Sublinearly Morse boundaries from the viewpoint of combinatorics
aa r X i v : . [ m a t h . G T ] J a n SUBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OFCOMBINATORICS
MERLIN INCERTI-MEDICI AND ABDUL ZALLOUM
Abstract.
We prove that the sublinearly Morse boundary of every known cubulated group con-tinuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for allCAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorialdescription using the hyperplanes crossed by such sequences. As an application of this combinato-rial description, we show that a certain subspace of the Roller boundary continously surjects on thesubspace of the visual boundary consisting of sublinearly Morse geodesic rays. Introduction
Main Results.
Boundaries at infinity are a common tool in the study of large scale geometricproperties. When a group acts geometrically on a metric space and the space satisfies some suit-able curvature conditions, the boundary carries rich topological, dynamic, metric, quasi-conformal,measure-theoretic and algebraic structures that allow us to study the group we started with. A partic-ularly fruitful instance of this is the case of a Gromov-hyperbolic metric space and its visual boundary.Since any quasi-isometry between hyperbolic metric spaces induces a homeomorphism on the visualboundary, we can define the visual boundary of a hyperbolic group as a topological space up to home-omorphism (see [Gro87]). This is no longer true, when the group is acting on a non-positively curvedspace, e. g. a CAT(0) space. In [CK00], Croke and Kleiner provided an example of a group actinggeometrically on two different quasi-isometric CAT(0) spaces which have non-homeomorphic visualboundaries. So we cannot associate the visual boundary as a topological space up to homeomorphismto a group.Several attempts have recently been made to rectify this issue, most recent of which is work of Qing,Rafi and Tiozzo where they introduce the sublinearly Morse boundary and show that it’s a metrizablequasi-isometry invariant of any CAT(0) space [QRT19]. Given a sublinear function κ , a geodesic ray b is said to be κ -Morse if there exists a function m b : R + × R + → R + such that for any ( q , Q )-quasigeodesic α with end points on b , if p ∈ α, we have d ( p, b ) ≤ m b ( q , Q ) κ ( || p || ), where || p || = d ( p, b (0)) . Elements of the κ -Morse boundary are in a one to one correspondence with κ -Morse geodesic raysstarting at a fixed point o ∈ X. The advantage of the κ -Morse boundary over other hyperbolic-likeboundaries is that it’s generic with respect to various measures (see Subsection 1.3 on the history ofsuch boundaries). Therefore, statements about the κ -Morse boundary could be regarded as statementsabout almost every (quasi)-geodesic ray in X . Since the κ -Morse boundary of a CAT(0) space X is aquasi-isometry invariant, we may denote it by ∂ κ G whenever G is a group acting geometrically on X .Our first result is that if a group G admits a geometric action on a CAT(0) cube complex with a factorsystem, then the κ -Morse boundary of G continuously injects in the Gromov boundary of a certaingeodesic hyperbolic graph Γ . We remark that all known proper cocompact cube complexes admit afactor system, see [HS20] by Hagen and Susse.
Theorem 1.1.
Let G be a group acting geometrically on a CAT(0) cube complex X with a factorsystem. There exist a δ -hyperbolic graph Γ , depending only on X , and a projection map p : X → Γ such that for any sublinear function κ the following holds:(1) Every κ -Morse geodesic ray in X projects to an infinite unparameterized quasi-geodesic in Γ . (2) The κ -Morse boundary of G denoted ∂ κ G continuously injects in the Gromov boundary ∂ Γ . When G is a right-angled Artin group, the graph Γ is the contact graph of its corresponding Salvetticomplex. We remark that in the special case of the Morse boundary ( κ = 1), the above theorem is known tohold in the setting of hierarchically hyperbolic spaces due to Abbott, Behrstock, and Durham [ABD].More precisely, they show that if X is a hierarchically hyperbolic space with mild assumptions, thenthere exists a δ -hyperbolic space Y such that every Morse geodesic ray in X projects to an infiniteunparameterized quasi-geodesic in Y defining a continuous injection between the Morse boundary of X and the Gromov boundary of Y. Therefore, it’s natural to wonder if the same conclusion of Theorem1.1 holds in the settings of hierarchically hyperbolic spaces. Using work of Rafi and Verberne [RV18],we show that the answer to this questions is negative.
Proposition 1.2.
There exists a hierarchically hyperbolic space X such that the following holds. If Y is the maximal δ -hyperbolic space associated to X , and p : X → Y is the projection map to Y , thenall Morse geodesic rays in X project to infinite unparameterized quasi-geodesics in Y . Nonetheless,there exists a κ -Morse geodesic ray in X which projects to an infinite diameter set that is not anunparameterized quasi-geodesic in Y . Denote the boundary of CAT(0) space X as a set by ∂X . On the other hand, when we equip thisset ∂X with the standard cone topology, we denote it by ∂ ∞ X and refer to it as the visual boundary.When considering a CAT(0) cube complex X , one can define the following topology on the set ∂X :Fix a vertex o ∈ X as a base point and let h , . . . , h n be distinct hyperplanes in X . V o ,h ,...,h n := { ξ ∈ ∂X | The unique geodesic representative of ξ basedat o crosses the hyperplanes h , . . . , h n } . The collection B = { V o ,h ,...,h n | n ∈ N , h , h , .., h n are hyperplanes } forms the basis of a topologywhich we denote HYP .It is worth noting that the
HYP -topology is usually very different from the cone topology. Forexample, in R , the cone topology on ∂ R gives a circle, however, the HYP -topology on ∂ R is noteven Hausdorff.While the cone topology is different from the HYP -topology on the set ∂X , our second main resultstates that the two topologies agree when restricted to the subset of ∂X consisting of all κ -Morsegeodesic rays, we denote this subset by ∂ κ X ⊆ ∂X. Theorem 1.3.
Let X be a finite dimensional CAT(0) cube complex and let κ be a sublinear function.The restrictions of the cone topology and HYP to the subset ∂ κ X ⊂ ∂X are equal. The combinatorial nature of
HYP enables us to understand some connections between the κ -Morseboundary and the Roller boundary ∂ R X (see Section 2.5 for a precise definition). The Roller boundarycan be thought of as a combinatorial counterpart to the visual boundary. Fixing a vertex o as basepoint in X , every point x in the Roller boundary can be represented by a combinatorial geodesic ray α x in the 1-skeleton of X that starts at o . In general, such a combinatorial geodesic ray doesn’t uniquelydetermine a point in the visual boundary of X. However, we show that when the combinatorial geodesicray α x is κ -Morse, every other associated combinatorial geodesic ray β x is also κ -Morse, and that x determines a unique point in the visual boundary. We denote the set of points in the Roller boundarythat can be represented by a κ -Morse combinatorial geodesic ray by ∂ κR X . We also denote the subsetof the visual boundary consisting of points that can be represented by κ -Morse geodesic rays by ∂ κ ∞ X when equipped with the subspace topology. Theorem 1.4.
Let X be a finite dimensional CAT(0) cube complex. There exists a well-definedcontinuous surjective map
Φ : ∂ κR X → ∂ κ ∞ X . For all x, y ∈ ∂ κR X , we have that Φ( x ) = Φ( y ) if andonly if [ x | y ] o = ∞ . (See Definition 2.39 for a definition of [ ·|· ] o ).Furthermore, the induced quotient map Φ : ∂ κR X → ∂ κ ∞ X is a homeomorphism. UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 3
We remark that the above theorem provides an extension of Theorem D of Beyrer, Fioravantiin [BF18] and Theorem 1.1 of Zalloum in [Zal18]. More precisely, Theorem D in [BF18] shows Theorem1.1 for the special case where κ = 1 (the Morse boundary of a CAT(0) cube complex). On the otherhand, Theorem 1.1 in [Zal18] shows that for any proper geodesic metric space X , an appropriatesubspace of the horofunctions boundary of X continuously surjects on the Morse boundary of X .When X is a CAT(0) cube complex, the horofunctions boundary with respect to the l -metric of X isexactly the Roller boundary (Proposition 6.20 of [FLM18] by Fern´os, L´ecureux and Math´eus).1.2. Sketch of the proof of Theorem 1.1.
We fix a finite dimensional CAT(0) cube complex X .A collection of three disjoint hyperplanes in X is said to form a facing triple if none of them separatesthe other two. Two disjoint hyperplanes h , h in X are said to be k -well-separated if the number ofhyperplanes meeting them both and containing no facing triple is bounded above by k (for a detailedcomparison between the notion of k -well-separation versus the Charney and Sultan’s k -separationnotion, see Section 1.2 of [MQZ20]). In [Gen20b], starting with a CAT(0) cube complex X , AnthonyGenevois constructs a (non-geodesic) hyperbolic metric space ( Y k , d k ) where Y k is the collection ofvertices X (0) and d k ( x, y ) is the cardinality of the maximal collection of k -well-separated hyperplanesseparating x, y , in fact, he shows that when k = 0 , the space Y k is quasi-isometric to the contactgraph of Hagen [Hag13]. An essential result due to Genevois in [Gen20b] is that for a cocompactCAT(0) cube complex X with a factor system, there exists an L depending only on X , such that if twohyperplanes are k -well-separated for some k, then k ≤ L. That is to say, two hyperplanes are eithernot well-separated or L -well-separated. We will refer to this constant L as the separation constant of X . Let G be a group admitting a geometric action on a CAT(0) cube complex X with a factor system,and let L be the separation constant, our steps for the proof are as follows:(1) Denote ∂ κ X the quasi-isometry invariant κ -Morse boundary of X given in [QRT19], and ∂ κ ∞ X the subspace of the visual boundary of X consisting of κ -Morse geodesic rays with the subspacetopology. We first show that the map ι : ∂ κ X → ∂ κ ∞ X, is continuous. That is to say, the quasi-isometry invariant topology of Qing, Rafi and Tiozzo[QRT19] is finer than the subspace topology induced on the collection of κ -Morse geodesicrays. This is Lemma 2.12.(2) We construct a graph (Γ , d Γ ) which is bilipschitz equivalent to ( Y L , d L ) . The graph is built asfollows: the vertices are the same as the vertices of Y L which are simply the vertices of X (0) ,and we connect two vertices x, y with an edge if the number of L -well-separated hyperplanesseparating them is at most 10 L + 4 . This is Lemma 6.14.(3) We show that every (CAT(0) or combinatorial) geodesic of X projects to an unparameterizedquasi-geodesic in ( Y L , d L ) and hence to (Γ , d Γ ) . Since the argument for this is short and combi-natorial, we sketch it here. Let c be a geodesic in X and let x, y, z be the points c ( t ) , c ( t ) , c ( t )for t < t < t . Suppose that d k ( x, y ) = L and that d k ( y, z ) = L . That is to say, if H , H are the maximal collections of L -well-separated hyperplanes separating x, y and y, z respec-tively, then |H | = L and |H | = L . We will think of the sets H , H as ordered sets basedon the order they are crossed by c. We will show that d L ( x, z ) ≥ d L ( x, y ) + d L ( y, z ) − ( L + 2) . Let h last denote the last hyperplane in H and h ′ , h ′ denote the first and second hyperplanesof H . If h last and h ′ are disjoint, there is nothing to prove. Otherwise, if h ′ crosses somecollection of H , then h ′ must also cross the same collection as h ′ , h ′ are disjoint. However,since the collection H consists of L -well-separated hyperplanes, there can be at most L hy-perplanes crossed by both h ′ , h ′ (see Figure 1). Denote such hyperplanes h , h , . . . , h L . Theset H ∪ H \ { h , h , . . . , h L , h ′ , h ′ } is a collection of L -well-separated hyperplanes separat-ing x, z whose cardinality is at least L + L − ( L + 2). This shows that a geodesic c in X projects to an unparameterized quasi-geodesic in Y L , and by step 2, the geodesic c projects toan unparameterized quasi-geodesic in Γ. This is Lemma 6.7. MERLIN INCERTI-MEDICI AND ABDUL ZALLOUM | {z } L h ′ h ′ Figure 1.
