Assume that d ( x , y ) > δ .Then for all connected path γ ∶ [ , T ] → H such that γ ( ) = x , γ ( T ) = y and h + ( γ ) ≤ h ( x ) we have: l ( γ ) ≥ ∆ h ( x, x ) + ∆ h ( y, y ) + − δ d ( x ,y ) − δ. (8)For trees when δ = this Lemma still makes sense. Indeed, if δ tends to then the length of thepath described in this Lemma tends to infinity, which is consistent with the fact that such a path doesnot exist in trees. The proof would use the fact that in Proposition 3.4 we have d ( x, im ( c )) = when δ = since -hyperbolic spaces are real trees. Proof.
One can follow the idea of the proof on Figure 4. We will consider γ to be parametrised byarclength. Let B ( x, ∆ h ( x , x )) ⊂ H be the ball of radius h ( x ) − h ( x ) centred on x , and let m ∈ B ( x, ∆ h ( x , x )) be a point in this ball. Then: d r ( m, x ) = d ( m, x ) − ∆ h ( m, x ) ≤ ∆ h ( x, x ) − ∆ h ( m, x ) ≤ ∆ h ( x , m ) . Let us first assume that h ( m ) ≥ h ( x ) , then: h ( m ) + d r ( m, x ) ≤ h ( m ) + ∆ h ( x , m ) ≤ h ( m ) + h ( x ) − h ( m ) = h ( x ) + h ( m ) ≤ h ( x ) . (9)By Lemma 3.2 we have: d ( V x ( h ( m ) + d r ( m, x ) ) , V m ( h ( m ) + d r ( m, x ) )) ≤ δ. We now assume that h ( m ) ≤ h ( x ) , then: h ( x ) + d r ( x, m ) ≤ h ( x ) + d ( x, m ) ≤ h ( x ) + ∆ h ( x, x ) ≤ h ( x ) . d ( V x ( h ( x ) + d r ( m, x ) ) , V m ( h ( x ) + d r ( m, x ) )) ≤ δ. Since H is a Busemann space, the function t → d ( V x ( t ) , V m ( t )) is convex. Furthermore t → d ( V x ( t ) , V m ( t )) is bounded on [ +∞[ as H is Gromov hyperbolic, hence t → d ( V x ( t ) , V m ( t )) is a non increasing func-tion. Therefore both cases h ( m ) ≤ h ( x ) and h ( x ) ≤ h ( m ) give us that: d ( x , V m ( h ( x )) ) = d ( V x ( h ( x )) , V m ( h ( x ))) ≤ δ. (10)In other words, all points of B ( x, ∆ h ( x , x )) belong to a vertical geodesic passing nearby x . By thesame reasoning we have ∀ n ∈ B ( y, ∆ h ( y , y )) : d ( y , V n ( h ( y )) ) ≤ δ. (11)Then by the triangular inequality: d ( V m ( h ( x )) , V n ( h ( y ))) ≥ − d ( x , V m ( h ( x )) ) + d ( x , y ) − d ( y , V n ( h ( y )) )≥ δ − δ − δ ≥ δ. (12)Specifically d ( V m ( h ( x )) , V n ( h ( y ))) = d ( V m ( h ( x )) , V n ( h ( x ))) > which implies that m ≠ n .Then B ( x, ∆ h ( x , x )) ∩ B ( y, ∆ h ( y , y )) = ∅ . By continuity of γ we deduce the existence of the twofollowing times t x ≤ t y such that: t x = inf { t ∈ [ , T ] ∣ d ( γ ( t ) , x ) = ∆ h ( x, x )} ,t y = sup { t ∈ [ , T ] ∣ d ( γ ( t ) , y ) = ∆ h ( y, y )} . In order to have a lower bound on the length of γ we will need to split this path into three parts: γ = γ ∣[ ,t x ] ∪ γ ∣[ t x ,t y ] ∪ γ ∣[ t y ,T ] . As γ is parametrised by arclength and d ( γ ( ) , γ ( t x )) = ∆ h ( x, x ) we have that: l ( γ ∣[ ,t x ] ) ≥ ∆ h ( x, x ) . (13)For similar reasons we also have: l ( γ ∣[ t y ,T ] ) ≥ ∆ h ( y, y ) . (14)We will now focus on proving a lower bound for the length of γ ∣[ t x ,t y ] .We want to construct a path γ ′ joining x = V γ ( t x ) ( h ( x )) to y = V γ ( t y ) ( h ( x )) , that stays below h ( x ) and such that γ ∣[ t x ,t y ] is contained in γ ′ . Let x ∶= V γ ( t x ) ( h ( x )) and y ∶= V γ ( t y ) ( h ( x )) . Weconstruct γ ′ by gluing paths together: γ ′ = ⎧⎪⎪⎪⎨⎪⎪⎪⎩ V γ ( t x ) from x to γ ( t x ) γ from γ ( t x ) to γ ( t y ) V γ ( t y ) from γ ( t y ) to y Applying inequalities (10) and (11) used on γ ( t x ) and γ ( t y ) we get: d ( x , x ) ≤ δ, (15) d ( y , y ) ≤ δ. (16)12n order to apply Proposition 3.4 to γ ′ we need to check that there exists a point A of the geodesicsegment [ x , y ] such that h ( A ) ≥ h ( x ) . Applying Lemma 3.2 to [ x , y ] and since h ( x ) = h ( y ) weget: h + ([ x , y ]) ≥ d r ( x , y ) + h ( x ) − δ = d ( x , y ) + h ( x ) − δ. Thanks to the triangular inequality and inequalities (15) and (16): h + ([ x , y ]) ≥ d ( y , x ) − d ( x , x ) − d ( y , y ) + h ( x ) − δ ≥ d ( x , y ) + h ( x ) − δ. Since by hypothesis d ( x , y ) > δ , there exists a point A of [ x , y ] exactly at the height: h ( A ) = d ( x , y ) + h ( x ) − δ. We can then apply Proposition 3.4 to get: δ ∣ log ( l ( γ ′ ))∣ + ≥ d ( A, γ ′ ) ≥ ∆ h ( A, x ) ≥ d ( x , y ) + h ( x ) − δ − h ( x )≥ d ( x , y ) − δ. Since δ ≥ , last inequality implies that l ( γ ′ ) ≥ − δ d ( x ,y ) . Now we use this inequality to have alower bound on the length of γ ∣[ t x ,t y ] : l ( γ ∣[ t x ,t y ] ) ≥ l ( γ ′ ) − ∆ h ( γ ( t x ) , x ) − ∆ h ( γ ( t y ) , y )≥ − δ d ( x ,y ) − ∆ h ( γ ( t x ) , x ) − ∆ h ( γ ( t y ) , y ) . (17)We claim that l ( γ ∣[ t x ,t y ] ) ≥ ∆ h ( γ ( t x ) , x ) + ∆ h ( γ ( t y ) , y ) − δ , hence: l ( γ ∣[ t x ,t y ] ) ≥ − δ d ( x ,y ) − δ, (18)which ends the proof by combining inequality (18) with inequalities (13) and (14).Proof of the claim. Inequality (12) with m = γ ( t x ) and n = γ ( t y ) gives d ( x , y ) ≥ δ . We want toprove that h + ([ γ ( t x ) , γ ( t y )]) ≥ h ( x ) − δ . First, by Lemma 2.8 we have that [ γ ( t x ) , γ ( t y )] ∪ V γ ( t x ) ∪ V γ ( t y ) is a δ -slim triangle. Then there exist three times t , t and t such that d ( V γ ( t x ) ( t ) , γ ( t )) ≤ δ and such that d ( V γ ( t y ) ( t ) , γ ( t )) ≤ δ . Then: ∣ t − t ∣ = ∆ h ( V γ ( t x ) ( t ) , V γ ( t y ) ( t )) ≤ d ( V γ ( t x ) ( t ) , V γ ( t y ) ( t ))≤ d ( V γ ( t x ) ( t ) , γ ( t )) + d ( γ ( t ) , V γ ( t y ) ( t )) ≤ δ. (19)We will show by contradiction that either t = h ( V γ ( t x ) ( t )) ≥ h ( x ) or t = h ( V γ ( t y ) ( t )) ≥ h ( x ) .Assume that t < h ( x ) and t < h ( x ) . Then by the triangular inequality: d ( V γ ( t x ) ( t ) , V γ ( t y ) ( t )) ≥ d ( V γ ( t y ) ( t ) , V γ ( t x ) ( t )) − d ( V γ ( t x ) ( t ) , V γ ( t x ) ( t ))≥ d ( V γ ( t y ) ( t ) , V γ ( t x ) ( t )) − δ , since ∣ t − t ∣ ≤ δ by equation (19).As H is a Busemann space, the function t ↦ d ( V γ ( t x ) ( t ) , V γ ( t y ) ( t )) is non increasing. Furthermore, h ( x ) ≥ t hence: δ ≥ d ( V γ ( t x ) ( t ) , V γ ( t x ) ( t )) ≥ d ( V γ ( t x ) ( t ) , V γ ( t y ) ( t )) − δ ≥ d ( V γ ( t x ) ( h ( x )) , V γ ( t y ) ( h ( x ))) − δ ≥ d ( x , y ) − δ ≥ d ( x , y ) − d ( x , x ) − d ( y , y ) − δ ≥ d ( x , y ) − δ , by inequalities ( ) and ( ) , ≥ δ , since d ( x , y ) ≥ δ by hypothesis , D ∆∆ t t V ( D ) α V ( t ) V ( t ) V ( D − ∆ ) ∆ V ( D − t ) V ( D − ∆ − t ) Figure 5: Proof of Lemma 3.6which is impossible. Therefore t ≥ h ( x ) or t ≥ h ( x ) . We assume without loss of generality that t ≥ h ( x ) , then: ∆ h ( γ ( t ) , V γ ( t x ) ( t )) ≤ d ( γ ( t ) , V γ ( t x ) ( t )) ≤ δ, which implies: h + ([ γ ( t x ) , γ ( t y )]) ≥ h ( γ ( t )) ≥ h ( V γ ( t x ) ( t )) − ∆ h ( γ ( t ) , V γ ( t x ) ( t )) ≥ h ( x ) − δ, and gives us: l ( γ ∣[ t x ,t y ] ) ≥ h + ([ γ ( t x ) , γ ( t y )]) − h ( γ ( t x )) + h + ([ γ ( t x ) , γ ( t y )]) − h ( γ ( t y ))≥ h ( x ) − δ − h ( γ ( t x )) + h ( x ) − δ − h ( γ ( t y ))≥ ∆ h ( γ ( t x ) , x ) + ∆ h ( γ ( t y ) , y ) − δ. (20)Next lemma shows that we are able to control the relative distance of a couple of points travellingalong two vertical geodesics. Lemma 3.6 (Backwards control) . Let δ ≥ and H be a proper, δ -hyperbolic, Busemann space. Let V and V be two vertical geodesics of H . Then for all couple of times ( t , t ) and for all t ∈ [ , d r ( V ( t ) , V ( t ))] : ∣ d r ( V ( t + d r ( V ( t ) , V ( t )) − t ) , V ( t + d r ( V ( t ) , V ( t )) − t )) − t ∣ ≤ δ. Proof.