The first two blue hyperplanes h ′ , h ′ can cross at most L of the redhyperplanes.(4) In [MQZ20], Murray, Qing and Zalloum show that a geodesic ray is κ -Morse if and only if thereexists a constant n such that c crosses an infinite sequence of hyperplanes h i at points c ( t i )with d ( t i , t i +1 ) ≤ n κ ( t i +1 ) and h i , h i +1 are L -well-separated. Hence, by part (3), a κ -Morsegeodesic ray projects to an infinite unparameterized quasi-geodesic in Γ . This defines a map ι : ∂ κ ∞ X → ∂ Γ , we show this map is continuous and injective. Continuity and injectivity ofthe map are Lemma 6.18 and Lemma 6.12 respectively.(5) Composing the maps from step (1) and step (4), we get a continuous injection ι = ι ι ι : ∂ κ X → ∂ Γ . Since ∂ κ X is independent of the space X on which G admits a geometric action, it may bedenoted by ∂ κ G. History of hyperbolic-like boundaries and cube complexes.
The κ -Morse boundary ofQing, Rafi and Tiozzo was preceded by a few other hyperbolic-like boundaries which are also invariantunder quasi-isometries. Charney and Sultan introduced the Morse boundary for CAT(0) spaces whosehomeomorphism class is invariant under quasi-isometry (see [CS15b]). The Morse boundary was latergeneralized by Cordes to all proper geodesic metric spaces (see [Cor17]). In either case, this boundaryconsists of equivalence classes of geodesic rays that have similar properties to geodesic rays in hyperbolicspaces, indicating that these are the ‘hyperbolic’ directions in the space under consideration.The Morse boundary of Charney-Sultan and Cordes is equipped with a quasi-isometry invarianttopology by using a direct limit construction. This direct limit construction was later refined to adirect limit of Gromov hyperbolic boundaries by Cordes and Hume (see [CH17]). Group actions ontheir Morse boundaries have similar dynamical properties to that of the action of a hyperbolic groupon it’s Gromov’s boundary (see [Liu19] by Liu). However, this topology is not first countable ingeneral. Concretely, this topology is not first countable for the group Z ∗ Z , an example also knownas the ‘tree of flats’ (see [Mur19] by Murray). Cashen and Mackay introduced a refined topology forthe Morse boundary (see [CM19]) and were able to show that the Morse boundary, with respect totheir topology is a metrizable quasi-isometry invariant. The κ -Morse boundary was introduced byQing, Rafi and Tiozzo in [QRT19] and sought to rectify some of the shortcomings found in the Morseboundary, for example, unlike the Morse boundary of Charney-Sultan, Cordes and Cashen-Mackay,the κ -Morse boundary serves as a topological model of the Poisson boundary for right-angled Artingroups (see [QT19] and [QRT19]). Furthermore, it was recently announced by Gekhtman, Qing andRafi [GQR] that κ -Morse geodesics are generic in rank-1 CAT(0) spaces in various natural measures.In this article, we restrict our attention to the κ -Morse boundary of CAT(0) cube complexes. Cubecomplexes were introduced by Gromov in [Gro87] and have become a central object in geometricgroup theory over the last decade due to their fruitfulness in solving problems in group theory andlow-dimensional topology and due to the fact that many interesting groups are cubulable, i. e. they actproperly and cocompactly on a CAT(0) cube complex. The class of groups that are cubulable includes UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 5
Right-angled Artin groups, hyperbolic 3-manifold groups ( [BW12]), most non-geometric 3-manifoldgroups ( [PW14], [HP15], [PW18]), small cancelation groups ( [Wis04]) and many others. Cubulatedgroups played a key role in Agol’s and Wise’s proof of the virtual Haken and virtual fibered conjecture( [Wis11], [Ago13]). The additional structure of CAT(0) cube complexes allows us to study sublinearlyMorse boundaries via more combinatorial methods.
Outline of the paper.
Section 2 contains some background material on CAT(0) spaces, cube com-plexes, sublinearly Morse boundaries, Roller boundaries and the various topologies these boundariesare equipped with. In Section 3 we show that any two combinatorial geodesic rays which are crossed byan infinite sequence of hyperplanes must fellow travel, provided that at least one of them is κ -Morse.This will be essential to establishing well-definness of the map in Theorem 1.4. Section 4 proves The-orem 1.3 giving a combinatorial description of convergence to sublinearly Morse geodesic rays, thiswill be used in proving continuity of the map in Theorem 1.4. In Section 5, we prove Theorem 1.4.In Section 6, we describe the hyperbolic graph Γ and we prove Theorem 1.1. In Section 7 we proveProposition 1.2. We remark that Sections 6 and 7 are independent of other sections and can be readseparately. Acknowledgement.
The authors would like to thank Carolyn Abbott, Ruth Charney, MatthewDurham, Elia Fioravanti, Anthony Genevois, Mark Hagen, Qing Liu, Kasra Rafi and Jacob Russellfor fruitful discussions. 2.
Preliminaries
CAT(0) geometry.Definition 2.1 (Quasi-geodesics) . A geodesic ray in a metric space X is an isometric embedding b : [0 , ∞ ) → X . We fix a base-point o ∈ X and always assume that b (0) = o , that is, a geodesicray is always assumed to start from this fixed base-point. A quasi-geodesic ray is a continuous quasi-isometric embedding β : [0 , ∞ ) → X starting from o . The additional assumption that quasi-geodesicsare continuous is not necessary, but is added to make the proofs simpler.Let ( X, d X ) be a metric space. A metric space is called proper if closed balls are compact. It iscalled geodesic if any two points x, y ∈ X can be connected by a geodesic segment. A proper, geodesicmetric space ( X, d X ) is CAT(0) if geodesic triangles in X are at least as thin as triangles in Euclideanspace with the same side lengths. To be precise, for any given geodesic triangle △ pqr , consider theup to isometry unique triangle △ pqr in the Euclidean plane with the same side lengths. For anypair of points x, y on the triangle, for instance on edges [ p, q ] and [ p, r ] of the triangle △ pqr , if wechoose points x and y on edges [ p, q ] and [ p, r ] of the triangle △ pqr so that d X ( p, x ) = d E ( p, x ) and d X ( p, y ) = d E ( p, y ), then d X ( x, y ) ≤ d E ( x, y ) . For the remainder of the paper, we assume (
X, d ) is a proper CAT(0) space. Here, we list someproperties of proper CAT(0) spaces that are needed later (see [BH99]).
Lemma 2.2.
A proper
CAT(0) space X has the following properties:(1) It is uniquely geodesic, that is, for any two points x, y in X , there exists exactly one geodesicconnecting them.(2) The nearest-point projection from a point x to a geodesic line b is a unique point denoted x b .The closest-point projection map π b : X → b is Lipschitz.(3) Convexity: For any convex set Z ∈ X , the distance function f : X → R + given by f ( x ) = d ( x, Z ) is convex. MERLIN INCERTI-MEDICI AND ABDUL ZALLOUM
The cone topology on the visual boundary.Definition 2.3 (space of geodesic rays) . As a set, the boundary of X , denoted by ∂X is defined to bethe collection of equivalence classes of all infinite geodesic rays. Let b and c be two infinite geodesicrays, not necessarily starting at the same point. We define an equivalence relation as follows: b and c are in the same equivalence class, if and only if there exists some n ≥ d ( b ( t ) , c ( t )) ≤ n forall t ∈ [0 , ∞ ) . We denote the equivalence class of a geodesic ray b by b ( ∞ ) . Notice that by Proposition 8.2 in the CAT(0) boundary section of [BH09], for each b representing anelement of ∂X , and for each x ′ ∈ X , there is a unique geodesic ray b ′ starting at x ′ with b ( ∞ ) = b ′ ( ∞ ) . Now we describe the cone topology on this boundary:
Definition 2.4. (Cone topology) Fix a base point o and let b be a geodesic ray starting at o . Aneighborhood basis for b ( ∞ ) is given by sets of the form: V r,ǫ ( b ( ∞ )) := { c ( ∞ ) ∈ ∂X | c (0) = o and d ( b ( t ) , c ( t )) < ǫ for all t < r } . The topology generated by the above neighborhood basis is called the the cone topology and denotedby
CON E . When we equip ∂X with CON E , we denote it ∂ ∞ X and call it the visual boundary. Noticethat two geodesic rays are close together in the cone topology if they have representatives starting atthe same point which stay close (are at most ǫ apart) for a long time (at least r ).While the above definition of the visual boundary ∂ ∞ X made a reference to a base point o , Proposi-tion 8.8 in the CAT(0) boundaries section of [BH09] shows that the topology of the visual boundary isa base point invariant. It’s also worth mentioning that when X is proper, then the space X = X ∪ ∂ ∞ X is compact.2.3. Sublinearly Morse geodesics.
In this section we review the definition and properties of κ -Morse geodesic rays needed for this paper. For further details, see [QRT19]. We fix a function κ : [0 , ∞ ) → [1 , ∞ )that is monotone increasing, concave, and sublinear, that islim t →∞ κ ( t ) t = 0 . Note that using concavity, for any a >
1, we have(1) κ ( at ) ≤ a (cid:18) a κ ( at ) + (cid:18) − a (cid:19) κ (0) (cid:19) ≤ a κ ( t ) . It is worth noting that the assumption that κ is increasing and concave makes certain argumentscleaner, otherwise they are not really needed. One can always replace any sublinear function κ , withanother sublinear function κ ′ so that κ ( t ) ≤ κ ′ ( t ) ≤ c κ ( t ) for some constant c and κ ′ is monotoneincreasing and concave. Notation 2.5.
We fix a base point o ∈ X once and for all and denote for all x ∈ X , || x || := d ( o , x ).To simplify notation, we often drop k·k . That is, for x ∈ X , we define κ ( x ) := κ ( k x k ) . If x, y are within κ ( x ) of each other, then κ ( x ) and κ ( y ) are multiplicatively the same. Proposition 2.6 ( [QRT19, Lemma 3.2]) . For any d > , there exists d , d > depending only on d and κ so that for x, y ∈ X , we have d ( x, y ) ≤ d κ ( x ) = ⇒ d κ ( x ) ≤ κ ( y ) ≤ d κ ( x ) . Definition 2.7. ( κ -contracting) A geodesic ray b is said to be κ -contracting if there exists a constant n ≥ B centered at x with B ∩ b = ∅ , we have diam ( π b ( B )) < n κ ( x ) . UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 7 o bxx b n · κ ( || x || ) || x || ( κ, n )–neighbourhood of b Figure 2. A κ -neighbourhood of a geodesic ray b with multiplicative constant n . Definition 2.8 ( κ –neighborhood) . For a closed set Z and a constant n define the ( κ, n )–neighbourhoodof Z to be N κ ( Z, n ) = n x ∈ X (cid:12)(cid:12)(cid:12) d X ( x, Z ) ≤ n · κ ( || x || ) o . Definition 2.9 ( κ –fellow travelling) . Let α and β be two quasi-geodesic rays in X . If α is containedin some κ –neighbourhood of β and β is contained in some κ –neighbourhood of α , we say that α and βκ –fellow travel each other. This defines an equivalence relation on the set of quasi-geodesic rays in X . Definition 2.10 ( κ –Morse) . A set Z is κ –Morse if there is a function m Z : R → R + so that if β : [ s, t ] → X is a ( q , Q )–quasi-geodesic with end points on Z then β [ s, t ] ⊂ N κ (cid:0) Z, m b ( q , Q ) (cid:1) . We refer to m Z as the Morse gauge for Z . For convenience, we may always assume that m Z ( q , Q ) isthe largest element in the set { q , Q , m Z ( q , Q ) } . It is worth noting that when κ = 1 we recover thestandard definition of a Morse set . For a good reference on Morse geodesics, see [Cor19] by Cordes.We remark that we will be mostly interested in the case where the set Z is a geodesic ray b . Definition 2.11 (Sublinearly Morse boundary) . Let κ be a sublinear function and let X be a CAT(0)space. We define the κ -Morse boundary, as a set, by ∂ κ X := { all κ -Morse quasi-geodesics } /κ -fellow travellingThe κ -Morse boundary is equipped with the following topology. Fix a base point o , let ξ ∈ ∂ κ X , andlet b be the unique geodesic representative of ξ that starts at o . For all r >
0, we define U κ ( b, r ) to bethe set of all points η ∈ ∂ κ X such that for every ( q , Q )-quasi-geodesic β representing η , starting at o ,and satisfying m b ( q , Q ) ≤ r κ ( r ) , we have β | [0 ,r ] ⊂ N κ ( b, m b ( q , Q )) (see [QRT19]). We will denote thetopology induced by these neighbourhood bases by SM .We compare the SM -topology to the cone topology induced by the visual boundary: Lemma 2.12. ( CON E ⊆ SM over κ -Morse rays) There is an injective map i of the set of pointsin ∂ κ X into the visual boundary ∂ ∞ X . Furthermore, the map i : ∂ κ X → ∂ ∞ X is continuous. Whenequipping the image of this map with the subspace topology we denote it ∂ κ ∞ X . In other words, ∂ κ ∞ X is the subspace of ∂ ∞ X consisting of all κ -Morse geodesic rays emanating from a fixed based point. We note that these topologies are likely not equal in general. This stems from the fact that SM hasa lot of similarities with the topology FQ that Cashen-Mackay introduced for the Morse boundary (see[CM19]). Thus, the example that shows that CON E 6 = FQ in [IM20] likely generalises to sublinearlyMorse boundaries. Proof.