To simplify the computations, we use the following notation, D ∶= t + d r ( V ( t ) , V ( t )) and ∆ = ∣ t − t ∣ . The term ∆ is the difference of height between V ( t ) and V ( t ) since verticalgeodesics are parametrised by their height. Then we have to prove that ∀ t ∈ [ , d r ( V ( t ) , V ( t ))] , ∣ d r ( V ( D − ∆ − t ) , V ( D − t ))− t ∣ ≤ δ . We can assume without loss of generality that t ≤ t . Lemma3.2 applied with x = V ( t ) and with y = V ( t ) gives us d ( V ( D ) , V ( D )) ≤ δ . Furthermore, therelative distance is smaller than the distance, hence d r ( V ( D ) , V ( D )) ≤ δ . Now if we move the14 x y [ x, y ] α h ( y ) h ( x ) V x h ( y ) + d r ( x, y ) − ∆ H ∆ H V y Figure 6: Proof of Lemma 3.7two points backward from V ( D − ∆ ) and V ( D ) along V and V , we have for t ∈ [ , D ] : d r ( V ( D − ∆ − t ) , V ( D − t )) = d ( V ( D − ∆ − t ) , V ( D − t )) − ∆ (21) ≤ d ( V ( D − ∆ − t ) , V ( D − ∆ )) + d ( V ( D − ∆ ) , V ( D ))+ d ( V ( D ) , V ( D − t )) − ∆ , furthermore V and V are geodesics, then: ≤ t + d ( V ( D − ∆ ) , V ( D )) + d ( V ( D ) , V ( D )) + t − ∆ ≤ t + ∆ + δ + t − ∆ ≤ t + δ. (22)Let us consider a geodesic α between V ( t ) and V ( t ) . Since H is a Busemann space, and thanks toLemma 3.2 we have d ( V ( D − ∆ − t ) , α ( D − ∆ − t − t )) ≤ δ and d ( V ( D − t ) , α ( D − t + t )) ≤ δ . Then the second part of our inequality follows: d r ( V ( D − ∆ − t ) , V ( D − t )) = d ( V ( D − ∆ − t ) , V ( D − t )) − ∆ ≥ d ( α ( D − ∆ − t − t ) , α ( D − t + t ))− d ( V ( D − ∆ − t ) , α ( D − ∆ − t − t ))− d ( V ( D − t ) , α ( D − t + t )) − ∆ ≥ d ( α ( D − ∆ − t − t ) , α ( D − t + t )) − δ − ∆ ≥ t + ∆ − δ − ∆ ≥ t − δ. (23)The next lemma is a slight generalisation of Lemma 3.5. The difference is we control the length ofa path with its maximal height instead of the distance between the projection of its extremities on ahorosphere. Lemma 3.7.
Let δ ≥ and H be a proper, δ -hyperbolic, Busemann space. Let x, y ∈ H such that h ( x ) ≤ h ( y ) . Let α be a path connecting x to y with h + ( α ) ≤ h ( y ) + d r ( x, y ) − ∆ H and where ∆ H is a positivenumber such that ∆ H > δ . Then: l ( α ) ≥ d ( x, y ) + − δ ∆ H − H − δ. roof. This proof is illustrated on Figure 6. Since h + ( α ) ≥ h ( y ) we have that d r ( x, y ) ≥ ∆ H . Apply-ing Lemma 3.6 with V = V x , V = V y , t = h ( x ) , t = h ( y ) and t = ∆ H we have: ∣ d r ( V x ( h ( x ) + d r ( x, y ) − ∆ H ) , V y ( h ( y ) + d r ( x, y ) − ∆ H )) − H ∣ ≤ δ. Then we have: d r ( V x ( h ( x ) + d r ( x, y ) − ∆ H ) , V y ( h ( y ) + d r ( x, y ) − ∆ H )) ≥ H − δ. Furthermore, Lemma 3.3 applied on V x ( h ( x ) + d r ( x, y ) − ∆ H ) and V y ( h ( y ) + d r ( x, y ) − ∆ H ) gives (notice that the only difference between the two sides of the following inequality is the height inthe vertical geodesic V x ): d r ( V x ( h ( x ) + d r ( x, y ) − ∆ H ) , V y ( h ( y ) + d r ( x, y ) − ∆ H ))≤ d ( V x ( h ( y ) + d r ( x, y ) − ∆ H ) , V y ( h ( y ) + d r ( x, y ) − ∆ H )) + δ. Then: d ( V x ( h ( y ) + d r ( x, y ) − ∆ H ) , V y ( h ( y ) + d r ( x, y ) − ∆ H )) ≥ H − δ > δ. (24)Let us denote t = h ( y ) + d r ( x, y ) − ∆ H . Thanks to inequality (24) the hypothesis of Lemma 3.5 holdswith x = V x ( h ( y ) + d r ( x, y ) − ∆ H ) and y = V y ( h ( y ) + d r ( x, y ) − ∆ H ) . Applying this lemmaon α provides: l ( α ) ≥ ∆ h ( x, x ) + ∆ h ( y, y ) + − δ d ( x ,y ) − δ ≥ h ( y ) + d r ( x, y ) − ∆ H − h ( x ) + h ( y ) + d r ( x, y ) − ∆ H − h ( y ) + − δ d ( x ,y ) − δ ≥ ∆ h ( y, x ) + d r ( y, x ) − H + − δ d ( x ,y ) − δ ≥ d ( x, y ) − H + − δ ( H − δ ) − δ , by equation (24). ≥ d ( x, y ) + − δ ∆ H − H − δ. This previous lemma tells us that a path needs to reach a sufficient height for its length not toincrease to much. We give now a generalization of Lemma 3.7, where the path reaches a given lowheight before going to its end point. This lemma will be the central result for the understanding of thegeodesic shapes in a horospherical product.
Lemma 3.8.
Let δ ≥ and H be a proper, δ -hyperbolic, Busemann space. Let x, y, m ∈ H such that h ( m ) ≤ h ( x ) ≤ h ( y ) and let α ∶ [ , T ] → H be a path connecting x to y such that h − ( α ) = h ( m ) . Withthe notation ∆ H = h ( y ) + d r ( x, y ) − h + ( α ) we have: l ( α ) ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − − max ( , H ) − δ. Proof.
This proof is illustrated on Figure 7. We first assume that ∆ H > δ , we postpone the othercases to the end of this proof. Let V x and V m be vertical geodesics respectively containing x and m .We call x = V x ( h ( y )) and m = V m ( h ( y )) the points of V x and V m at height h ( y ) . First, Lemma 3.3provides ∣ d ( x , y ) − d r ( x, y )∣ ≤ δ . Then we consider a geodesic triangle between the three points x , m and y . Lemma 3.2 tells us that h + ([ x , y ]) ≥ h ( y ) + d r ( x , y ) − δ ≥ h ( y ) + d r ( x, y ) − δ .16 x ym [ x, y ] α m x m h ( y ) h ( x ) V x V m ∆ h ( x, y ) ∆ h ( x, m ) ∆ H ∆ H h + ( α ) h ( y ) + d r ( x, y ) h ( m ) Figure 7: Proof of Lemma 3.8Since [ x , y ] is included in the δ -neighbourhood of the two other sides of the geodesic triangle, one ofthe two following inequalities holds: ) h + ([ x , m ]) ≥ h ( y ) + d r ( x, y ) − δ ) h + ([ m , y ]) ≥ h ( y ) + d r ( x, y ) − δ. We first assume ) that h + ([ x , m ]) ≥ h ( y ) + d r ( x, y ) − δ , hence: d ( x , m ) ≥ d r ( x, y ) − δ. (25)Let us denote m = V m ( h ( x )) the point of V m at height h ( x ) . By considering the δ -slim quadrilat-eral between the points x, x , m , m we have that [ x , m ] is in the δ - neighbourhood of [ x , x ] ∪[ x, m ] ∪ [ m , m ] . Furthermore d r ( x, y ) ≥ ( h + ( α ) − h ( y )) + H ≥ H ≥ δ by assumption,then h + ([ x , m ]) ≥ h ( y ) + d r ( x, y ) − δ ≥ h ( y ) + δ . Since h + ([ x , x ]) = h + ([ m , m ]) = h ( y ) we have that h + ([ x, m ]) ≥ h + ([ x , m ]) − δ ≥ h ( y ) + δ . Moreover: d r ( x, m ) = d ( x, m ) ≥ h + ([ x, m ]) − h ( x ) ≥ h ( y ) − h ( x ) + δ ≥ ∆ h ( x, y ) + δ, which allows us to use Lemma 3.6 on V x and V m with t = d r ( x, m )− ∆ h ( x, y ) ≥ and t = t = h ( x ) .It gives: ∣ d r ( V x ( h ( x ) + ∆ h ( x, y )) , V m ( h ( x ) + ∆ h ( x, y ))) − d r ( x, m ) + h ( x, y )∣ ≤ δ, which implies in particular: d r ( V x ( h ( y )) , V m ( h ( y ))) + h ( x, y ) − δ ≤ d r ( x, m ) . (26)Combining inequalities (25) and (26) we have d ( x, m ) = d r ( x, m ) ≥ d r ( x, y ) + h ( x, y ) − δ .Lemma 3.3 used on x and m then gives: d r ( x, m ) ≥ d ( x, m ) − δ ≥ d r ( x, y ) + h ( x, y ) − δ. (27)17et us denote α the part of α linking x to m and α the part of α linking m to y . We have: h + ( α ) ≤ h + ( α ) ≤ h ( y ) + d r ( x, y ) − ∆ H ≤ h ( x ) + ∆ h ( x, y ) + d r ( x, y ) − ∆ H ≤ h ( x ) + ( h ( x, y ) + d r ( x, y )) − ∆ H ≤ h ( x ) + ( d r ( x, m ) + δ ) − ∆ H , by inequality ( ) . ≤ h ( x ) + d r ( x, m ) + δ − ∆ H ≤ h ( x ) + d r ( x, m ) − ∆ H ′ , with ∆ H ′ = ∆ H − δ . By assumption ∆ H > δ , hence ∆ H ′ > δ which allows us to applyLemma 3.7 on α . It follows: l ( α ) ≥ d ( x, m ) + − δ ∆ H ′ − H ′ − δ ≥ ∆ h ( x, m ) + d r ( x, m ) + − δ ∆ H − H − δ , since ∆ H ′ = ∆ H − δ. ≥ ∆ h ( x, m ) + d r ( x, y ) − δ + − δ ∆ H − H − δ , by inequality (27) ≥ ∆ h ( x, m ) + d r ( x, y ) + − δ ∆ H − H − δ. We use in the following inequalities that l ( α ) ≥ d ( m, y ) ≥ ∆ h ( m, y ) , we have: l ( α ) ≥ l ( α ) + l ( α ) ≥ ∆ h ( x, m ) + d r ( x, y ) + − δ ∆ H − H − δ + ∆ h ( m, y )≥ h ( x, m ) + ∆ h ( x, y ) + d r ( x, y ) + − δ ∆ H − H − δ ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − H − δ ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − − H − δ, ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − − max ( , H ) − δ , since ∆ H > δ ≥ , which ends the proof for case 1).Now assume that ) holds, which is h + ([ m , y ]) ≥ h ( y ) + d r ( x, y ) − δ . It implies d ( m , y ) ≥ d r ( x, y ) − δ , then: h + ( α ) ≤ h + ( α ) ≤ h ( y ) + d r ( x, y ) − ∆ H ≤ h ( y ) + d r ( m , y ) + δ − ∆ H ≤ h ( y ) + d r ( m , y ) − ∆ H ′′ , with ∆ H ′′ = ∆ H − δ . Lemma 3.3 provides us with: d r ( m, y ) ≥ d ( m , y ) − δ ≥ d r ( x, y ) − δ. (28)Since ∆ H > δ , we have ∆ H ′′ > δ which allows us to apply Lemma 3.7 on α . It follows that: l ( α ) ≥ d ( y, m ) + − δ ∆ H ′′ − H ′′ − δ ≥ ∆ h ( y, m ) + d r ( y, m ) + − δ ∆ H − H − δ , since ∆ H ′′ = ∆ H − δ. ≥ ∆ h ( y, m ) + d r ( x, y ) + − δ ∆ H − H − δ , by inequality (26) . Hence: l ( α ) ≥ l ( α ) + l ( α ) ≥ ∆ h ( x, m ) + ∆ h ( y, m ) + d r ( x, y ) + − δ ∆ H − H − δ ≥ h ( x, m ) + ∆ h ( y, x ) + d r ( x, y ) + − δ ∆ H − H − δ ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − H − δ ≥ h ( x, m ) + d ( x, y ) + − δ ∆ H − − max ( , H ) − δ. ∆ H ≤ δ , where ∆ H = h ( y ) + d r ( x, y ) − h + ( α ) . Let n denotea point of α such that h ( n ) = h + ( α ) . If m comes before n , we have l ( α ) ≥ d ( x, m ) + d ( m, n ) + d ( n, y ) .Otherwise n comes before m and we have l ( α ) ≥ d ( x, n ) + d ( n, m ) + d ( m, y ) . Since h ( m ) ≤ h ( x ) ≤ h ( y ) ≤ h ( n ) we always have: l ( α ) ≥ ∆ h ( x, m ) + ∆ h ( m, n ) + ∆ h ( n, y )≥ ∆ h ( x, m ) + ∆ h ( m, x ) + ∆ h ( x, y ) + ∆ h ( y, n ) + ∆ h ( y, n ) ≥ h ( x, m ) + ∆ h ( x, y ) + ( h + ( α ) − h ( y ))≥ h ( x, m ) + ∆ h ( x, y ) + d r ( x, y ) − H ≥ h ( m, x ) + d ( x, y ) − δ. Furthermore ∆ H ≤ δ , then − δ ∆ H ≤ . Therefore: l ( α ) ≥ h ( m, x ) + d ( x, y ) + − δ ∆ H − − max ( , H ) − δ, which ends the proof for the remaining case. In this part we generalize the definition of horospherical product, as seen in [5, Eskin, Fisher, Whyte] fortwo trees or two hyperbolic planes, to any pair of proper, geodesically complete, Gromov hyperbolic,Busemann spaces. We recall that given a proper, δ -hyperbolic space H with distinguished a ∈ ∂H and w ∈ H , we defined the height function on H in Definition 2.1 from the Busemann functions withrespect to a and w . Definition 4.1 (Horospherical product) . Let H p and H q be two δ − hyperbolic spaces. We fix the basepoints w p ∈ H p , w q ∈ H q and the directions in the boundaries a p ∈ ∂H p , a q ∈ ∂H q . We consider theirheights functions h p and h q respectively on H p and H q . We define the horospherical product of H p and H q , denoted H p & H q = H , by: H ∶= {( x p , x q ) ∈ H p × H q / h p ( x p ) + h q ( x q ) = } . From now on, with slight abuse, we omit the base points and fixed points on the boundary in theconstruction of the horospherical product. The metric space H refers to a horospherical product of twoGromov hyperbolic Busemann spaces. We choose to denote H p and H q the two components in orderto identify easily which objects are in which component.One of our goals is to understand the shape of geodesics in H according to a given distance on it. In acartesian product the chosen distance changes the behaviour of geodesics. However we show that in ahoropsherical product the shape of geodesics does not change for a large family of distances, up to anadditive constant.We will define the distances on H p & H q = H as length path metrics induced by distances on H p × H q .A lot of natural distances on the cartesian product H p × H q come from norms on the vector space R .Let N be such a norm and let us denote d N ∶= N ( d H p , d H q ) , the length l N ( γ ) of a path γ = ( γ p , γ q ) inthe metric space ( H p × H q , d N ) is defined by: l N ( γ ) = sup θ ∈ Θ ([ t ,t ]) ( n θ − ∑ i = d N ( γ ( θ i ) , γ ( θ i + ))) . Where Θ ([ t , t ]) is the set of subdivisions of [ t , t ] . Then the N -path metrics on H is: Definition 4.2 (The N -path metrics on H ) . Let N be a norm on the vector space R . The N -path metricon H ∶= H p & H q , denoted by d H ,N , is the length path metric induced by the distance N ( d H p , d H q ) on H p × H q . For all x and y in H we have: d H ,N ( x, y ) = inf { l N ( γ )∣ γ path in H linking x to y } . (29)19ny norm N on R can be normalized such that N ( , ) = . We call admissible any such normwhich satisfies an additional condition. Definition 4.3 (Admissible norm) . Let N be a norm on the vector space R such that N ( , ) = . Thenorm N is called admissible if and only if for all real a and b we have: N ( a, b ) ≥ a + b . (30) Since all norms are equivalent in R , there exists a constant C N ≥ such that: N ( a, b ) ≤ C N a + b . (31)As an example, any l p norm with p ≥ is admissible. Property 4.4.
Let N be an admissible norm on the vector space R . Let γ ∶= ( γ p , γ q ) ⊂ H p × H q be aconnected path. Then we have: l H p ( γ p ) + l H q ( γ q ) ≤ l N ( γ ) ≤ C N l H p ( γ p ) + l H q ( γ q ) . Proof.
Let γ ∶= ( γ p , γ q ) ∶ [ t , t ] → H p × H q be a connected path and θ a subdivision of [ t , t ] , then bythe definition of the length: l N ( γ ) ≥ n θ − ∑ i = d N ( γ ( θ i ) , γ ( θ i + )) = n θ − ∑ i = N ( d H p ( γ p ( θ i ) , γ p ( θ i + )) , d H q ( γ q ( θ i ) , γ q ( θ i + )))≥ n θ − ∑ i = ( d H p ( γ p ( θ i ) , γ p ( θ i + )) + d H q ( γ q ( θ i ) , γ q ( θ i + ))) , since N is admissible . ≥ ( n θ − ∑ i = d H p ( γ p ( θ i ) , γ p ( θ i + )) + n θ − ∑ i = d H q ( γ q ( θ i ) , γ q ( θ i + ))) . Any couple of subdivision θ and θ can be merge into a subdivision θ that contains θ and θ . Fur-thermore the last inequality holds for any subdivision θ , hence by taking the supremum on all thesubdivisions we have: l N ( γ ) ≥ l H p ( γ p ) + l H q ( γ q ) . Furthermore, we have that ∀ a, b ∈ R , N ( a, b ) ≤ C N a + b , hence: n θ − ∑ i = d N ( γ ( θ i ) , γ ( θ i + )) ≤ C N ( n θ − ∑ i = d H p ( γ p ( θ i ) , γ ( θ i + )) + n θ − ∑ i = d H q ( γ q ( θ i ) , γ q ( θ i + )))≤ C N l H p ( γ p ) + l H p ( γ p ) Since last inequality holds for any subdivision θ , we have that l N ( γ ) ≤ C N l Hp ( γ p )+ l Hp ( γ p ) .The definition of height on H p and H q is used to construct a height function on H p & H q . Definition 4.5 (Height on H ) . The height h ( x ) of a point x = ( x p , x q ) ∈ H p & H q is defined as h ( x ) = h p ( x p ) = − h q ( x q ) . On Gromov hyperbolic spaces we have that de distance between two points is greater than theirheight difference. The same occurs on horospherical products given with an admissible norm. Let x and y be two points of H , and let us denote ∆ h ( x, y ) ∶= ∣ h ( x ) − h ( y )∣ their height difference.20 emma 4.6. Let N be a admissible norm, and let d H ,N the distance on H = H p & H q induced by N . Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H , we have: ∀ x, y ∈ H , d H ,N ( x, y ) ≥ ∆ h ( x, y ) . (32) Proof.
Since N is admissible we have: d H ,N ( x, y ) ≥ d H p ( x p , y p ) + d H q ( y p , y q ) ≥ ∆ h ( x p , y p ) + ∆ h ( x q , y q ) ≥ ∆ h ( x p , y p ) = ∆ h ( x, y ) . Following Proposition 2.10, we define a notion of vertical paths in a horospherical product.
Definition 4.7 (Vertical paths in H ) . Let V ∶ R → H be a connected path. We say that V is vertical ifand only if there exists a parametrisation by arclength of V such that h ( V ( t )) = t for all t . Actually, a vertical path of a horospherical product is a geodesic.
Lemma 4.8.
Let N be an admissible norm. Let V ∶ R → H be a vertical path. Then V is a geodesic of (H , d H ,N ) .Proof. Let t , t ∈ R . The path V is vertical therefore ∆ h ( V ( t ) , V ( t )) = ∣ t − t ∣ . Since V is connectedand parametrised by arclength, we have that: ∣ t − t ∣ = l N ( V ∣[ t ,t ] ) ≥ d H ,N ( V ( t ) , V ( t ))≥ ∆ h ( V ( t ) , V ( t )) = ∣ t − t ∣ . Then d H ,N ( V ( t ) , V ( t )) = ∣ t − t ∣ , which ends the proof.Such geodesics are called vertical geodesics. Next proposition tells us that vertical geodesics of H p & H q are exactly couples of vertical geodesics of H p and H q . Proposition 4.9.