The proof of the existence of the set map is immediate by the first part of Theorem 2.35. Notethat, if two topologies T , T ′ are defined by neighbourhood bases, we have that, whenever every basicneighbourhood U of a point x with respect to T admits a basic neighbourhood U ′ of x with respect MERLIN INCERTI-MEDICI AND ABDUL ZALLOUM to T ′ such that U ′ ⊆ U , then T ′ ⊇ T . Our strategy is to find a suitable neighbourhood basis for thetopology CON E .Fix a base point o ∈ X . Let ξ ∈ ∂ κ X , R, ǫ >
0, and ξ o the unique geodesic representative of ξ starting at o . By definition of the cone topology, the sets V R,ǫ ( ξ ) := { η ∈ ∂ κ X | d ( η o ( R ) , ξ o ( R )) < ǫ } define a neighbourhood basis of the cone topology pulled back via the inclusion i . We also define theset V κ ( ξ o , r ) to be the set of all points η ∈ ∂ κ X such that η o | [0 ,r ] ⊂ N κ ( ξ o , m ξ o (1 , r satisfying m ξ o (1 , ≤ r κ ( r ) , we have U κ ( ξ o , r ) ⊆ V κ ( ξ o , r ). We conclude that, if the sets V κ ( ξ o , r )form a neighbourhood basis for CON E , then
FQ ⊇ CON E .We are left to show that, for r large, we find R ′ , such that V κ ( ξ o , r ) ⊇ V R ′ , ( ξ ), as this wouldprove that the V κ ( ξ o , r ) are a neighbourhood basis for CON E . (The other inclusion is immediateas κ ≥ β be a geodesic ray starting at o , respresenting a point η ∈ ∂ κ X and suppose that β | [0 ,r ] * N κ ( ξ o , m ξ o (1 , κ -neighbourhoods in CAT(0) spaces are convex, this implies that d ( β ( r ) , ξ o ( r )) > m ξ o (1 , κ ( k β ( r ) k ) . Using convexity of distance functions, we conclude that for all r ′ ≥ r , we have d ( β ( r ′ ) , ξ o ( r ′ )) > r ′ r · m ξ o (1 , κ ( k β ( r ) k ) . Choosing r ′ sufficiently large, we find R ′ such that R ′ r · m ξ o (1 , κ ( k β ( r ) k ) > . We conclude that for all η / ∈ V κ ( ξ o , r ), we have η / ∈ V R ′ , ( ξ ). Therefore, V R ′ , ( ξ ) ⊆ V κ ( ξ o , r ). Thisimplies that the sets V κ ( ξ o , r ) form a neighbourhood basis of CON E , and
SM ⊇ CON E . (cid:3) CAT(0) cube complexes.
For a more detailed introduction to CAT(0) cube complexes, see[Sag14]. Our goal for this section is to fix notation and to recall some definitions and facts that willbe used in the remainder of the paper.
Definition 2.13. (Cubes and midcubes) A cube is a Euclidean unit cube [0 , n for some n ≥
0. A midcube of c is a subspace obtained by restricting exactly one coordinate in [0 , n to . Definition 2.14 (Cube complexes) . For n ≥
0, let [0 , n be an n -cube equipped with the Euclideanmetric. We obtain a face of a n –cube by choosing some indices in { , . . . , n } and considering the subsetof all points where we for each chosen index i , we fix the i -th coordinate either to be zero or to be one.A cube complex is a topological space obtained by glueing cubes together along faces, i.e. every gluingmap is an isometry between faces. One may think of a cube complex as a CW-complex whose cellsare equipped with the geometry of Euclidean cubes and whose glueing maps are isometries of faces. Definition 2.15. (local/uniform local finiteness) Let X be a CAT(0) cube complex and x ∈ X (0) avertex. The degree , denoted deg( x ) of x , is defined to be the number of edges incident to x . We havethe following: • X is said to be finite dimensional if there is an integer v X such that whenever [0 , n is a cubein X , we have n ≤ v X . • X is said to be locally finite if every vertex x ∈ X (0) has finite degree. • X is said to be uniformly locally finite if there is an integer u X such that for every vertex x ∈ X (0) , we have deg( x ) ≤ u X .Any cube complex can be equipped with a metric as follows. The n –cubes are equipped with theEuclidean metric, which allows us to define the length of continuous paths inside the cube complex.(Simply partition every path into segments that lie entirely within one cube and use the Euclideanmetric of that cube.) We define d ( x, y ) := inf { length ( γ ) | γ a continous path from x to y } . UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 9
The map d defines a metric on X . We sometimes call d the metric induced by the Euclidean metricon each cube. Definition 2.16 (CAT(0) cube complexes) . Let X be a cube complex and d the metric induced bythe Euclidean metric on each cube. We say that X is a CAT(0) cube complex if and only if ( X, d ) isa CAT(0) space.
Definition 2.17. (Hyperplanes, half spaces and separation) Let X be a CAT(0) cube complex. A hyperplane is a connected subspace h ⊂ X such that for each cube c of X , the intersection h ∩ c is eitherempty or a midcube of c . For each hyperplane h , the complement X \ h has exactly two components,called halfspaces associated to h . We usually denote these h + and h − . If h + is a halfspace, there isa unique hyperplane h such that h + is a component of X \ h , and we also say h is the hyperplaneassociated to h + . A hyperplane h is said to separate the sets U, V ⊆ X if U ⊆ h + and V ⊆ h − . Whenever X is a CAT(0) cube complex, we refer to the metric induced by the Euclidean metric oneach cube as the CAT(0)-metric . In contrast to the CAT(0) metric, we can also equip every n –cubewith the restriction of the l metric of R n and consider the induced path metric d (1) ( · , · ). We refer to d (1) as the combinatorial metric (or l –metric ). The following lemma is a standard lemma about howthe l -metric relates to the CAT(0) metric of a CAT(0) cube complex, for example, see [CS11]. Lemma 2.18. If X is a finite-dimensional CAT(0) cube complex, then the
CAT(0) metric d and thecombinatorial metric d (1) are bi-Lipschitz equivalent and complete. In particular, if all cubes in X have dimension ≤ m , then d ≤ d (1) ≤ √ md . Furthermore, for two vertices x, y ∈ X (0) , we have d (1) ( x, y ) = |{ hyperplanes h ⊆ X which separate the vertices x, y }| . Since we now have two metrics on the CAT(0) cube complex X , we need to distinguish betweenterminologies for the two metrics. A path in X is called a geodesic when it is a geodesic with respectto the CAT(0) metric. Definition 2.19. (Combinatorial geodesics) A path in the 1–skeleton of X is called a combinatorialgeodesics if it is a geodesic between vertices of X with respect to the combinatorial metric. Combina-torial geodesic rays are defined analogously.If all cubes in X have dimension at most m , every combinatorial geodesic is a ( √ m, Lemma 2.20 (Proposition 2.8 in [BF18]) . Let X be a finite-dimensional CAT(0) cube complex ofdimension at most m . Every CAT(0) geodesic ray b starting at a vertex o is at Hausdorff-distance atmost m from a combinatorial geodesic α with the same origin. Furthermore, α can be chosen such thatit crosses the same hyperplanes as b . Notation 2.21.
We adopt the following notations: • We denote the set of all hyperplanes and of all halfspaces of X by W ( X ) and H ( X ) respectively. • Given a halfspace s , denote the other halfspace bounded by the same hyperplane by s ∗ . • Given a hyperplane h , we call a choice of halfspace bounded by h an orientation of h . • We sometimes denote the two orientations of h by { h + , h − } . • Given two subsets U , V ⊂ X , we define W ( U | V ) to be the set of all hyperplanes that separate U from V . Given a (combinatorial) geodesic α , we write W ( α ) for the set of hyperplanescrossed by α .When X is fixed, we will simply use H instead of H ( X ) . The set H is endowed with the orderrelation given by inclusions; the involution ∗ is order reversing. The triple ( H , ⊆ , ∗ ) is thus a pocset (see [Sag14]). Definition 2.22. (Chain). A chain in X is a (possibly infinite) collection of mutually disjoint hyper-planes which are associated to a collection of nested half-spaces. Definition 2.23. (transverse hyperplanes) Two hyperplanes are called transverse if they intersect.Analogously, two halfspaces s , s ′ ∈ H are called transverse , if their bounding hyperplanes are trans-verse. Equivalently, they are transverse if and only if the four intersections s ∩ s ′ , s ∩ s ′∗ , s ∗ ∩ s ′ , s ∗ ∩ s ′∗ are non-empty. A hyperplane is transverse to a halfspace s if it is transverse to the hyperplane thatbounds s .Note that every intersection h ∩ · · · ∩ h k of pairwise transverse hyperplanes h , . . . , h k inherits aCAT(0) cube complex structure. Its cells are precisely the intersections h ∩ · · · ∩ h k ∩ c for any cube c ⊆ X . Alternatively, h ∩ · · · ∩ h k can be viewed as a subcomplex of the cubical subdivision of X . Definition 2.24. (Convex subcomplexes) Let X be a CAT(0) cube complex. We define the following:(1) A subcomplex Y ⊆ X is said to be a convex subcomplex with respect to the CAT(0) metric ifwhenever x, y ∈ Y , we have [ x, y ] ∈ Y where [ x, y ] is the unique CAT(0) geodesic connecting x, y. (2) A subcomplex Y ⊆ X is said to be a convex subcomplex with respect to the combinatorialmetric if whenever x, y ∈ Y , we have [ x, y ] ∈ Y where [ x, y ] denotes the collection of combi-natorial geodesics from x to y. Definition 2.25. (Convex hulls) Let α be a combinatorial geodesic ray in a CAT(0) cube complex X. The convex hull of α, denoted Hull( α ), is defined to be the intersection of all half spaces which containthe combintorial geodesic ray α . Notice that for any combinatorial geodesic ray α , the space Hull ( α )is convex in both the CAT(0) and the combinatorial metric. An essential property of a convex hull ofa combinatorial geodesic ray α is that a hyperplane h meets Hull ( α ) if and only if h meets α. The following is a standard lemma about CAT(0) cube complexes.
Lemma 2.26. (the Helly property) Any finite collection of convex subcomplexes of a finite dimensionalCAT(0) cube complex X satisfy the Helly property. That is to say, any finite collection of pairwiseintersecting convex subcomplexes has nonempty total intersection. Definition 2.27. (Combinatorial projection) Given a subset S ⊂ X that is closed and convex withrespect to the combinatorial metric, we can define the combinatorial projection g S : X → S , whichsends every point x ∈ X to the unique point in S that is closest to S . If x is a vertex and S is aconvex subcomplex, then g S ( x ) can be characterised as the unique vertex s ∈ S , such that for everyhyperplane h ∈ W ( S ) the points s, x are contained in the same halfspace bounded by h . That is tosay, a hyperplane h separates x from it’s combinatorial g S ( x ) if and only if h separates x from S . Sincehyperplanes are closed and convex, they in particular have a combinatorial projection. Definition 2.28. ( HYP -topology on ∂X ) For a CAT(0) cube complex X , recall that ∂X denotes theset of all equivalence classes of CAT(0) geodesic rays modulo finite Hausdorff. We define the followingtopology on the set ∂X : Fix a vertex o ∈ X as a base point and let h , . . . , h n be distinct hyperplanesin X . Define V o ,h ,...,h n := { ξ ∈ ∂X | The unique geodesic representative of ξ basedat o crosses the hyperplanes h , . . . , h n } . The collection B = { V o ,h ,...,h n | n ∈ N , h , h , .., h n are hyperplanes } forms the basis of a topologywhich we denote HYP .We now introduce some notions that are helpful to detect Morse and κ -Morse geodesic rays inCAT(0) cube complexes. Two hyperplanes are said to be k -separated if the number of hyperplanescrossing both of them is bounded from above by k . In [CS15a], Charney and Sultan give the followingcharacterization for 1–Morse geodesic rays in CAT(0) cube complexes. UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 11
Theorem 2.29 ( [CS15a, Theorem 4.2]) . Let X be a uniformly locally finite CAT(0) cube complexand c ≥ . There exist r > , k ≥ (depending only on c and X ) such that a geodesic ray b in X is –Morse with Morse gauge c (cf. Definition 2.10) if and only if b crosses an infinite sequence ofhyperplanes h , h , h , . . . at points x i = b ∩ h i satisfying:(1) h i , h i +1 are k -separated and(2) d ( x i , x i +1 ) ≤ r. Since 1–Morse geodesic rays in CAT(0) spaces always admit a constant function as a Morse gauge,this provides a characterisation of all 1–Morse geodesics in CAT(0) cube complexes. We now providean analogous characterisation of κ -Morse geodesic rays. Definition 2.30 (Facing Triples) . A collection of three hyperplanes h , h , h is said to form a facingtriple if they are disjoint and none of the three hyperplanes separates the other two.Notice that if a (combinatorial or CAT(0)) geodesic b crosses three disjoint hyperplanes h , h , and h in that order, then h separates h and h . In particular, a geodesic b cannot cross a facing triple.Conversely, if C is a collection of hyperplanes which contains no facing triple, then there is a geodesicwhich crosses a definite proportion of the hyperplanes in C : Lemma 2.31 ( [Hag20, Corollary 3.4]) . Let X be a CAT(0) cube complex of dimension at most m .There exists a constant k , depending only on m , such that the following holds. If C is a collection ofhyperplanes which contains no facing triple, then there exists a (combinatorial or CAT(0) ) geodesicwhich crosses at least |C| k hyperplanes from the collection C . Definition 2.32 (Well-separated hyperplanes) . Two disjoint hyperplanes h , h are said to be k -well-separated if any collection of hyperplanes intersecting both h , h , which does not contain any facingtriple, has cardinality at most k . We say that h and h are well-separated if they are k -well-separatedfor some k . Theorem 2.33 ( [MQZ20], Theorem B) . Let X be a finite dimensional CAT(0) cube complex. Ageodesic ray b ∈ X is κ -contracting if and only if there exists c > such that b crosses an infinitesequence of hyperplanes h , h , ... at points b ( t i ) satisfying:(1) d ( t i , t i +1 ) ≤ c κ ( t i +1 ) . (2) h i , h i +1 are c κ ( t i +1 ) -well-separated. Definition 2.34 ( κ -excursion geodesics) . A geodesic ray b ∈ X is said to be a κ -excursion geodesic if there exists c > b crosses an infinite sequence of hyperplanes h , h , ... at points b ( t i )satisfying:(1) d ( t i , t i +1 ) ≤ c κ ( t i +1 ) . (2) h i , h i +1 are c κ ( t i +1 )-well-separated.The constant c will be referred to as the excursion constant and the hyperplanes { h i } will be referredto as the excursion hyperplanes . Theorem 2.35.
Let X be a proper CAT(0) space and let ζ be an equivalence class of κ -fellow travellingquasi-geodesics in X . If ζ contains a κ -Morse quasi-geodesic ray, then ζ contains a unique geodesicray b starting at b (0) . Furthermore, if η is an equivalence class of κ -fellow travelling quasi-geodesics,then the following are equivalent: • η contains a κ -Morse geodesic ray. • η contains a κ -contracting geodesic ray. • η contains a κ -excursion geodesic ray. • Every quasi-geodesic ray in η is κ -Morse. • There exists a quasi-geodesic ray in η which is κ -Morse.In particular, a geodesic ray b is κ -Morse if and only if it is κ -contracting if and only if it is a κ -excursion geodesic. Proof.
The statement that if ζ contains a κ -Morse quasi-geodesic ray, then ζ contains a unique geodesicray b starting at b (0) is precisely Lemma 3.5 and Proposition 3.10 of [QRT19]. The rest of the statementis exactly Theorem 3.8 in [QRT19] and Theorem 2.33. (cid:3) The Roller boundary.
Recall that for a CAT(0) cube complex X , the set H ( X ), or simply H , is the set of all half spaces of X . Given a vertex x ∈ X (0) , we denote by σ x ⊆ H the set of allhalfspaces containing the point x . Definition 2.36. (The Roller boundary) Let ι : X → Q h ∈W { h + , h − } denote the map that takes eachpoint x to the set of all half spaces containing x given by σ x . Endowing Q h ∈W { h + , h − } with theproduct topology, we can consider the closure ι ( X ). Equipped with the subspace topology, the space ι ( X ) is a compact Hausdorff space known as the Roller compactification of X . The Roller boundary ∂ R X is defined as the difference ι ( X ) \ ι ( X ). If X is locally finite, ι ( X ) is open in X and ∂ R X iscompact; however, this is not true in general.The idea of the following definition is to abstract the properties met by the collection of hyperplanes σ x for some x ∈ X (0) Definition 2.37. (Ultrafilters) A collection of half spaces σ ⊆ H is called a DCC ultrafilter if itsatisfies the following three conditions:(1) given any two halfspaces s , s ′ ∈ σ , we have s ∩ s ′ = ∅ ;(2) for any hyperplane h ∈ W , a side of h lies in σ ;(3) every descending chain of halfspaces in σ is finite.We refer to a set σ ⊆ H satisfying only (1) and (2) simply as an ultrafilter . Remark 2.38.
We remark that the image ι ( X ) is precisely the collection of all DCC ultrafilters.Furthermore, ι ( X ) coincides with the set of all ultrafilters. We prefer to imagine ∂ R X as a set ofpoints at infinity, represented by combinatorial geodesic rays in X , rather than a set of ultrafilters.We will therefore write x ∈ ∂ R X for points in the Roller boundary and employ the notation σ x ⊆ H to refer to the ultrafilter representing x .We say that a hyperplane h separates two points x and y in the Roller boundary if σ x and σ y donot contain the same halfspace bounded by h . In other words, they induce opposite orientation on h .Recall that for subsets U, V of X , we use the notation W ( U | V ) to denote the set of all hyperplanesthat separate U from V . Definition 2.39 (Combinatorial Gromov product, cf. [BFIM18]) . Let x, y ∈ ∂ R X and o ∈ X (0) be avertex. We define the combinatorial Gromov product of x and y with respect to o to be[ x | y ] o := W ( o | x, y ) , i.e. the number of hyperplanes h , for which σ o does not contain the same halfspace bounded by h as σ x and σ y . Definition 2.40. (Components) Two points x , y ∈ ∂ R X lie in the same component if and only ifthere are only finitely many hyperplanes that separate x from y . This defines an equivalence relationon ∂ R X and partitions the Roller boundary into equivalence classes, called components. We denotethe component containing a point x ∈ ∂ R X by C ( x ).Each component inherits the structure of a CAT(0) cube complex whose hyperplanes are a strictsubset of the set of hyperplanes of X . We say that a hyperplane k ∈ W ( X ) intersects a component C whenever it corresponds to a hyperplane in C . Note that for any two hyperplanes h , k that intersecta component C , there exist infinitely many h i ∈ W ( X ) that intersect both h and k .Given two components C ( x ) , C ( y ), we say that C ( x ) ≺ C ( y ) if and only if the combinatorialGromov product [ x | y ] o = ∞ and any descending chain of half spaces containing y and not containing x UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 13 terminates. (This partial order was introduced by Guralnik. See for example the appendix of [Gen20a]for more information.)
Definition 2.41. (Median) Let x, y, z ∈ X (0) ∪ ∂ R X . We define the median of the triple x, y, z tobe the unique point m ( x, y, z ) ∈ X (0) ∪ ∂ R X obtained by orienting every hyperplane of X towardsits halfspace that contains the majority of the points x, y, z . This orientation yields an ultrafilter andtherefore a point in the Roller compactification of X , making the median well-defined.We know consider a special subspace of the Roller boundary: Definition 2.42. ( κ -Morse points in the Roller boundary) Let X be finite dimensional. A point x ∈ ∂ R X is called κ –Morse if it admits a combinatorial geodesic representative that is κ –Morse.Denote the set of all κ –Morse points in the Roller boundary of X by ∂ κR X .Since we often deal with different topologies on the same boundary of the CAT(0) space X , wemake explicit which notations we use for which boundary: Notation 2.43. (Various boundaries) • We denote the equivalence classes of the CAT(0) geodesic rays in X as a set by ∂X . • When equipping the set ∂X with the cone topology given in Definition 2.3, we denote it ∂ ∞ X and refer to it as the visual boundary . • As a subset of the visual boundary, the collection of all κ -Morse geodesic rays as a set isdenoted by ∂ κ X . • When equipping the subset ∂ κ X with the subspace topology from the visual boundary, wedenote it by ∂ κ ∞ X . • The set ∂ κ X is the set of all equivalence classes of κ -Morse quasi geodesic rays with respectto the κ -fellow travelling relation from Definition 2.9. • When we equip the set ∂ κ X with the SM -topology from Definition 2.11, we will still denoteit by ∂ κ X. • The subspace ∂ κR X denotes the subspace of the Roller boundary consisting of points that canbe represented by κ -Morse combinatorial geodesic rays (see Definition 2.42). • For a geodesic ray b , we will often confuse b with its image im ( b ). Since we deal with bothCAT(0) and combinatorial rays, we will use the letters a, b, c, . . . denote CAT(0) geodesic rays,and the letters α, β, γ, . . . to denote combinatorial geodesic rays.3. Combinatorial Gromov products and sublinear fellow-traveling
In this section, we prove a few useful statements which will be used in the proof of the main theorems.The following lemma is the main lemma of this section, and its essential to prove well-definedness ofthe map in Theorem 1.4.
Lemma 3.1.
Let α, β be two combinatorial geodesic rays in a
CAT(0) cube complex X such that α (0) = β (0) =: o and [ α | β ] o = ∞ . If α is κ -Morse, then β is also κ -Morse. Furthermore, α and βκ -fellow travel. In preparation for the proof of Lemma 3.1, we prove several other lemmas first.