Let N be an admissible norm and let V = ( V p , V q ) ∶ R → H be a geodesic of (H , d H ,N ) .The two following properties are equivalent:1. V is a vertical geodesic of (H , d H ,N ) V p and V q are respectively vertical geodesics of H p and H q .Proof. Let us first assume that V be a vertical geodesic, we have for all real t that h ( V p ( t )) = h ( V ( t )) = t , hence ∀ t , t ∈ R : d H p ( V p ( t ) , V p ( t )) ≥ ∆ h ( V p ( t ) , V p ( t )) = ∣ t − t ∣ . (33)Similarly we have that d H q ( V q ( t ) , V q ( t )) ≥ ∣ t − t ∣ . Using that N is admissible and that V is ageodesic we have: d H p ( V p ( t ) , V p ( t )) = d H p ( V p ( t ) , V p ( t )) + d H q ( V q ( t ) , V q ( t )) − d H q ( V q ( t ) , V q ( t ))≤ d H ,N ( V ( t ) , V ( t )) − ∣ t − t ∣ = ∣ t − t ∣ . d H p ( V p ( t ) , V p ( t )) = ∣ t − t ∣ , hence V p is a vertical geodesicof H p . Similarly, V q is a vertical geodesic H q .Let us assume that V p and V q are vertical geodesics of H p and H q . Let t , t ∈ R , we have: d H ,N ( V ( t ) , V ( t )) = sup θ ∈ Θ ([ t ,t ]) ( n θ − ∑ i = d N ( V ( θ i ) , V ( θ i + )))= sup θ ∈ Θ ([ t ,t ]) ( n θ − ∑ i = N ( d H p ( V p ( θ i ) , V p ( θ i + )) , d H q ( V q ( θ i ) , V q ( θ i + ))))= sup θ ∈ Θ ([ t ,t ]) ( n θ − ∑ i = N ( ∆ h ( V p ( θ i ) , V p ( θ i + )) , ∆ h ( V q ( θ i ) , V q ( θ i + ))))= sup θ ∈ Θ ([ t ,t ]) ( N ( , ) n θ − ∑ i = ∆ h ( V p ( θ i ) , V p ( θ i + )))= N ( , ) ∆ h ( V p ( t ) , V p ( t )) = ∣ t − t ∣ , since N ( , ) = . Where Θ ([ t , t ]) is the set of subdivision of [ t , t ] . Hence the proposition is proved.This previous result is the main reason why we are working with distances which came from ad-missible norms. Definition 4.10.
A geodesic ray of
H = H p & H q is called vertical if it is a subset of a vertical geodesic. A metric space is called geodesically complete if all its geodesic segments can be prolonged intogeodesic lines. If H p and H q are proper hyperbolic geodesically complete Busemann spaces, theirhorospherical product H is connected. Property 4.11.
Let H p and H q be two proper, geodesically complete, δ -hyperbolic, Busemann spaces. Let H = H p & H q be their horospherical product. Then H is connected, furthermore ( d H p + d H q ) ≤ d H ≤ C N ( d H p + d H q ) .Proof. Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H . From Property 2.11, there exists a verticalgeodesic V x q such that x q is in the image of V x q , and there exists a vertical geodesic V y p such that y p isin the image of V y p . Let y ′ q be the point of V x q at height h ( y q ) . Let α p be a geodesic of H p linking x p to y p and let α ′ q be a geodesic of H q linking y ′ q to y q . We will connect x to y with a path composed withpieces of α p , α ′ q , V x q and V y p .We first link ( x p , x q ) to ( y p , y ′ q ) with α p and V x q . It is possible since V x q is parametrised by its height.More precisely we construct the following path c : ∀ t ∈ [ , d ( x p , y p )] , c ( t ) = ( α p ( t ) , V x q ( − h ( α p ( t )))) . Since V x q is parametrised by its height, we have h ( V x q ( − h ( α p ( t )))) = − h ( α p ( t )) which implies c ( t ) ∈ H . Furthermore, using the fact that the height is 1-Lipschitz, we have ∀ t , t ∈ [ , d ( x p , y p )] : d H q ( V x q ( − h ( α p ( t ))) , V x q ( − h ( α p ( t )))) = ∣ h ( α p ( t )) − h ( α p ( t ))∣ ≤ d H p ( α p ( t ) , α p ( t )) . Hence c ,q ∶ t ↦ V x q ( − h ( α p ( t ))) is a connected path such that l ( c ,q ) ≤ l ( α p ) ≤ d H p ( x p , y p ) . Hence c is a connected path linking ( x p , x q ) to ( y p , y ′ q ) . Using Property 4.4 on c provides us with: l N ( c ) ≤ C N ( l ( c ,q ) + l ( α p )) ≤ C N l ( α p )≤ C N d H p ( x p , y p ) -1-2 1 2 H p ∶ H q ∶ a p ∈ ∂ H p a q ∈ ∂ H q H p ⋈ H q ∶ x p x p x p − x p − x p x p x p ( , − ) x q x q x q − x q − x q x q x q ( , ) ( x p , x q − )( x p , x q − )( x p , x q )( x p − , x q )( x p − , x q )( x p − , x q ) ( x p , x q − )( x p ( , − ) , x q )( x p ( , − ) , x q ( , ) )( x p − , x q ( , ) ) -1 0-1 -1-2 1 32-3 -1 Figure 8: Example of horospherical product which is not connected. The number in a vertex is theheight of that vertex.We recall that by definition y ′ q = V x q ( h ( y q )) . We show similarly that c ∶ t ↦ ( V y p (− h ( α ′ q ( t ))) , α ′ q ( t )) is a connected path linking ( y p , y ′ q ) to ( y p , y q ) such that: l ( c ) ≤ C N d H q ( y ′ q , y q ) ≤ C N ( d H q ( y ′ q , x q ) + d H q ( x q , y q ))= C N ( ∆ h ( x q , y q ) + d H q ( x q , y q )) , since y ′ q = V x q ( h ( y q ))≤ C N d H q ( x q , y q ) . Hence, there exists a connected path c = c ∪ c linking x to y such that: l ( c ) ≤ C N d H p ( x p , y p ) + C N d H q ( x q , y q ) ≤ C N ( d H p ( x p , y p ) + d H q ( x q , y q )) . (34)However if the two components H p and H q are not geodesically complete, H may not be connected. Example 4.12.
Let H p and H q be two graphs, constructed from an infinite line Z (indexed by Z ) with anadditional vertex glued on the for H p and on the − for H q . Their construction are illustrated on figure8. They are two 0-hyperbolic Busemann spaces which are not geodesically complete. Let w p ∈ H p be thevertex indexed by in H p , and let w q ∈ H q be the vertex indexed by − in H q . We choose them to be thebase points of H p and H q . Since ∂H p and ∂H q contain two points each, we fix in both cases the point ofthe boundary a p or a q to be the one that contains the geodesic ray indexed by N . On figure 8, we denotedthe height of a vertex inside this one. Then the horospherical product H = H p & H q taken with the (cid:96) pathmetric is not connected. Since some vertices of H p and H q are not contained in a vertical geodesic, one maynot be able to adapt its height correctly while constructing a path joining ( x p − , x q ( , ) ) to ( x p ( , − ) , x q ( , ) ) . It is not clear that a horospherical product is still connected without the hypothesis that H p and H q are Busemann spaces. In that case we would need a "coarse" definition of horospherical product.Indeed, the height along geodesics would not be smooth as in Proposition 2.10, therefore the conditionrequiring to have two exact opposite heights would not suits. A first example of horospherical product is the family of Diestel-Leader graphs. They are by construc-tion horospherical products of two trees. 23 efinition 4.13 (Diestel-Leader graph DL ( p, q ) ) . Let p ≥ and q ≥ be two integers. Let T p be the p -homogeneous tree and T q be the q -homogeneous tree. The two graphs T p and T q are -hyperbolic propergeodesically complete Busemann spaces. The Diestel-Leader graph DL ( p, q ) is defined by DL ( p, q ) = T p & T q . We see T p and T q as connected metric spaces with the usual distance on them. By choosing half ofthe (cid:96) path metric on DL ( p, q ) , this horospherical product becomes a graph with the usual distance onit. Indeed, the set of vertices of DL ( p, q ) is then defined by the subset of couples of vertices of T p × T q included in DL ( p, q ) . In this horospherical product, two points ( x p , x q ) and ( y p , y q ) of DL ( p, q ) areconnected by an edge if and only if x p and y p are connected by an edge in T p and if x q and y q areconnected by an edge in T q . Furthermore, when p = q , there is a one-to-one correspondance between DL ( q, q ) and the Cayley graph of the lamplighter group Z q ≀ Z , see [13, Woess] for further details.The SOL geometry is the Riemannian manifold with coordinates ( x, y, z ) ∈ R and with the Rieman-nian metric ds = dz + e z dx + e − z dy . It is the horospherical product of two hyperbolic planes, it isdescribed in [14, Woess]. Let us consider H the Log model of the hyperbolic plane, defined as the Rie-mannian manifold with coordinates ( x, z ) ∈ R and with the Riemannian metric ds = dz + e − z dx .We fix w = ( , ) as the base point of H and the "upward" direction a as the point on the boundary. Inthat case the height function in regards to ( a, w ) taken on a point ( x, z ) ∈ H is h ( a,w ) ( x, z ) = z . Wenow look at the horospherical product H & H ∶= {( x , z , x , z ) ∈ R × R ∣ z = − z } taken with the (cid:96) path metric. Since the second and the fourth variable are exactly opposite, we merge them into one.Hence we have that H & H is isometric to the space {( x , x , z ) ∈ R } with the metric ds = dz + e − z dx + dz + e z dx = dz + e − z dx + e z dx . Changing the coordinates by dividing x and x by two tells us that this space is isometric to SOL.Depending on the case, we either used the (cid:96) path metric or the (cid:96) path metric. Proposition 4.18 tellsus that it does not matter, up to an additive uniform constant. Quasi-isometric rigidity results havebeen proved in the Diestel-Leader graphs and the SOL geometry with the same techniques in [5, Eskin,Fisher, Whyte] and [6, E,F,W].The horospherical product of a hyperbolic plane and a regular tree has been studied as the 2-complex ofBaumslag-Solitar groups in [1, Bendikov, Saloff-Coste, Salvatori, Woess]. They are called the treebolicspaces. The distance they choose on the treebolic spaces is similar to ours. In fact our Proposition 4.17and their Proposition . page 9 (in [1]) tell us they are equal up to an additive constant. Rigidity resultson the treebolic spaces were brought up in [7, Farb, Mosher] and [8, F,M].The previous examples were already known, however our construction still works for many otherspaces. As an example, a geodesically complete manifold with a curvature lower than a negative con-stant could be used as the component H p or H q in the horospherical product. H From now on, unless otherwise specified, H p and H q will always be two proper, geodesically complete, δ -hyperbolic, Busemann spaces with δ ≥ , and N will always be an admissible norm. Let x and y betwo points of H ∶= H p & H q , and let α be a geodesic of H connecting them. We first prove an upperbound on the length of α by computing the length of a path γ ⊂ H linking x to y Lemma 4.14.
Let x and y be points of the horospherical product H = H p & H q . There exists a path γ connecting x = ( x p , x q ) to y = ( y p , y q ) such that: l N ( γ ) ≤ d r ( x q , y q ) + d r ( x p , y p ) + ∆ h ( x, y ) + δC N . roof. Without loss of generality, we assume h ( x ) ≤ h ( y ) . One can follow the idea of the proof on Fig-ure 9. We consider V x p and V y p two vertical geodesics of H p containing x p and y p respectively. Similarlylet V x q and V y q be two vertical geodesics of H q containing x q and y q respectively. We will use them toconstruct γ . Let A be the point of the vertical geodesic ( V x p , V x q ) ⊂ H at height h ( x ) − d r ( x q , y q ) and A be the point of the vertical geodesic ( V x p , V y q ) ⊂ H at the same height h ( x ) − d r ( x q , y q ) . Let A be the point of the vertical geodesic ( V x p , V y q ) at height h ( y ) + d r ( x p , y p ) and A be the point ofthe vertical geodesic ( V y p , V y q ) at the same height h ( y ) + d r ( x p , y p ) . Then γ ∶= γ ∪ γ ∪ γ ∪ γ ∪ γ is constructed as follows:- γ is the part of ( V x p , V x q ) linking x to A .- γ is a geodesic linking A to A . Such a geodesic exists by Property 4.11.- γ is the part of ( V x p , V y q ) linking A to A .- γ is a geodesic linking A to A . Such a geodesic exists by Property 4.11.- γ is the part of ( V y p , V y q ) linking A to y .In fact A and A are close to each other. Indeed, the two points A = ( A ,p , A ,q ) and A = ( A ,p , A ,q ) are characterised by the two geodesics ( V x p , V x q ) and ( V x p , V y q ) . Then, because − h ( y ) = h q ( y q ) ≤ h q ( x q ) , Lemma 3.2 applied on x q and y q in H q gives us d H q ( A ,q , A ,q ) ≤ δ . Furthermore Property4.11 provides us with d H ,N ≤ C N ( d H p + d H q ) , however we have that A ,p = A ,p hence: d H ,N ( A , A ) ≤ δC N . (35)Lemma 3.2 applied on x p and y p provides similarly: d H ,N ( A , A ) ≤ δC N , (36)which gives us: l N ( γ ) = l N ( γ ) + l N ( γ ) + l N ( γ ) + l N ( γ ) + l N ( γ )= d H ,N ( x, A ) + d H ,N ( A , A ) + d H ,N ( A , A ) + d H ,N ( A , A ) + d H ,N ( A , y ) Since γ , γ and γ are vertical geodesics, we have: = ∆ h ( x, A ) + d H ,N ( A , A ) + ∆ h ( A , A ) + d H ,N ( A , A ) + ∆ h ( A , y )= d r ( x q , y q ) + d H ,N ( A , A ) + d r ( x q , y q ) + d r ( x p , y p ) + ∆ h ( x, y ) + d H ,N ( A , A ) + d r ( x p , y p )≤ d r ( x q , y q ) + d r ( x p , y p ) + ∆ h ( x, y ) + δC N , by inequalities (35) and (36).We are aiming to use Lemma 3.8 on the two components α p ⊂ H p and α q ⊂ H q of α to obtain lowerbounds on their lengths. We hence need the following lemma to ensure us that when α is a geodesic,the exponential term in the inequality of Lemma 3.8 will be small. Lemma 4.15.
Let C = δC N + and let e ∶ R → R be a map defined by ∀ t ∈ R , e ( t ) = C C − t − ( , t ) . Then ∀ t ∈ R :1. e ( t ) ≥ − C ( e ( t ) ≤ δC N ) ⇒ ( t ≤ C ) .Proof. For all time t , we have that e ( t ) = C C − t − ( , t ) ≤ C C − t − t =∶ e ( t ) . The derivative of e is e ′ ( t ) = log ( ) C C − t − , which is non negative ∀ t ≥ C log ( ( ) C ) and non positive otherwise.Then ∀ t ∈ R : e ( t ) ≥ e ( log ( ( ) C )) ≥ C log ( ) − C log ( ( ) C ) ≥ C log ( ) − C log (√ ( ) C )≥ C log ( ) − √ ( ) C ≥ − √ ( ) C ≥ − C . yh γ γ γ A A A A γ γ ∆ h ( x, y ) ≤ δ ≤ δ Figure 9: Construction of the path γ when h ( x ) ≤ h ( y ) for Lemma 4.14.Since C ≥ ( ) we have C ≥ C log ( C ) ≥ C log ( ( ) C ) , then e is non decreasing on [ C log ( C ) ; +∞[ . We show that e ( C ) ≥ δC N : e ( C ) ≥ e ( C log ( C )) = C C log2 ( C ) C − C log ( C ) = C ( C − ( C )) . Since C ≥ we have C − ( C ) ≥ and since C ≥ δC N we have that e ( C ) ≥ C × ≥ δC N which provides ∀ t ∈ [ C ; +∞[ we have e ( t ) ≥ δC N . Furthermore ∀ t ∈ R + , e ( t ) = e ( t ) , hence ∀ t ∈ [ C ; +∞[ we have e ( t ) ≥ δC N which implies point . of this lemma.The following lemma provides us with a lower bound matching Lemma 4.14, and a first control onthe heights a geodesic segment must reach. Lemma 4.16.
Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q such that h ( x ) ≤ h ( y ) .Let α = ( α p , α q ) be a geodesic segment of H linking x to y . Let C = ( δC N + ) , we have:1. l ( α ) ≥ ∆ h ( x, y ) + d r ( x q , y q ) + d r ( x p , y p ) − C h + ( α ) ≥ h ( y ) + d r ( x p , y p ) − C h − ( α ) ≤ h ( x ) − d r ( x q , y q ) + C .Proof. Let us denote ∆ H + = h ( y ) + d r ( x p , y p ) − h + ( α ) and ∆ H − = h − ( α ) − ( h ( x ) − d r ( x q , y q )) .Let m be a point of α at height h − ( α ) = h ( x ) − d r ( x q , y q ) + ∆ H − , and n be a point of α at height h + ( α ) = h ( y ) + d r ( x p , y p ) − ∆ H + . Then Lemma 3.8 used on α p gives us: l ( α p ) ≥ h ( x p , m p ) + d ( x p , y p ) + − δ ∆ H + − − ( , ∆ H + ) − δ ≥ h ( x p ) − ( h ( x p ) − d r ( x q , y q ) + ∆ H − ) + d ( x p , y p ) + − δ ∆ H + − − ( , ∆ H + ) − δ ≥ d r ( x q , y q ) + d r ( x p , y p ) + ∆ h ( x, y ) + − δ ∆ H + − − ( , ∆ H + ) − H − − δ. h ( x q ) ≥ h ( y q ) and h ( n q ) = h ( y q )− d r ( x p , y p )+ ∆ H + , Lemma 3.8 used on α q provides similarly: l ( α q ) ≥ d r ( x p , y p ) + d r ( x q , y q ) + ∆ h ( x, y ) + − δ ∆ H − − − ( , ∆ H − ) − H + − δ. Hence by Property 4.4: l N ( α ) ≥ ( l ( α p ) + l ( α q )) ≥ d r ( x p , y p ) + d r ( x q , y q ) + ∆ h ( x, y ) − δ + − δ ∆ H − + − δ ∆ H + − ( , ∆ H − ) − ( , ∆ H + ) − . (37)Furthermore, we know by Lemma 4.14 that l N ( α ) ≤ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + δC N .Since C N ≥ we have: δC N ≥ − δ ∆ H − − ( , ∆ H − ) + − δ ∆ H + − ( , ∆ H + ) − . Let us denote S ∶= max { ∆ H − , ∆ H + } . Therefore we have − δ S − ( , S ) − ≤ δC N . Byassumption δ ≥ hence − δ S − ( , S ) ≤ δC N . Furthermore, for C = δC N + , wehave both − ≥ C and δ ≥ C . Then we have C SC − ( , S ) ≤ δC N . Lemma 4.15 provides S ≤ C = C which implies points . and . of our lemma. Lemma 4.15 also provides us with: − C ≤ − δ ∆ H − − ( , ∆ H − ) + − δ ∆ H + − ( , ∆ H + ) . Last inequality is a lower bound of the term we want to remove in inequality (37). The first point ofour lemma hence follows since δ + ≤ C .Combining Lemma 4.14 and 4.16 we get the following corollary. Corollary 4.17.
Let N be an admissible norm and let C = ( δC N + ) . The length of a geodesicsegment α connecting x to y in (H , d H ,N ) is controlled as follows: ∣ l N ( α ) − ( ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ))∣ ≤ C , which gives us a control on the N -path metric, for all points x and y in H we have: ∣ d H ,N ( x, y ) − ( ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ))∣ ≤ C . This result is central as it shows that the shape of geodesics does not depend on the N -path metricchosen for the distance on the horospherical product. Corollary 4.18.
Let r ≥ . For all x and y in H = H p & H q we have: ∣ d H ,(cid:96) r ( x, y ) − d H ,(cid:96) ( x, y )∣ ≤ ( δ + ) . Proof.
The (cid:96) r norm inequalities provide us with: r √ d H p r + d H q r ≤ d H p + d H q ≤ r − r r √ d H p r + d H q r . Hence we have r √ ( d H p + d H q ) ≤ r √ d H p r + d H q r ≤ d H p + d H q . Then the (cid:96) r norms are admissible normswith C (cid:96) r ≤ , which ends the proof.The next corollary tells us that changing this distance does not change the large scale geometry of H . Corollary 4.19.
Let N and N be two admissible norms. Then the metric spaces (H , d H ,N ) and (H , d H ,N ) are quasi-isometric. The control on the distances of Lemma 4.17 will help us understand the shape of geodesic segmentsand geodesic lines in a horospherical product. 27
Shapes of geodesics and visual boundary of H In this section we focus on the shape of geodesics. We recall that in all the following H p and H q areassumed to be two proper, geodesically complete, δ -hyperbolic, Busemann spaces with δ ≥ , and N isassumed to be an admissible norm.The next lemma gives a control on the maximal and minimal height of a geodesic segment in ahorospherical product. It is similar to a traveller problem, who needs to walk from x to y passing by m and n . This result follows from the inequalities on maximal and minimal heights of Lemma 4.16combined with Lemma 4.14. Lemma 5.1.
Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q such that h ( x ) ≤ h ( y ) .Let N be an admissible norm and let α = ( α p , α q ) be a geodesic of (H , d H ,N ) linking x to y . Let C =( δC N + ) , we have:1. ∣ h − ( α ) − ( h ( x ) − d r ( x q , y q ))∣ ≤ C ∣ h + ( α ) − ( h ( y ) + d r ( x p , y p ))∣ ≤ C .Proof. Let us consider a point m of α such that h ( m ) = h − ( α ) and a point n of α such that h ( n ) = h + ( α ) . Then m comes before n or n comes before m . In both cases, since h ( m ) ≤ h ( x ) ≤ h ( y ) ≤ h ( n ) and by Lemma 4.6 we have: l N ( α ) ≥ ∆ h ( x, y ) + ( h ( x ) − h − ( α )) + ( h + ( α ) − h ( y ))≥ ∆ h ( x, y ) + ( h ( x ) − h − ( α )) + d r ( x p , y p ) − C , by Lemma 4.16 . Furthermore Lemma 4.14 provides l N ( α ) ≤ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C , hence: ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C ≥ ∆ h ( x, y ) + ( h ( x ) − h − ( α )) + d r ( x p , y p ) − C , which implies ( h ( x ) − d r ( x q , y q )) − h − ( α ) ≤ C . In combination with the third point of Lemma 4.16it proves the first point of our Lemma 5.1. The second point is proved similarly. Lemma 5.2.