Lemma 3.2.
Let α be a combinatorial geodesic ray and let Y = Hull ( α ) . Suppose that Y containsa κ -excursion CAT(0) geodesic ray b . If { h i } i ∈ N is the collection of excursion hyperplanes for b , p i = h i ∩ im ( α ) , and q i = im ( b ) ∩ h i , then there exist constants d , d , depending only on κ , theexcursion constant c and on the dimension v X such that:(1) |W ( p i | p i +1 ) | ≤ d κ ( t i ) ,(2) |W ( p i | q i ) | ≤ d κ ( t i ) ,where t i = d ( q i , b (0)) . Proof.
Let W i denote the collection of hyperplanes meeting h i . To simplify notation, denote W = W ( p i | p i +1 ) . Every hyperplane in W must live in either W ( q i | q i +1 ) , W i or W i +1 . Thus, we have |W| ≤ |W ∩ W ( q i | q i +1 ) | + |W ∩ W i | + |W ∩ W i +1 | . Using Lemma 2.18, there exists a constant c ′ such that |W ( q i | q i +1 ) | ≤ c ′ d ( q i , q i +1 ) ≤ c ′ c κ ( t i +1 ). Noticethat every h ∈ W ∩ W i must live in either W i − or W ( q i − , q i ) (see Figure 3). In other words, we havethe following |W ∩ W i | ≤ |W ∩ W i ∩ W i − | + |W ∩ W i ∩ W ( q i − | q i ) | . The collection of hyperplanes
W ∩ W i ∩ W i − is one that contains no facing triple since W ⊆ W ( α ).Therefore, as h i − , h i are c κ ( t i )-well-separated, we have |W ∩ W i ∩ W i − | ≤ c κ ( t i ). By Lemma 2.18,there exists a constant c ′ such that |W ( q i − | q i ) | ≤ c ′ d ( q i − , q i ) ≤ c ′ c κ ( t i ). Hence, we have |W ∩ W i | ≤ |W ∩ W i ∩ W i − | + |W ∩ W i ∩ W ( q i − | q i ) |≤ c κ ( t i ) + c ′ c κ ( t i )= c (1 + c ′ ) κ ( t i ) . Similarly, we have |W ∩ W i +1 | ≤ c (1 + c ′ ) κ ( t i +1 ) . Thus, we have |W| ≤ |W ∩ W ( q i | q i +1 ) | + |W ∩ W i | + |W ∩ W i +1 |≤ c ′ c κ ( t i +1 ) + c (1 + c ′ ) κ ( t i ) + c (1 + c ′ ) κ ( t i +1 ) ≤ c ′ c κ ( t i +1 ) + c (1 + c ′ ) κ ( t i +1 ) + c (1 + c ′ ) κ ( t i +1 )= ( c ′ c + c (1 + c ′ ) + c (1 + c ′ )) κ ( t i +1 ) . Letting d = c ′ c + 2 c (1 + c ′ ) gives that |W ( p i | p i +1 ) | ≤ d κ ( t i +1 ), however, since t i , t i +1 satisfy d ( t i , t i +1 ) ≤ c κ ( t i +1 ), Proposition 2.6 implies that κ ( t i +1 ) ≤ d ′ κ ( t i ) for a constant d ′ dependingonly on κ . Taking d = d d ′ gives that |W ( p i | p i +1 ) | ≤ d κ ( t i ) which finishes the first part of thestatement.For the second part, let W ′ = W ( p i | q i ). Notice that any hyperplane h ∈ W ′ lives in either W ( q i − | q i ), W ( p i − | q i − ) , or W ( p i − | p i ). In other words, we have |W ′ | ≤ |W ′ ∩ W ( q i − | q i ) | + |C ′ ∩ W ( p i − | q i − ) | + |C ′ ∩ W ( p i − | p i ) | . Now, using part 1 of this lemma and since b is a κ -excursion geodesic with constant c we get: |W ′ | ≤ |W ′ ∩ W ( q i − | q i ) | + |W ′ ∩ W ( p i − | q i − ) | + |W ′ ∩ W ( p i − | p i ) |≤ c ′ c κ ( t i +1 ) + c κ ( t i +1 ) + d κ ( t i +1 )= ( c ′ c + c + d ) κ ( t i +1 ) . See Figure 4 for the decomposition of W ′ . Taking c = c ′ c + c + d gives that |W ( p i | q i ) | ≤ c κ ( t i +1 ).Using Proposition 2.6, as d ( t i , t i +1 ) ≤ c κ ( t i +1 ), there exists a constant c ′ , depending only on κ , suchthat κ ( t i +1 ) ≤ c ′ κ ( t i ). Therefore, letting d = c c ′ gives that |W ( p i | q i ) | ≤ d κ ( t i ). (cid:3) UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 15 bαh i − h i h i +1 h i +2 hp i p i +1 q i q i +1 Figure 3.
The five possibilities how a hyperplane h separating p i from p i +1 mayinteract with h i − , h i , h i +1 and h i +2 . The number of appearances of each possibility isbounded in terms of κ because of κ -well-separatedness and the fact that the d ( q j , q j +1 )is κ -bounded for all j . bαh i − h i hp i − p i q i − q i Figure 4.
The three possibilities how a hyperplane h separating p i from q i mayinteract with p i − , q i − and h i − . The number of appearances of each possibility isbounded in terms of κ because of κ -well-separatedness, the fact that the d ( q j , q j +1 ) is κ -bounded for all j and inequality (1). Lemma 3.3.
Let b be a CAT(0) κ -excursion geodesic ray, H = { h i } be the collection of excursionhyperplanes crossed by b and let q i = im ( b ) ∩ h i . There exists a constant d such that the followingholds:(1) If c is a geodesic ray crossing h j − , h j , h j +1 ∈ H at points p j − , p j , p j +1 , then d ( p j , q j ) ≤ d κ ( t j ) , where t j = d ( b (0) , q j ) . (2) Suppose h is a hyperplane crossing h j − , h j , h j +1 ∈ H . Let p j be the closest point projectionof q j to the intersection h ∩ h j and let p j − , p j +1 be points in h ∩ h j − , h ∩ h j +1 respectively.We have d ( p j , q j ) ≤ d κ ( t j ) . In particular, if the geodesic c or the hyperplane h crosses infinitely many hyperplanes from H , then b lives in the κ -neighborhood of c or h respectively.Proof. The proof of this lemma is very similar to the proof of Lemma 3.2, however, for completeness,we give the proof. Let c be the excursion constant. Every hyperplane in W ( p j , q j ) must meet either[ q j − , q j ] , [ p j − , q j − ] or [ p j − , p j ] . This implies that(1) | W ( p j , q j ) | ≤ | W ( q j , q j − ) | + | W ( q j − , p j − ) | + | W ( p j − , p j ) | . Now we consider the collection W ( p j − , p j ). Notice that every hyperplane in W ( p j − , p j ) mustmeet either [ q j , q j +1 ] or [ q j +1 , p j +1 ]. That is(2) | W ( p j − , p j ) | ≤ | W ( q j , q j +1 ) | + | W ( q j +1 , p j +1 ) | . Now, combining the equations 1 and 2, and using the fact that b is a κ -excursion hyperplane withexcursion constant c , we get the following: | W ( p j , q j ) | ≤ | W ( q j , q j − ) | + | W ( q j − , p j − ) | + | W ( p j − , p j ) |≤ c κ ( t j ) + c κ ( t j ) + | W ( p j − , p j ) |≤ c κ ( t j ) + c κ ( t j +1 ) + c κ ( t j +1 )= 4 c κ ( t j +1 ) . Now, using Proposition 2.6, there exists a constant d depending only on c and κ such that κ ( t j +1 ) ≤ d κ ( t j ). Taking d = 4 cd gives that | W ( p j , q j ) | ≤ d κ ( t j ) . However, since the CAT(0) distance is bi-lipschitz equivalent to the d (1) -distance (Lemma 2.18), theresult follows.The proof of part 2 is identical with a hyperplane h replacing the role of the geodesic c . (cid:3) Lemma 3.4.
Let α be a combinatorial geodesic ray in a CAT(0) cube complex X and let Y = Hull ( α ) .If Y contains a CAT(0) geodesic ray b which is κ -Morse, then the CAT(0) quasi-geodesic α and b must κ -fellow travel. In particular, α is κ -Morse.Proof. Using Theorem 2.35, the geodesic ray b must be κ -excursion. Part 2 of Lemma 3.2 gives that α and b κ -fellow travel each other. In other words, both α and b live in the same equivalence class B .Since b is κ -Morse, Theorem 2.35 implies that α is κ -Morse. (cid:3) The following corollary is Corollary 3.5 in [Hag20], it’s a strengthened version of Lemma 2.1 in[CS11].
Corollary 3.5. (Chains in geodesics) Let X be a CAT(0) cube complex of dimension v X < ∞ . Forany k ∈ N , if α is a geodesic (in the combinatorial or CAT(0) metric) that crosses at least v X · k hyperplanes, then the set of hyperplanes crossing α contains a chain of cardinality k . In particular,every geodesic line crosses a bi-infinite chain of hyperplanes. We use the above corollary to show the following.
Lemma 3.6.
Let α be a combinatorial geodesic ray in a CAT(0) cube complex X and let Y = Hull ( α ) .If ∂ ∞ Y contains a visibility point, then | ∂ ∞ Y = 1 | . Proof.
Recall from Definition 2.25 that a hyperplane h meets Hull ( α ) if and only if it meets α . We firstclaim that, since Y = Hull ( α ), there exists no bi-infinite chain ( h n ) n ∈ Z of hyperplanes in Y . For everyhyperplane in Y , we can choose an orientation (i.e. an associated halfspace) that contains the vertex α (0). Since α (0) ∈ X (0) , we know that every descending sequence of halfspaces in this orientationterminates in finite time. However, since the sequence ( h n ) n ∈ Z is crossed by the combinatorial geodesicray α , orienting all of them towards α (0) forms (after reordering the hyperplanes appropriately) aninfinite descending chain of halfspaces, a contradiction. We conclude that Y contains no bi-infinitechain of hyperplanes.We now prove the lemma. Let ζ be the visibility point in ∂ ∞ Y and let b be a CAT(0) geodesic rayin Y with [ b ] = ζ . Suppose for the sake of contradiction that there exists a geodesic ray b ′ in Y suchthat b ′ (0) = b (0) but b = b ′ . Since ζ is a visibility point, there exists a geodesic line l connecting [ b ] to[ b ′ ]. Since l lies in Y , every hyperplane meeting l also meets the combinatorial geodesic α . Since l isa bi-infinite geodesic line in a finite dimensional CAT(0) cube complex, using Corollary 3.5, the line l must cross a bi-infinite sequence of mutually disjoint hyperplanes. This contradicts the claim that weproved in the first half of this proof and implies the lemma. (cid:3) UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 17
Proposition 3.7 ( [QRT19, Proposition 3.10 ]) . If α : [0 , ∞ ) → X is a κ -Morse quasi geodesic ray,then there exists a geodesic ray b such that α and b κ -fellow travel each other. Furthermore, b is κ -Morse. The following corollary states that κ -Morse geodesic rays define visibility points in the visual bound-ary. Corollary 3.8 ( [Zal20, Corollary 1.2 ]) . Let Y be a proper CAT(0) space. If b is a κ -Morse geodesicray in Y , then b ( ∞ ) is a visibility point in ∂ ∞ Y. Corollary 3.9.