Let N be an admissible norm and let C = ( δC N + ) . Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q . Let α = ( α p , α q ) be a geodesic of (H , d H ,N ) linking x to y .Then there exist two points a = ( a p , a q ) , b = ( b p , b q ) of α such that h ( a ) = h ( x ) , h ( b ) = h ( y ) with thefollowing properties:1. If h ( x ) ≤ h ( y ) − C then:(a) h − ( α ) = h − ([ x, a ]) and h + ( α ) = h + ([ b, y ]) (b) ∣ d r ( x q , a q ) − d r ( x q , y q )∣ ≤ C and d r ( x p , a p ) ≤ C (c) ∣ d r ( y p , b p ) − d r ( x p , y p )∣ ≤ C and d r ( y q , b q ) ≤ C (d) ∣ d H ,N ( a, b ) − ∆ h ( a, b )∣ ≤ C .2. If h ( y ) ≤ h ( x )− C then ( a ) , ( b ) , ( c ) and ( d ) hold by switching the roles of x and y and switchingthe roles of a and b .3. If ∣ h ( x ) − h ( y )∣ ≤ C at least one of the two previous conclusions is satisfied. Lemma 5.2 is illustrated on Figure 10. Its notations will be used in all section 5.28 q h x y H p nm ba h + ( α ) = h ( n ) h ( y ) = h ( b ) h ( x ) = h ( a ) h − ( α ) = h ( m ) α Figure 10: Notations of Lemma 5.2.
Proof.
Let us consider a point m of α such that h ( m ) = h − ( α ) and a point n of α such that h ( n ) = h + ( α ) . We first assume that m comes before n in α oriented from x to y . Let us call a the first pointbetween m and n at height h ( x ) and b the last point between m and n at height h ( y ) . Property ( a ) of our Lemma is then satisfied. Let us denote α the part of α linking x to a , α the part of α linking a to b and α the part of α linking b to y . We have that m is a point of α and that n is a point of α .Inequalities . and . of Lemma 4.16 used on α provide l N ( α ) ≥ d ( x, m ) + d ( m, a ) ≥ h ( x, m ) ≥ d r ( x q , y q ) − C and similarly l N ( α ) ≥ d r ( x p , y p ) − C . Furthermore we have l N ( α ) ≥ ∆ h ( x, y ) .Combining l N ( α ) = l N ( α ) − l N ( α ) − l N ( α ) and Lemma 4.14 we have: l N ( α ) ≤ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C − ∆ h ( x, y ) − d r ( x p , y p ) + C ≤ d r ( x q , y q ) + C . (38)We have similarly that l N ( α ) ≤ d r ( x p , y p ) + C and that d H ,N ( a, b ) = l N ( α ) ≤ ∆ h ( x, y ) + C . Itgives us ∣ d H ,N ( a, b ) − ∆ h ( x, y )∣ ≤ C , point ( d ) of our lemma. Furthermore, using Lemma 5.1 on α and α provides: ∣ h − ( α ) − ( h ( x ) − d r ( x q , y q ))∣ ≤ C , ∣ h − ( α ) − ( h ( x ) − d r ( x q , a q ))∣ ≤ C . Since h − ( α ) = h − ( α ) we have: ∣ d r ( x q , a q ) − d r ( x q , y q )∣ ≤ C , (39)which is the first inequality of ( b ) . Using the first point of Lemma 4.16 on α in combination withinequality (38) gives us: d r ( x q , y q ) + C ≥ l N ( α ) ≥ ∆ h ( x, a ) + d r ( x p , a p ) + d r ( x q , a q ) − C ≥ d r ( x p , a p ) + d r ( x q , a q ) − C ≥ d r ( x p , a p ) + d r ( x q , y q ) − C , by inequality (39).Then d r ( x p , y p ) ≤ C the second inequality of point ( b ) holds. We prove similarly the inequality ( c ) of this lemma. This ends the proof when m comes before n . If n comes before m , the proof is still29orking by orienting α from y to x hence switching the roles between x and y .We will now prove that if h ( x ) ≤ h ( y ) − C then m comes before n on α oriented from x to y .Let us assume that h ( x ) ≤ h ( y ) − C . We will proceed by contradiction, let us assume that n comesbefore m , using h ( m ) ≤ h ( x ) ≤ h ( y ) ≤ h ( n ) it implies: l N ( α ) ≥ d H ,N ( x, n ) + d H ,N ( n, m ) + d H ,N ( m, y ) ≥ ∆ h ( x, n ) + ∆ h ( n, m ) + ∆ h ( m, y )≥ ∆ h ( x, y ) + ∆ h ( y, n ) + ∆ h ( m, x ) + ∆ h ( x, y ) + ∆ h ( y, n ) + ∆ h ( m, x ) + ∆ h ( x, y )≥ h ( x, y ) + ∆ h ( x, y ) + h ( m, x ) + ( y, n )≥ C + ∆ h ( x, y ) + ( h ( x ) − h − ( α )) + ( h + ( α ) − h ( y )) . However Lemma 4.16 applied on α provides h + ( α ) ≥ h ( y ) + d r ( x p , y p ) − C and h − ( α ) ≤ h ( x ) − d r ( x q , y q ) + C . Then: l N ( α ) ≥ C + ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) − C ≥ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C , which contradict Lemma 4.14. Hence, if h ( x ) ≤ h ( y ) − C , the point m comes before the point n andby the first part of the proof, . holds. Similarly, if h ( y ) ≤ h ( x ) − C then n comes before m and then . holds. Otherwise when ∣ h ( x ) − h ( y )∣ ≤ C both cases could happened, then . or . hold.This previous lemma essentially means that if x is sufficiently below y , the geodesic α first travelsin a copy of H q in order to "loose" the relative distance between x q and y q , then it travels upward usinga vertical geodesic from a to b until it can "lose" the relative distance between x p and y p by travellingin a copy of H p . It looks like three successive geodesics of hyperbolic spaces, glued together. The ideais that the geodesic follows a shape similar to the path γ we constructed in Lemma 4.14. We formalizethis in the following theorem, which tells us that a geodesic segment is in the constant neighbourhoodof three vertical geodesics. It can be understood as an extension of the fact that in a hyperbolic space,a geodesic segment is in a constant neighbourhood of two vertical geodesics. Theorem 5.3.
Let N be an admissible norm. Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q and let α be a geodesic segment of (H , d H ,N ) linking x to y . Let C = ( δC N + ) , thereexist two vertical geodesics V = ( V ,p , V ,q ) and V = ( V ,p , V ,q ) such that:1. If h ( x ) ≤ h ( y ) − C then α is in the C C N -neighbourhood of V ∪ ( V ,p , V ,q ) ∪ V
2. If h ( x ) ≥ h ( y ) + C then α is in the C C N -neighbourhood of V ∪ ( V ,p , V ,q ) ∪ V
3. If ∣ h ( x ) − h ( y )∣ ≤ C then at least one of the conclusions of . or . holds.Specifically V and V can be chosen such that x is close to V and y is close to V . Figure 11 pictures the C C N -neighbourhood of such vertical geodesics when h ( x ) ≤ h ( y )− C .When ∣ h ( x ) − h ( y )∣ ≤ C , there are two possible shapes for a geodesic segment. In some cases, twopoints can be linked by two different geodesics, one of type and one of type . Proof.
Let m = ( m p , m q ) be a point of α such that h ( m ) = h − ( α ) , and n = ( n p , n q ) be a point of α such that h ( n ) = h + ( α ) . Then by Lemma 5.1 we have: ∣ ∆ h ( x, m ) − d r ( x q , y q )∣ ≤ C . (40)We show similarly that: ∣ ∆ h ( y, n ) − d r ( x p , y p )∣ ≤ C . (41)30 V ( V ,p , V ,q ) H p H q h N C C N ( V ) x y h ( y ) α Figure 11: Theorem 5.3. The neighbourhood’s shapes are distorted since when going upward, distancesare contracted in the "direction" H p and expanded in the "direction" H q .In the first case we assume that h ( x ) ≤ h ( y ) − C . With notations as in Lemma 5.2, and by inequality(38), we have that l N ([ x, a ]) ≤ d r ( x q , y q ) + C , hence: l N ([ x, m ]) = l N ([ x, a ]) − l N ([ a, m ]) ≤ d r ( x q , y q ) + C − ∆ h ( a, m )≤ d r ( x q , y q ) + C , since ∆ h ( x, m ) = ∆ h ( a, m ) . (42)It follows from this inequality that: d H p ( x p , m p ) = d H p × H q ( x, m ) − d H q ( x q , m q ) ≤ d H ,N ( x, m ) − d H q ( x q , m q )≤ l N ([ x, m ]) − d H q ( x q , m q ) ≤ d r ( x q , y q ) + C − ∆ h ( x, m ) ≤ d r ( x q , y q ) + C . Then: d r ( x p , m p ) = d H p ( x p , m p ) − ∆ h ( x, m ) ≤ d r ( x q , y q ) + C − ∆ h ( x, m )≤ C , by inequality ( ) . Similarly d r ( x q , m q ) ≤ C . Let us consider the vertical geodesic V m p of H p containing m p , and thevertical geodesic V x q of H q containing x q . Let us denote x ′ p the point of V m p at the height h ( x ) . Since d r ( x p , m p ) ≤ C , Lemma 3.3 applied on x p and m p provides d H p ( x p , x ′ p ) ≤ C . We will thenconsider two paths of H p . The first one is α ,p = [ x p , m p ] , the part of α p linking x p to m p . The secondone is [ m p , x ′ p ] a piece of vertical geodesic linking m p to x ′ p . We show that these two paths have closelength. Using Property 4.4 with inequalities (40) and (42) provides us with: l H p ([ x p , m p ]) ≤ l N ([ x, m ]) − l H q ([ x q , m q ]) ≤ ( d r ( x q , y q ) + C ) − ∆ h ( x, m )≤ ∆ h ( x, m ) + C Furthermore l H p ([ x p , m p ]) ≥ ∆ h ( x, m ) and we know that l H p ([ m p , x ′ p ]) = ∆ h ( x, m ) , hence: ∣ l H p ([ x p , m p ]) − l H p ([ m p , x ′ p ])∣ ≤ C
31e already proved that their end points are also close to each other d ( x p , x ′ p ) ≤ C . Since δ ≤ C , theproperty of hyperbolicity of H p gives us that α ,p is in the ( + + ) C = C -neighbourhood of [ m p , x ′ p ] , a part of the vertical geodesic V m p . We show similarly that α ,q is in the C -neighbourhoodof V x q . Since N is an admissible norm, Property 4.11 gives us that α is in the C C N -neighbourhoodof ( V m p , V x q ) . We show similarly that α , the portion of α linking n to y , is in the C C N -neighbourhoodof ( V y p , V n q ) . We now focus on α , the portion of α linking m to n . Let us denote [ m p , n p ] the path α ,p and [ m q , n q ] the path α ,q . Then Lemma 5.1 provides us with: ∣ ∆ h ( m, n ) − ( ∆ h ( x, y ) + d r ( x q , y q ) + d r ( x p , y p ))∣ ≤ C . (43)However from Lemma 4.14 and since δC N ≤ C : l N ( α ) = l N ( α ) − l N ( α ) − l N ( α )≤ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C − ∆ h ( x, m ) − ∆ h ( n, y )≤ ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C , by inequalities ( ) and ( ) . It follows from this inequality and the fact that N is admissible that: d H p ( m p , n p ) ≤ l N ( α ) − d H q ( m q , n q ) ≤ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) + C − ∆ h ( m, n )≤ ∆ h ( m, n ) + C , by inequality ( ) . Thus: d r ( m p , n p ) = d H p ( m p , n p ) − ∆ h ( m, n ) ≤ C . In the same way we have d r ( m q , n q ) ≤ C . Let us denote n ′ p the point of V m p at the height h ( n p ) .Since d r ( x p , m p ) ≤ C , Lemma 3.3 applied on m p and n p provides: d H p ( m p , n ′ p ) ≤ C (44)Hence we have proved that α ,p and [ m p , n ′ p ] have their end points close to each other. Let us now provethat these paths have close lengths. We have that l H p ([ m p , n ′ p ]) = ∆ h ( m, n ) , and from inequalities(40) and (41) we have: l H p ([ m p , n p ]) ≤ l N ( α ,p ) − l H q ([ m q , n q ]) = ( l N ( α ) − l N ( α ) − l N ( α )) − ∆ h ( m, n )≤ ( C + ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) − ∆ h ( x, m ) − ∆ h ( n, y )) − ∆ h ( m, n )≤ ( ∆ h ( x, y ) + d r ( x p , y p ) + d r ( x q , y q ) − ∆ h ( x, m ) − ∆ h ( n, y )) − ∆ h ( m, n )≤ ( ∆ h ( x, y ) + ∆ h ( x, m ) + ∆ h ( n, y ) + C ) − ∆ h ( m, n ) + C ≤ ∆ h ( m, n ) + C As l H p ([ m p , n p ]) ≥ ∆ h ( m, n ) we obtain: ∣ l H p ([ m p , n p ]) − l H p ([ m p , n ′ p ])∣ ≤ C (45)Then by similar arguments as for the path α ,p , inequalities (44) and (45) show that α ,p is in the ( + + ) C = C neighbourhood of V m p . Similarly we prove that α ,q is in the C neigh-bourhood of V n q . Since N is an admissible norm, Property 4.11 gives us that α is in the C C N -neighbourhood of ( V m p , V n q ) .In the second case, we assume that h ( y ) ≤ h ( x ) − C . Then by switching the role of x and y , Lemma5.2 gives us the result identically.In the third case, we assume that ∣ h ( x ) − h ( y )∣ ≤ C . Then Lemma 5.2 tells us that on of the twoprevious situations prevails, which proves the result.32 .2 Coarse monotonicity The fact that a geodesic is following a vertical geodesic is related to the next definition.