Let α be a κ -Morse combinatorial geodesic ray in a CAT(0) cube complex and let Y = Hull ( α ) . There exists a κ -Morse geodesic ray b ⊂ Y such that b (0) = α (0) and b and α κ -fellowtravel. Furthermore, b is the unique geodesic ray in Y with b (0) = α (0) .Proof. Since α is a κ -Morse combinatorial geodesic in the d -metric of X , it must be a κ -Morsequasi geodesic in the CAT(0)-metric of X . Thus, α is a κ -Morse quasi-geodesic in Y, and hence byProposition 3.7 applied to α and to Y, there exists a κ -Morse geodesic ray b ∈ Y such that α and bκ -fellow travel each other. Using Theorem 3.8, since b defines a visibility point in the visual boundary ∂ ∞ Y, Lemma 3.6 gives the desired statement. (cid:3)
We are now ready to prove the main lemma of this section. We restate it here.
Lemma 3.10.
Let α, β be two combinatorial geodesic rays in a
CAT(0) cube complex X such that α (0) = β (0) =: o and [ α | β ] o = ∞ . If α is κ -Morse, then β is also κ -Morse. Furthermore, α and βκ -fellow travel.Proof. Denote Y = Hull ( α ) and Y = Hull ( β ). Since [ α | β ] o = ∞ , there exists an infinite sequence ofhyperplanes { h i } intersecting both α and β . For any i, since h i ∩ Y , h i ∩ Y , Y ∩ Y are all non-empty,there has to exist a point x i ∈ Y ∩ Y ∩ h i by the Helly property. Notice that since x i ∈ h i andsince X is proper, the sequence x i is unbounded (its not stuck in any compact subset of X ). ApplyingArzel`a–Ascoli to the CAT(0) geodesics [ o , x i ] in Y ∩ Y yields a CAT(0) geodesic ray b ∈ Y ∩ Y with b (0) = o . Applying the previous Corollary to α and Y , we get that α and b κ -fellow travel and that b is κ -Morse. Since b is κ -Morse, the point defined by b in the visual boundary of Y is a visibility point,hence by Lemma 3.6, we have | ∂ ∞ Y | = 1. Now, since b ∈ Y is κ -Morse, Lemma 3.4 implies that β isalso κ -Morse, and that b and β κ -fellow travel. Since α and b κ -fellow travel and β and b also κ fellowtravel, we conclude that α and β κ -fellow travel, in other words, [ α ] = [ β ] . (cid:3) Comparing HYP to the subspace topology on sublinearly Morse geodesic rays
In this section, we prove that the topology
HYP on the sublinear boundary is the same as thesubspace topology induced by the cone topology. We begin with the following Lemma.
Lemma 4.1.
Let X be a finite dimensional CAT(0) cube complex and let o ∈ X . The open sets { U o ,k } form a basis for the topology HYP on the set ∂ κ X. Proof.
Suppose that h , . . . , h n is a finite set of hyperplanes crossed by a κ -Morse geodesic ray b . Let( k i ) i be a family of excursion hyperplanes for b . We wish to find some hyperplane k that doesn’tintersect h , . . . , h n and is crossed by b after crossing h , . . . , h n . Suppose by contradiction that everyhyperplane crossed by b after crossing h , . . . , h n intersects one of the h i . This implies that one of thosehyperplanes, we will denote it by h , is crossed by an infinite family ( k i j ) j of excursion hyperplanes thatare crossed by b after crossing h . Note that, since all k i are mutually disjoint and ordered in the orderin which they are crossed by b , we have that if h intersects k i and k j for i < j , then h intersects k l forall i < l < j . Therefore, we have that h intersects all elements of the sequence ( k i ) i for i sufficientlylarge.Let q i := b ∩ k i and let p i ∈ h ∩ k i be the closest point to q i . Define t i := d ( b (0) , q i ). By Lemma 3.3,there exists a constant c ′ , such that d ( q i , p i ) ≤ c ′ κ ( t i ) for all i sufficiently large. Since q i = b ( t i ) , we have d ( b ( t i ) , h ) ≤ cκ ( t i ), which is a sublinear function. However, the function t d ( b ( t ) , h ) is convex.Since all non-negative, sublinear, convex functions are non-increasing functions, d ( b ( t ) , h ) has to benon-increasing, which contradicts the fact that b intersects h . The lemma follows. (cid:3) Theorem 4.2.
Let X be a finite dimensional CAT(0) cube complex. The restrictions of the conetopology and
HYP to the set of all geodesic rays that can be represented by κ -Morse geodes rays,denoted ∂ κ X , are equal.Proof. Let ξ ∈ ∂ ∞ X . Fix a base point o ∈ X . Denote the unique geodesic representative of ξ startingat o by ξ o . We denote the basic open sets of the cone topology, restricted to ∂ κ X by U R,ǫ ( ξ ) := { η ∈ ∂ ∞ X | d ( ξ o ( R ) , η o ( R )) < ǫ } ∩ ∂ κ X. We first show that the cone topology is finer than
HYP . Let U o ,k be a basic open set of HYP andlet ξ ∈ U o ,k . We need to find R, ǫ >
0, such that ξ ∈ U R,ǫ ( ξ ) ⊂ U o ,k .Since ξ ∈ U o ,k , we have that ξ o crosses the hyperplane k . Therefore, we find some R >
0, such thatfor all r ≥ R , d ( ξ o ( r ) , k ) ≥
1. We claim that U R, ( ξ ) ⊂ U o ,k . Suppose, η ∈ U R, ( ξ ). We concludethat the geodesic from ξ o ( R ) to η o ( R ) is contained in the ball B ( ξ o ( R )) and cannot intersect k . Thisimplies that k separates o and η o ( R ). Therefore, η o crosses k and η ∈ U o ,k . This implies that U o ,k isopen with respect to the cone topology. Since the sets { U o ,k } k generate HYP , we conclude that thecone topology is finer than
HYP on ∂ κ X .We now prove that HYP is finer than the cone topology on ∂ κ X . Let ξ ∈ ∂ κ X , let U R,ǫ ( ξ ) be abasic open set with respect to the cone topology and let η ∈ U R,ǫ ( ξ ). Since for every ξ ∈ ∂ ∞ X , the sets { U R,ǫ ( ξ ) } R,ǫ form a neighbourhood basis for ξ , we can assume without loss of generality that η = ξ .We need to find a hyperplane k , such that ξ ∈ U o ,k ⊂ U R,ǫ ( ξ ).Denote the geodesic representative of ξ based at o by b . Since ξ is κ -Morse, we find a family ofexcursion hyperplanes ( h i ) i for b . Let η ∈ ∂ κ X and c the geodesic representative of η based at o . Forany n , we denote p n := b ∩ h n and (if it exists) q n := c ∩ h n . Suppose that c crosses h n +1 . By Lemma3.3 there exists a constant d , which depends only on the excursion constant and κ , such that d ( p n , q n ) ≤ d κ ( t n ) . We have shown that for the geodesic representative b of ξ based at o and any geodesic ray c basedat o that intersects h n +1 , we can bound the distance d ( p n , q n ). Suppose that d ( b ( R ) , c ( R )) ≥ ǫ . Sincedistance functions are convex in a CAT(0) space, this implies that, for all t ≥ R , d ( b ( t ) , c ( t )) ≥ tR ǫ .Since κ is sublinear, we find some T , such that for all t ≥ T , 2 d κ ( t ) < tR ǫ . Define N to be the firstnumber such that d ( o , p N ) ≥ T . We conclude that, for all η ∈ U o ,h N +1 with geodesic representative c , d ( b ( t N ) , c ( t N )) ≤ d ( p N , q N ) ≤ d κ ( t N ) < t N R ǫ.
The first inequality follows from the fact that c can cross h N only at a time s N that satisfies | t N − s N | ≤ d ( p N , q N ) (because it is a geodesic and starts at the same point as the geodesic b ). By the discussionon convexity above, this implies that d ( b ( R ) , c ( R )) < ǫ . Therefore, U o ,h N +1 ⊂ U R,ǫ ( ξ ). The lemmafollows. (cid:3) Continuity of the Roller map
Let X be a finite dimensional CAT(0) cube complex and let m be an upper bound for the degreeof every vertex. Fix a base point o ∈ X (0) . Recall that we defined ∂ κR X = { x ∈ ∂ R X | x can be represented by a κ − Morse combinatorial geodesic ray } . Let x ∈ ∂ κR X and let α , β be combinatorial geodesic rays starting at o that represent x . Suppose α is κ -Morse. Since α and β both represent x and start at the same vertex, we have [ α | β ] o = ∞ . ByLemma 3.1, this implies that α and β are κ -fellow traveling and β is κ -Morse as well. In particular, if UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 19 we see α and β as κ -Morse quasi-geodesics, they represent the same point in ∂ κ X and thus a uniquepoint in ∂ κ ∞ X using Lemma 2.12. This way, we obtain a mapΦ : ∂ κR X → ∂ κ ∞ X. We show that the map Φ is a continuous surjection.
Theorem 5.1.
The natural map
Φ : ∂ κR X → ∂ κ ∞ X is a continuous surjection. For all x, y ∈ ∂ κR X , wehave that Φ( x ) = Φ( y ) if and only if [ x | y ] o = ∞ . Furthermore, the induced quotient map Φ : ∂ κR X → ∂ κ ∞ X is a homeomorphism. Remark 5.2.
We can express the fibres of the map Φ using the notion of components in ∂ R X andthe partial order ≺ between components (see Definition 2.40 and the paragraphs after). The fibreΦ − ( ξ ) of any point ξ ∈ ∂ κ ∞ X is the union of a component C that is maximal with respect to ≺ andall components satisfying C ′ ≺ C . In particular, we obtain that, whenever a component C in ∂ R X is represented by κ –Morse combinatorial geodesic rays, then C can be contained in only one maximalelement with respect to ≺ . Proof.