Definition 5.4.
Let C be a non negative number. A geodesic α ∶ I → H of H = H p & H q is called C -coarsely increasing if ∀ t , t ∈ I : ( t > t + C ) ⇒ ( h ( α ( t )) > h ( α ( t )) ) . The geodesic α is called C -coarsely decreasing if ∀ t , t ∈ I : ( t > t + C ) ⇒ ( h ( α ( t )) < h ( α ( t )) ) . The next lemma links the coarse monotonicity and the fact that a geodesic segment is close tovertical geodesics.
Lemma 5.5.
Let N be an admissible norm and let C = ( δC N + ) . Let x = ( x p , x q ) and y = ( y p , y q ) be two points of H = H p & H q and let α be a geodesic segment of (H , d H ,N ) linking x to y .Let m ∈ α and n ∈ α be two points in H such that h − ( α ) = h ( m ) and h + ( α ) = h ( n ) . We have:1. If h ( x ) ≤ h ( y ) − C , then α is C -coarsely decreasing on [ x, m ] and C -coarsely increasingon [ m, n ] and C -coarsely decreasing on [ n, y ] .2. If h ( x ) ≥ h ( y ) + C , then α is C -coarsely increasing on [ x, n ] and C -coarsely decreasingon [ n, m ] and C -coarsely increasing on [ m, n ] .3. If ∣ h ( x ) − h ( y )∣ ≤ C then the conclusions of . or . holds.Proof. Assume that h ( x ) ≤ h ( y ) − C . Then from inequality (42) in the proof of Theorem 5.3, l N ([ x, m ]) ≤ d r ( x q , y q ) + C . Furthermore Lemma 5.1 gives us that ∣ ∆ h ( x, m ) − d r ( x q , y q )∣ ≤ C . Then: l N ([ x, m ]) ≤ ∆ h ( x, m ) + C . (46)We will proceed by contradiction, assume that [ x, m ] is not C -coarsely decreasing, then there exists i ∈ α , i ∈ α such that h ( i ) = h ( i ) and l ([ i , i ]) > C . Hence: l N ([ x, m ]) ≥ l N ([ x, i ]) + l N ([ i , i ]) + l N ([ i , m ]) ≥ ∆ h ( x, i ) + l N ([ i , i ]) + ∆ h ( i , m )> ∆ h ( x, m ) + C , which contradicts inequality (46). Then [ x, m ] is C -coarsely decreasing. We show in a similar waythat [ m, n ] is C -coarsely increasing and that [ n, y ] is C -coarsely decreasing. This proves thefirst point of our lemma. The second point is proved by switching the roles of x and y . We now assume ∣ h ( x ) − h ( y )∣ ≤ C , as in the proof of Theorem 5.3 the inequality (42) or a corresponding inequalityholds, which ends the proof. In this section we are focusing on using the previous results to get informations on the shapes ofgeodesic rays and geodesic lines. We first link the coarse monotonicity of a geodesic ray to the factthat it is close to a vertical geodesic. Let λ ≥ and c ≥ , a ( λ, c ) -quasigeodesic of the metric space (H , d H ,N ) is the image of a function φ ∶ R → H verifying that ∀ t , t ∈ R : ∣ t − t ∣ λ − c ≤ d H ,N ( φ ( t ) , φ ( t )) ≤ λ ∣ t − t ∣ + c (47)33 emma 5.6. Let N be an admissible norm and let C = ( δC N + ) . Let α = ( α p , α q ) be ageodesic ray of (H , d H ,N ) and let K be a positive number such that α is K -coarsely monotone. Then α p and α q are ( , C + K ) -quasigeodesics.Proof. Let t and t be two times. Let us denote x = ( x p , x q ) = α ( t ) and y = ( y p , y q ) = α ( t ) . Weapply Lemma 5.2 on the part of α linking x to y denoted by [ x, y ] . By K -coarse monotonicity of α wehave that d ( x, a ) H ,N ≤ K and d H ,N ( b, y ) ≤ K . Hence using d ) of Lemma 5.2: ∆ h ( x, y ) ≤ d H ,N ( x, y ) ≤ d H ,N ( x, a ) + d H ,N ( a, b ) + d H ,N ( b, y ) ≤ K + ∆ h ( a, b ) + C + K ≤ ∆ h ( x, y ) + ∆ h ( x, a ) + ∆ h ( b, y ) + C + K ≤ ∆ h ( x, y ) + C + K. Furthermore, d H p ( x p , y p ) ≥ ∆ h ( x p , y p ) = ∆ h ( x, y ) and d H q ( x q , y q ) ≥ ∆ h ( x, y ) . Since N is an admis-sible norm we have: ∆ h ( x, y ) ≤ d H p ( x p , y p ) = d H p × H q ( x, y ) − d H q ( x q , y q ) ≤ d H ,N ( x, y ) − d H q ( x q , y q )≤ h ( x, y ) + C + K − ∆ h ( x, y ) ≤ ∆ h ( x, y ) + C + K. Hence: d H ,N ( x, y ) − C − K ≤ d H p ( x p , y p ) ≤ d H ,N ( x, y ) + C + K, By definition we have x p = α p ( t ) , y p = α p ( t ) and d H ,N ( x, y ) = ∣ t − t ∣ . Then α p is a ( , C + K ) -quasigeodesic ray. We prove similarly that α q is a ( , C + K ) -quasigeodesic ray.We will now make use of the rigidity property of quasi-geodesics in Gromov hyperbolic spaces,presented in Theorem 3.1 p.41 of [3, Coornaert, Delzant, Papadopoulos]. Theorem 5.7 ([3]) . Let H be a δ -hyperbolic geodesic space. If f ∶ R → H is a ( λ, k ) -quasi geodesic, thenthere exists a constant κ > depending only on δ, λ and k such that the image of f is in the κ -neighbourhoodof a geodesic in H . Lemma 5.8.
Let N be an admissible norm and let T and T be two real numbers. Let α = ( α p , α q ) ∶[ T , +∞[→ H be a geodesic ray of (H , d H ,N ) . Let K be a positive number such that α is K -coarselymonotone. Then there exists a constant κ > depending only on K , δ and N such that α is in the κ -neighbourhood of a vertical geodesic ray V ∶ [ T ; +∞[→ H and such that d H ,N ( α ( T ) , V ( T )) ≤ κ .Proof. We assume without loss of generality that lim t →+∞ h ( α ( t )) = +∞ . Let C = ( δC N + ) ,by Lemma 5.6, α p is a ( , C + K ) -quasi geodesic ray. Then Theorem 5.7 says there exists κ p > depending only on C + K and δ such that α p is in the κ p -neighbourhood of a geodesic V p . Since C depends only on δ and N , κ p depends only on K , δ and N . Then lim t →+∞ h ( α ( t )) = +∞ gives us lim t →+∞ h ( V p ( t )) = +∞ which implies that V p is a vertical geodesic of H p . We will now build the verticalgeodesic we want in H q . We have lim t →+∞ h ( α q ( t )) = −∞ and by Lemma 5.6: ∆ h ( α q ( t ) , α q ( t )) − C − K ≤ d H q ( α q ( t ) , α q ( t )) ≤ ∆ h ( α q ( t ) , α q ( t )) + C + K. Since H q is Busemann, there exists a vertical geodesic ray β starting at α q ( T ) . Since β is parametrisedby its height, α q ∪ β is also a ( , C + K ) -quasi geodesic, hence there exists κ q and V q dependingonly on K , δ and N such that α q ∪ β is in the κ q -neighbourhood of V q . Since lim t →−∞ h ( V q ( t )) = +∞ , V q isa vertical geodesic of H q . Furthermore, by Property 4.11, d H ,N ≤ C N ( d H p + d H q ) , hence there exists κ depending only on K , δ and N such that α is in the κ -neighbourhood (for d H ,N ) of ( V p , V q ) , a verticalgeodesic of (H , d H ,N ) . Since h ( α ( t )) ≥ h ( α ( T )) − C − K =∶ M , α is in the κ -neighbourhood of ( V p ([ M − κ ; +∞[) , V q (] − ∞ ; − M + κ ])) which is a vertical geodesic ray.We will now show that the starting points of α and V are close to each other. Let us denote T ′ a time34uch that d H ,N ( α ( T ) , V ( T ′ )) ≤ κ , then ∆ h ( α ( T ) , V ( T ′ )) ≤ κ , hence ∣ T ′ − M ∣ ≤ C + K + κ .Then by the triangular inequality: d H ,N ( α ( T ) , V ( M − κ )) ≤ d H ,N ( α ( T ) , V ( T ′ )) + d H ,N ( V ( T ′ ) , V ( M − κ ))≤ κ + C + K + κ + κ = C + K + κ Let us denote κ ′ ∶= C + K + κ ≥ κ and T ∶= M − κ . Hence α ∶ [ T ; +∞[→ H is in the κ ′ -neighbourhood of a vertical geodesic ray V ∶ [ T ∶ +∞[→ H , we have d H ,N ( α ( T ) , V ( T )) ≤ κ ′ and κ ′ depends only on δ and K . Lemma 5.9.