We first show surjectivity. Let ξ ∈ ∂ κ ∞ X and b the geodesic representative of ξ based at o . ByLemma 2.20, there exists a combinatorial geodesic ray α based at o that crosses the same hyperplanesas b and has finite Hausdorff distance to b . Let x ∈ ∂ κR X be the point represented by α . Clearly,Φ( x ) = ξ . Therefore, Φ is surjective.Next, we show that Φ( x ) = Φ( y ) if and only if [ x | y ] o = ∞ . Let x, y ∈ ∂ κR X , suppose [ x | y ] o = ∞ ,and choose combinatorial representatives α, β that start at o . Since α has to be κ -Morse, Lemma 3.1implies that α and β κ -fellow travel and, therefore, Φ( x ) = Φ( y ).Now suppose, [ x | y ] o < ∞ . Choose a combinatorial geodesic from o to the median m ( o , x, y ) andcombinatorial representatives of x and y respectively that start at m ( o , x, y ). Concatenating the geo-desic segment with one of the combinatorial geodesic rays yields representatives of x and y respectivelythat start at o and are identical until they have crossed all hyperplanes in W ( o | x, y ). Denote thesetwo representatives by α, β . We claim that, after separating, α and β diverge linearly. Let T be thetime when α ( T ) = m ( o , x, y ) = β ( T ). Since every hyperplane crossed by α after time T separates theunbounded part of α from β , we have d ( α ( T + n ) , β ) ≥ n √ m , which grows linearly in n . We conclude that α, β are not κ -fellow traveling. Therefore, Φ( x ) = Φ( y ),which implies that the fibres of Φ are as described above.We now prove continuity of Φ. Let ξ ∈ ∂ κ ∞ X and b the geodesic representative of ξ based at o . ByLemma 2.20, there exists a combinatorial geodesic ray α based at o that crosses the same hyperplanesas b and which has finite Hausdorff distance to b . Denote the point on the Roller boundary representedby α as x . Clearly, x ∈ Φ − ( ξ ).By Theorem 2.33 and Theorem 2.35, there exists a family of excursion hyperplanes ( h i ) i for b .Consider the sets U o ,h i = { η ∈ ∂ κ ∞ X | η o crosses h i } , where η o denotes the unique geodesic representative of η that starts at o . By Lemma 4.1, includingits proof, and Theorem 4.2 the family { U o ,h i } i forms a neighbourhood basis of the cone topology at ξ . In order to prove continuity of Φ, we are left to prove that for every i , we find a hyperplane k suchthat the set V o ,k := { y ∈ ∂ κR X | k ∈ W ( o | y ) } , which is open in ∂ κR X with the Roller topology, satisfiesΦ − ( ξ ) ⊂ V o ,k , Φ( V o ,k ) ⊂ U o ,h i . We claim that, for all i , Φ( V o ,h i +1 ) ⊂ U o ,h i and Φ − ( ξ ) ⊂ V o ,h i +1 . We first show that for all x ′ ∈ Φ − ( ξ ), any combinatorial representative α of x ′ starting at o has to cross all excursion hyperplanesof b . If α does not cross an excursion hyperplane h of b , then we have d ( b ( t ) , α ) ≥ d ( b ( t ) , h − ) , where h − denotes the halfspace of h containing o and thus α . Since b crosses h the function d ( b ( t ) , h − ) isat least linearly increasing (eventually), implying that d ( b ( t ) , α ) grows at least linearly. This contradictsthe fact that Φ( x ′ ) = ξ and therefore, Φ − ( ξ ) ⊂ V o ,h i +1 .We are left to prove that Φ( V o ,h i +1 ) ⊂ U o ,h i . Let y ∈ V o ,h i +1 and let ˜ β be a combinatorial geodesicray, representing y , starting at o . Since y ∈ ∂ κR X , ˜ β is κ -Morse. Therefore, there exists a κ -Morsegeodesic ray c , starting at o , that is κ -fellow traveling with ˜ β and a combinatorial geodesic ray β thatcrosses exactly the same hyperplanes as c . Suppose that c , and therefore β , does not cross h i . Wewill show that this implies that Φ([ β ]) = Φ([ ˜ β ]), which is a contradiction, as both of them are κ -fellowtraveling along c .Since β does not cross h i it cannot cross h i +1 either. Therefore, every k ∈ W ( β ) ∩ W ( ˜ β ) eithercrosses h i +1 or it crosses ˜ β before ˜ β crosses h i +1 . Since h i and h i +1 are well-separated and the seg-ment of ˜ β before crossing h i +1 is finite, we conclude that W ( β ) ∩ W ( ˜ β ) is a finite set. In other words,their combinatorial Gromov product satisfies h [ β ] | [ ˜ β ] i o < ∞ . From the characterisation of the fibresof Φ, we conclude that Φ([ β ]) = Φ([ ˜ β ]), a contradiction. This implies that β , and thus c crosses h i .Therefore, Φ( y ) = Φ([ ˜ β ]) ∈ U o ,h i , which implies continuity of Φ.Saying that two points in ∂ κR X are equivalent if and only if they lie in the same fibre of Φ, weobtain a quotient space, denoted ∂ κR X , and a bijective, continuous map Φ : ∂ κR X → ∂ κ ∞ X . To provethe Theorem, we are left to show that Φ − is continuous. We begin by proving the following result:Given points x n , x ∈ ∂ κR X and y ∈ ∂ R X such that x n → y and Φ( x n ) → Φ( x ), we have [ x | y ] o = ∞ .Let b n , b be geodesic representatives of Φ( x n ) and Φ( x ) respectively, all starting at o . By Lemma2.20, there exist combinatorial geodesics α n and α based at o that are asymptotic to and cross the samehyperplanes as b n and b respectively. Denote the points in ∂ R X represented by α n and α by z n and z respectively. By construction, Φ( x n ) = Φ( z n ) and Φ( x ) = Φ( z ) and, therefore, [ x n | z n ] o = [ x | z ] o = ∞ .By Theorem 2.33 and Theorem 2.35, we find a sequence of excursion hyperplanes ( h i ) i for b , whichare thus crossed by α . Let α x and α y be combinatorial geodesic rays based at o that represent x and y respectively. We claim that both α x and α y cross h i for all i .Suppose α x did not cross h i . Since [ x | z ] o = ∞ , there are infinitely many hyperplanes crossed byboth α and α x after α crosses h i +1 . All these hyperplanes have to intersect both h i and h i +1 , whichcontradicts the fact that h i and h i +1 are κ -well-separated. Therefore, α x crosses h i .Suppose α y did not cross h i . Let α x n be a combinatorial geodesic ray, starting at o representing x n .Since we assume that x n → y in Roller topology and Φ( x n ) → Φ( x ) in cone topology (which equals HYP ), we find a number N such that for all n ≥ N, α x n does not cross h i and b n crosses h i +1 . By thesame argument as above, there can be at most finitely many hyperplanes crossed by both α x n and b n after b n crosses h i +1 . Since b n crosses h i +1 after finite time, the total number of hyperplanes crossedby both α x n and b n has to be finite. Since b n and α n cross the same hyperplanes, we conclude that[ x n | z n ] o < ∞ , which is a contradiction. Therefore, α y crosses h i .We conclude that for all i , h i separates o from both x and y . Therefore, [ x | y ] o = ∞ .We now prove that Φ − is sequentially continuous. We denote the projection map onto the quotientby p : ∂ κR X → ∂ κR X . Let x n , x ∈ ∂ κR X such that Φ( x n ) → Φ( x ). Suppose that p ( x n ) p ( x ). Thenthere exists an open neighbourhood V ⊂ ∂ κR X of p ( x ) and a subsequence of x n , again denoted by x n , such that x n / ∈ p − ( V ) for all n . Since X is locally finite, ∂ R X is compact and, by moving to asubsequence if necessary, we can assume that x n → y ∈ ∂ R X . We have shown above that [ x | y ] o = ∞ .By Lemma 3.1, we see that y ∈ ∂ κR X and [ x | y ] o = ∞ . By our characterisation of the fibres of Φ, weconclude that Φ( y ) = Φ( x ) and p ( y ) = p ( x ) ∈ V . Therefore, lim n →∞ x n = y ∈ p − ( V ), contradicting UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 21 the assumption on x n . We conclude that p ( x n ) → p ( x ). Therefore, Φ − is sequentially continuousand, since ∂ κ ∞ X is first countable, it is continuous. This completes the proof. (cid:3) Embedding the sublinear boundary into a Gromov boundary
The main goal of this section is to prove Theorem 1.1.
Definition 6.1 (Parallel subcomplexes) . Let X be a finite-dimensional CAT(0) cube complex. Twoconvex subcomplexes H , H are said to be parallel if for any hyperplane h, we have h ∩ H = ∅ if andonly if h ∩ H = ∅ . Definition 6.2 (Factor systems [BHS17]) . Let X be a finite-dimensional CAT(0) cube complex. A factor system , denoted F , is a collection of subcomplexes of X such that:(1) X ∈ F .(2) Each F ∈ F is a nonempty convex subcomplex of X (3) There exists c ≥ x ∈ X (0) at most c elements of F contain x .(4) Every nontrivial convex subcomplex parallel to a hyperplane of X is in F .(5) There exists c such that for all F, F ′ ∈ F , either g F ( F ′ ) ∈ F or diam( g F ( F ′ )) ≤ c . Remark 6.3.
All known cocompact CAT(0) cube complexes admit factor systems [HS20].We also use the following generalization of the contact graph introduced by Genevois in [Gen20b]:
Definition 6.4 (The well-separation space) . Let X be a finite-dimensional CAT(0) cube complex. Foreach non-negative integer k , the k -well-separation space , denoted by Y k , is defined to be the set whoseelements are the vertices of X , with the following distance function. For x, y ∈ X (0) , the k -distancebetween x, y , denoted by d k ( x, y ), is defined to be the cardinality of a maximal collection of k -well-separated hyperplanes separating x, y . We will refer to the space ( Y k , d k ) as the k -well-separationspace.It’s worth remarking that when k = 0, the 0-well-separation space is quasi-isometric to the contactgraph (Fact 6.50 of [Gen20b]). Furthermore, we have the following: Proposition 6.5 ( [Gen20b, Proposition 6.54]) . Let X be a finite-dimensional CAT(0) cube complexand let Y k be the k -well-separation space as in Definition 6.4. For any non-negative integer k , themetric space ( Y k , d k ) is k + 2) hyperbolic. Remark 6.6.
For each x ∈ X, let n be the largest integer so that x ∈ [0 , n . We define v ( x ) to bea 0-cell (a vertex) of the cube [0 , n . For any x, y ∈ X , we denote the cardinality of the maximalcollection of k -well-separated hyperplanes separating x, y by l k ( x, y ). Since X is a finite dimensionalcube complex, there exists a constant c depending only on the dimension of X such that the followingholds: • We have: d k ( v ( x ) , v ( y )) − c ≤ l k ( x, y ) ≤ d k ( v ( x ) , v ( y )) + c . • We have d ( x, y ) − c ≤ d (1) ( v ( x ) , v ( y )) ≤ d ( x, y ) + c , where d (1) and d are the combinatorialand CAT(0) distance respectively. • We have d k ( v ( x ) , v ( y )) ≤ d ( x, y )+ c , where d k and d are the distances in Y k and X respectively. Lemma 6.7.
Let X be a CAT(0) cube complex and let ( Y k , d k ) be the k -well-separation space. Every( CAT(0) or combinatorial) geodesic in X projects to a uniform unparameterized quasi-geodesic in ( Y k , d k ) . More precisely, we have the following:(1) For a combinatorial geodesic α in X (1) , if x, y, z ∈ X (0) are points with x = α ( t ) , y = α ( t ) and z = α ( t ) with t ≤ t ≤ t , we have d k ( x, z ) ≥ d k ( x, y ) + d k ( y, z ) − k − .(2) For a CAT(0) geodesic c in X , if x, y, z ∈ X are points with x = c ( t ) , y = c ( t ) and z = c ( t ) with t ≤ t ≤ t , we have d k ( x, z ) ≥ d k ( x, y ) + d k ( y, z ) − k − c − , where c is a constantdepending only on the dimension of X (and not on the geodesic c ). Proof.
We only prove the first part of the assertion, the second part follows immediately by Remark6.6. Let L , L denote d k ( x, y ) , d k ( y, z ) respectively. By definition, L , L are the maximal cardinalityof k -well-separated hyperplanes separating x, y and y, z respectively. In other words, if W , W denotemaximal sets of k -well-separated hyperplanes separating x, y and y, z respectively, then |W | = L and |W | = L . Let W = { h , h , . . . h L } and define t i such that α ( t i ) = h i ∩ α and t i < t i +1 for all1 ≤ i ≤ L . Similarly, let W = { h ′ , h ′ , . . . h ′ L } where α ( s i ) = h i ∩ α and s i < s i +1 for all 1 ≤ i ≤ L . • Case 1: If h L and h ′ are disjoint, then any two hyperplanes h i ∈ C and h ′ j ∈ C −{ h ′ } must be k -well-separated since every hyperplane h meeting both h i , h ′ j must also meet h ′ . Therefore,every such h must meet the hyperplanes h ′ , h ′ j , but since h ′ , h ′ j are k -well-separated, therecan be at most k such hyperplanes containing no facing triple. That is to say, d k ( x, z ) ≥ d k ( x, y ) + d k ( y, z ) − • Case 2: (See figure 1.) If h L and h ′ are not disjoint, choose the smallest m so that h m ∩ h ′ = ∅ . Notice that, since h ′ , h ′ are disjoint, we have h m ∩ h ′ = ∅ . In particular, thehyperplanes h ′ , h ′ must both cross h m , h m +1 , . . . , h L . Since h m , h m +1 , . . . , h L all meet thegeodesic α , they can’t contain a facing triple and as h ′ , h ′ are k -well-separated, we have |{ h m , h m +1 , . . . , h L }| ≤ k. That is to say, m ≥ L − k . In particular, h ′ must be disjointfrom the hyperplanes h , h , . . . , h L − k − . Hence, proceeding as in argument for Case 1, thehyperplanes h , h , . . . , h L − k − , h ′ , h ′ , . . . h ′ L are pairwise k -well-separated. In other words, d k ( x, z ) ≥ ( L − k −
1) + ( L −
2) = L + L − k −
3. Notice that the projection is a uniformunparameterized quasi-geodesic as the constant k + 3 is independent of the projected geodesic α. (cid:3) We conclude from Lemma 6.7 that, if a geodesic ray crosses an infinite sequence of k -well-separatedhyperplanes, then it projects to an infinite, unparametrised quasi-geodesic and thus defines a point inthe Gromov boundary of ( Y k , d k ). More precisely, we have the following corollary. Corollary 6.8.