Let N be an admissible norm and let α ∶ R + → H be a geodesic ray of (H , d H ,N ) . Then α changes its C -coarse monotonicity at most once.Proof. Let α ∶ R + → H be a geodesic ray. Thanks to Lemma 5.5 α changes at most twice of C -coarsemonotonicity. Indeed, assume it changes three times, applying Lemma 5.5 on the geodesic segmentwhich includes these three times provides a contradiction. We will show in the following that it actu-ally only changes once.Assume α changes twice of C -coarse monotonicity. Then α must be first C -coarsely increas-ing or C -coarsely decreasing. We assume without loss of generality that α is first C -coarselydecreasing. Then there exist t , t , t ∈ R such that α is C -coarsely decreasing on [ α ( t ) , α ( t )] then C -coarsely increasing on [ α ( t ) , α ( t )] then C -coarsely decreasing on [ α ( t ) , α (+∞)[ .Hence Lemma 5.8 applied on [ α ( t ) , α (+∞)[ implies that there exists κ > depending only on δ (sincethe constant of coarse monotonicity depends only on δ ) and a vertical geodesic ray V = ( V p , V q ) suchthat [ α ( t ) , α (+∞)[ is in the κ -neighbourhood of V . Since h + ([ α ( t ) , α (+∞)[) < +∞ , we have that lim t →+∞ h ( α ( t )) = −∞ , hence there exists t ≥ t such that h ( α ( t )) ≤ h ( α ( t )) − C . Then Lemma 5.5tells us that α is first C -coarsely increasing, which contradicts what we assumed.We have classified the possible shapes of geodesic rays. Since geodesics lines are two geodesic raysglued together, we will be able to classify their shapes too. Definition 5.10.
Let N be an admissible norm and let α = ( α p , α q ) ∶ R → H be a path of (H , d H ,N ) .Let κ ≥ .1. α is called H p -type at scale κ if and only if:(a) α p is in a κ -neighbourhood of a geodesic of H p (b) α q is in a κ -neighbourhood of a vertical geodesic of H q .2. α is called H q -type at scale κ if and only if:(a) α q is in a κ -neighbourhood of a geodesic of H q (b) α p is in a κ -neighbourhood of a vertical geodesic of H p . The H p -type paths follow geodesics of H p , meaning that they are close to a geodesic in a copy of H p inside H . The H q -type paths follow geodesics of H q . Remark 5.11.
In a horospherical product, being close to a vertical geodesic is equivalent to be both H p -type and H q -type. Theorem 5.12.
Let N be an admissible norm. There exists κ ≥ depending only on δ and N such thatfor any α ∶ R → H geodesic of (H , d H ,N ) at least one of the two following statements holds.1. α is a H p -type geodesic at scale κ of (H , d H ,N ) α is a H q -type geodesic at scale κ of (H , d H ,N ) H p H qH p − type H q − typeV ertical geodesic Figure 12: Different type of geodesics in
H = H p & H q . Proof.
It follows from Lemma 5.9 that α changes its coarse monotonicity at most once. Otherwise therewould exist a geodesic ray included in α that changes at least two times of coarse monotonicity. Wecut α in two coarsely monotone geodesic rays α ∶ [ , +∞[→ H and α ∶ [ , +∞[→ H such that up to aparametrization α ( ) = α ( ) and α ∪ α = α . By Lemma 5.8 there exists κ and κ depending onlyon δ such that α is in the κ -neighbourhood of a vertical geodesic ray V = ( V ,p , V ,q ) ∶ [ +∞[→ H and such that α is in the κ -neighbourhood of a vertical geodesic ray V = ( V ,p , V ,q ) ∶ [ +∞[→ H .This lemma also gives us d H ,N ( α ( ) , V ( )) ≤ κ and d H ,N ( α ( ) , V ( )) ≤ κ .Assume that lim t →+∞ h ( V ,p ( t )) = lim t →+∞ h ( V ,p ( t )) = +∞ , then they are both vertical rays hence are closeto a common vertical geodesic ray. Furthermore lim t →+∞ h ( V ,q ( t )) = lim t →+∞ h ( V ,q ( t )) = −∞ in that case.Let W q be the non continuous path of H q defined as follows. W q ( t ) = { V ,q (− t ) ∀ t ∈] − ∞ ; 0 ] V ,q ( t ) ∀ t ∈] +∞[ We now prove that W q ∶ R → H q is a quasigeodesic of H q . Let t and t be two real numbers. Since V ,q and V ,q are geodesics, d H q ( W q ( t ) , W q ( t )) = ∣ t − t ∣ if t and t are both non positive or bothpositive. Thereby we can assume without loss of generality that t is non positive and that t is positive.We also assume without loss of generality that ∣ t ∣ ≥ ∣ t ∣ . The quasi-isometric upper bound is given by: d H q ( W q ( t ) , W q ( t )) = d H q ( V ,q (− t ) , V ,q ( t ))≤ d H q ( V ,q (− t ) , V ,q ( )) + d H q ( V ,q ( ) , V ,q ( )) + d H q ( V ,q ( ) , V ,q ( t ))≤ ∣ t ∣ + κ + κ + ∣ t ∣≤ ∣ t − t ∣ + κ + κ , since t and t have different signs.It remains to prove the lower bound of the quasi-geodesic definition on W q . d H q ( W q ( t ) , W q ( t )) = d H q ( V ,q (− t ) , V ,q ( t ))≥ C N d H ,N ( V (− t ) , V ( t )) − d H p ( V ,p (− t ) , V ,p ( t ))≥ C N d H ,N ( α ( t ) , α ( t )) − κ + κ C N − d H p ( V ,p (− t ) , V ,p ( t )) . (48)The Busemann assumption on H p provides us with: d H p ( V ,p (− t ) , V ,p (− t )) ≤ d H p ( V ,p ( ) , V ,p ( )) ≤ κ + κ . α is a geodesic and by using the triangular inequality on (48) we have: d H q ( W q ( t ) , W q ( t )) ≥ ∣ t − t ∣ C N − d H p ( V ,p (− t ) , V ,p (− t )) − d H p ( V ,p (− t ) , V ,p ( t )) − κ + κ C N ≥ ∣ t − t ∣ C N − ∆ h ( V ,q (− t ) , V ,q ( t ) − ( C N + ) ( κ + κ ) . Assume that ∆ h ( V ,q (− t ) , V ,q ( t )) ≤ ∣ t − t ∣ C N , then: d H q ( W q ( t ) , W q ( t )) ≥ ∣ t − t ∣ C N − ( C N + ) ( κ + κ ) . Hence W q is a ( C N , ( C N + ) ( κ + κ )) quasi-geodesic, which was the remaining case. Since κ and κ depend only on δ and N , there exists a constant κ ′ depending only on δ and N such that V ,q ∪ V ,q is in the κ ′ -neighbourhood of a geodesic of H q . The geodesic α is a H q -type geodesic in this case.Assume lim t →+∞ h ( V ,p ( t )) = lim t →+∞ h ( V ,p ( t )) = −∞ , we prove similarly that α is a H p -type geodesic.If a geodesic is both H p -type at scale κ and H q -type at scale κ , then it is in a κ -neighbourhood ofa vertical geodesic of H . H We will now look at the visual boundary of our horospherical products. This notion is described for theSOL geometry in the work of Troyanov [12, Troyanov] through the objects called geodesic horizons.We extend one of the definitions presented in page 4 of [12, Troyanov] for horospherical products.
Definition 5.13.
Two geodesic of a metric space X are called asymptotically equivalent if they are atfinite Hausdorff distance from each other. Definition 5.14.
Let X be a metric space and let o be a base point of X . The visual boundary of X is theset of asymptotic equivalence classes of geodesic rays α ∶ R + → such that α ( ) = o . It is denoted ∂ o X . We will use a result of [11, Papadopoulos] to describe the visual boundary of horospherical products.
Property 5.15 (Property . . p.234 of [11]) . Let X be a proper Busemann space, let q be a point in X and let r ∶ [ , +∞[→ X be a geodesic ray. Then, there exists a unique geodesic ray r ′ starting at q that isasymptotic to r . Theorem 5.16.
Let N be an admissible norm. We fix base points and directions ( w p , a p ) ∈ H p × ∂H p , ( w q , a q ) ∈ H q × ∂H q . Let H = H p & H q be the horospherical product with respect to ( w p , a p ) and ( w q , a q ) .Then the visual boundary of (H , d H ,N ) with respect to a base point o = ( o p , o q ) is given by: ∂ o H =(( ∂H p ∖ { a p }) × { a q }) ⋃ ({ a p } × ( ∂H q ∖ { a q }))=(( ∂H p × { a q }) ⋃ ({ a p } × ∂H q )) ∖ {( a p , a q )} The fact that ( a p , a q ) is not allowed as a direction in H is understandable since both heights in H p and H q would tend to +∞ , which is impossible by the definition of H . Proof.
Let α be a geodesic ray. Lemma 5.9 implies that there exists t ∈ R such that α is coarselymonotone on [ t , +∞[ . Then Lemma 5.8 tells us that α ([ t , +∞[) is at finite Hausdorff distance froma vertical geodesic ray V = ( V p , V q ) , hence α is also at finite Hausdorff distance from V . Since H p is Busemann and proper, Property 5.15 ensure us there exists V ′ p a vertical geodesic ray such that V p and V ′ p are at finite Hausdorff distance with V ′ p ( ) = o p . Similarly, there exists V ′ q a vertical geodesicray of H q with V ′ q ( ) = o q such that V q and V ′ q are at finite Hausdorff distance. Since there is at least37ne vertical geodesic ray V ′ = ( V ′ q , V ′ p ) in every asymptotic equivalence class of geodesic rays, ∂ o H isthe set of asymptotic equivalence classes of vertical geodesic rays starting at o . Hence an asymptoticequivalence class can be identified by the couple of directions of a vertical geodesic ray. Then ∂ o H canbe identified to: (( ∂H p ∖ { a p }) × { a q }) ⋃ ({ a p } × ( ∂H q ∖ { a q })) . the union between downward directions and upward directions, which proves the theorem. Example 5.17.
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