Every (
CAT(0) or combinatorial) geodesic ray crossing an infinite sequence of k -well-separated hyperplanes projects to an infinite unparameterized quasi-geodesic in ( Y k , d k ) with uniformquasi-geodesic constants (independent of the geodesic) defining a point in ∂Y k , the Gromov boundaryof Y k . The following lemma appears in the first version of [Gen20b] on the arXiv, however, due to technicalissues, it was omitted from the second version. With Genevois’s blessing, the statement and theargument were reproduced in [MQZ20].
Lemma 6.9 ( [MQZ20, Lemma 4.26]) . Let X be a cocompact CAT(0) cube complex with a factorsystem. There exists a constant L ≥ such that any two hyperplanes of X are either L -well-separatedor are not well-separated. The constant from the above lemma will be referred to frequently in this paper, therefore, we giveit a name:
Definition 6.10. (the separation constant) Let X be cocompact CAT(0) cube complex with a factorsystem. The the separation constant of X is the smallest integer L ≥ X are either L -well-separated or are not well-separated. Such a constant exists using Lemma 6.9.In light of Lemma 6.9, Corollary 6.8, Theorem 2.33, and Theorem 2.35 we get the following. Corollary 6.11.
Let X be a cocompact CAT(0) cube complex with a factor system and let L be theseparation constant. Every infinite κ -Morse ( CAT(0) or combinatorial) geodesic projects to an infiniteunparameterized quasi-geodesic in ( Y L , d L ) with uniform quasi-geodesic constants (independent of theprojected geodesic) defining a point in ∂Y L , the Gromov boundary of Y L . Proof.
The proof follows immediately from Lemma 6.9, Corollary 6.8, Theorem 2.33 and Theorem2.35. (cid:3)
UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 23
The previous corollary implies the existence of a map from the κ -Morse boundary of a CAT(0) cubecomplex to the Gromov boundary of the hyperbolic space Y L . The following lemma is to show thatthis map is injective. Lemma 6.12 (Injectivity of the map) . Let X be a cocompact CAT(0) cube complex with a factorsystem and let L be the separation constant. Let c, c ′ be two distinct CAT(0) κ -Morse geodesic rayswith c (0) = c ′ (0) = o . For each m , there exists t large enough so that the number of L -well-separatedhyperplanes separating c ( t ) from c ′ is at least m. Proof.
Notice that since c, c ′ are distinct, neither of them can be in the κ -neighborhood of the other.Let H = { h i } denote the excursion hyperplane’s for c and let c ( t i ) denote h i ∩ c . Suppose that the set H is ordered based on the order in which the excursion hyperplanes cross c . Using Lemma 3.3, onlyfinitely many hyperplanes of H can meet c ′ . Let j be the largest integer so that h j crosses c ′ . Define K = d ( o , h j ∩ c ). Notice that for t > K, if W t denotes the collection of L -well-separated excursionhyperplanes separating c ( t j ) from c ( t ), then |W t | → ∞ as t → ∞ . For a fixed t > K , since hyperplanesof W t can’t cross c ′ , they must cross every geodesic starting at c ( t ) and ending on c ′ . This impliesthat for each m , there exists a t such that the number of L -well-separated hyperplanes spearating c ( t )from c ′ is at least m . This gives the desired conclusion. (cid:3) The well-separation metric space given in Definition 6.4 is not a geodesic metric space. We will usethe well-separation space defined above to define a geodesic metric space with a comparable distancefunction. More precisely, for an integer k , let Γ k be the graph whose vertices are X (0) with two vertices x, y ∈ X (0) connected by an edge if and only if the number of k -well-separated hyperplanes separating x, y is at most 10 k + 4. Definition 6.13 (The well-separation graph) . Let X be a finite dimensional CAT(0) cube complex.For each integer k , the k -well-separation graph , denoted by Γ k is defined to be the graph whose verticesare the vertices of X with an edge connecting two distinct vertices x, y if the number of k -well-separatedhyperplanes separating x from y is bounded from above by 10 k + 4 . We now show that the spaces ( Y k , d k ) and (Γ k , d Γ k ) are bilipschitz equivalent to each other. Moreprecisely, we show the following. Lemma 6.14 (Bilipschitz equivalence) . Let X be a finite-dimensional CAT(0) cube complex and let ( Y k , d k ) , (Γ k , d Γ k ) be the k -well-separations space and graph respectively. For any two points x, y ∈ X (0) , we have k +4 d k ( x, y ) ≤ d Γ k ( x, y ) ≤ d k ( x, y ) .Proof. We first show that k +4 d k ≤ d Γ k . For x, y ∈ X (0) , let p be any edge path in Γ connecting x, y . Denote the vertices and the edges of this path by x = x, x , x , . . . x m = y and e , e , . . . , e m ,respectively. Let ˜ p denote some edge path in X (1) connecting x = x, x , x , . . . x m = y such that thesubpath ˜ p i of ˜ p connecting x i − to x i is a combinatorial geodesic for all i . Let W = { h , h , . . . , h n } denote a maximal collection of k -well-separated hyperplanes separating x, y. Every such hyperplanemust meet ˜ p , and since x i , x i +1 are connected by an edge e i in Γ k , each subpath ˜ p i can be crossed byat most 10 k + 4 hyperplanes of C . Therefore, n k +4 ≤ m which gives that k +4 d k ( x, y ) ≤ d Γ k ( x, y ).Now we show that d Γ k ( x, y ) ≤ d k ( x, y ). In order to do so, we will construct an edge path in Γ oflength d k ( x, y ) . Let α be a combinatorial geodesic in X connecting x, y and let W = { h , h , . . . , h n } be a maximal collection of k -well-separated hyperplanes separating x, y . Let x = x, x i = α ( t i )denote the vertex on α immediately after crossing the hyperplane h i for all 1 ≤ i ≤ n −
1. Weclaim that there is an edge in Γ connecting x i to x i +1 . In other words, we claim that the numberof k -well-separated hyperplanes separating x i , x i +1 is at most 10 k + 4 . Suppose not, that is, supposethat x i , x i +1 are separated by a collection of k -well-separated hyperplanes W ′ = { h ′ , h ′ , . . . , h ′ m } with m ≥ k + 5. Notice that h i +1 / ∈ W ′ as if h i +1 ∈ W ′ , then, since the hyperplanes h i − , h i are k -well-separated, at most k hyperplanes of the collection W ′ can cross them. Therefore, thecollection ( W−{ h i , h i − } ) ∪{ h ′ i ∈ W ′ | h ′ i does not cross both h i , h i − } is a collection of k -well-separatedhyperplanes separating x, y, whose cardinality is at least ( n −
2) + (10 k + 5 − k ) > n contradicting αh i − h i h i +1 h i +2 x i x i +1 h ′ i h ′ j h ′ l h ′ o h ′ p Figure 5.
Given five hyperplanes in W ′ that intersect neither h i − nor h i +2 , wecan take the “middle” three and obtain that W \ { h i , h i +1 } ∪ { h ′ j , h ′ l , h ′ o } is a set of k -well-separated hyperplanes separating x from y that is strictly larger than d k ( x, y ).the maximality assumption. Now, notice that for any five hyperplanes h ′ i , h ′ j , h ′ l , h ′ o , h ′ p ∈ W ′ orderedbased on the order they are crossed by α, at least one of them must cross either both h i , h i − or both h i +1 , h i +2 as otherwise one gets a larger collection of k -well-separated hyperplanes separating x, y. Inother words, if none of the five hyperplanes h ′ i , h ′ j , h ′ l , h ′ o , h ′ p cross both h i , h i − or both h i +1 , h i +2 , then,the set consisting of ( W − { h i , h i +1 } ) ∪ { h ′ j , h ′ l , h ′ o } is a set of k -well-separated hyperplanes separating x, y with cardinality n + 1 contradicting the maximality assumption, see Figure 5. Notice that 10 k + 5is exactly 5(2 k + 1), in other words, grouping elements of W ′ into subsets of cardinality exactly 5 yieldsexactly 2 k + 1 such subsets. Each of these subsets contains at least one hyperplane crossing eitherboth h i , h i − or both h i +1 , h i +2 . This implies that either h i , h i − or h i +1 , h i +2 is crossed by k + 1hyperplanes, which is a contradiction. The above argument shows that there is an edge connecting x i to x i +1 for all 2 ≤ i ≤ m −
2. It remains to show that there are three edges e , e , e connecting { x, x } , { x , x } , { x m − , y } respectively, but an almost identical argument to the above shows that. (cid:3) Corollary 6.15.
Let X be finite dimensional CAT(0) cube complex X and let k be a positive integer.There exists a constant c and a map v : ( X, d ) → (Γ k , d Γ k ) such that d Γ k ( v ( x ) , v ( y )) ≤ d k ( v ( x ) , v ( y )) ≤ d ( x, y ) + c . Proof.
This follows using Lemma 6.14 and part 3 of Remark 6.6. (cid:3)
Corollary 6.16.
Let X be a cocompact CAT(0) cube complex with a factor system and let L be theseparation constant. Every infinite κ -Morse ( CAT(0) or combinatorial) geodesic projects to an infiniteunparameterized quasi-geodesic in (Γ L , d Γ L ) with uniform quasi-geodesic constants (independent of theprojected geodesic) defining a point in ∂ Γ L , the Gromov boundary of Γ L . Proof.
The proof follows immediately from Lemma 6.9, Corollary 6.8, Theorem 2.33 and Theorem2.35. (cid:3)
Definition 6.17 (coarsely distance-decreasing) . A map f : ( X, d ) → ( Y, d ′ ) is said to be coarselydistance-decreasing if there exists constants K, C ≥ d ( f ( x ) , f ( y )) ≤ Kd ( x, y ) + C for all x, y ∈ X. Recall from Remark 6.6 that for a CAT(0) cube complex X , there is a coarsely distance decreasingmap v : X → X (0) and a constant c such that for any x, y ∈ X , if l k ( x, y ) denotes the cardinality of amaximal collection of k -well-separated hyperplanes separating x, y , then • We have: d k ( v ( x ) , v ( y )) − c ≤ l k ( x, y ) ≤ d k ( v ( x ) , v ( y )) + c . • We have d ( x, y ) − c ≤ d (1) ( v ( x ) , v ( y )) ≤ d ( x, y ) + c , where d (1) and d are the combinatorialand CAT(0) distance respectively. UBLINEARLY MORSE BOUNDARIES FROM THE VIEWPOINT OF COMBINATORICS 25
The following lemma is phrased in slightly more general terms than needed, but it will immediatelyimply that ∂ κ ∞ X continuously injects in the Gromov boundary of the geodesic hyperbolic metric spaceΓ L .Recall that convergence in the Gromov boundary of a δ -hyperbolic space is characterized as follows.If l ≥ δ + 1 , and c n , c are geodesic rays with c n (0) = c (0), then.:[ c n ] → [ c ] iff ∀ t, ∃ k = k t such that d ( c n ( t ) , c ( t )) ≤ l, ∀ n ≥ k, Lemma 6.18.
Let X be a CAT(0) space, Γ a geodesic hyperbolic space and p : X → Γ a coarselydistance decreasing map. Suppose that B ⊆ ∂ ∞ X is a subspace of the visual boundary such that everyelement of B is represented by a geodesic ray starting at o which projects to an infinite unparameterizedquasi-geodesics with uniform constants in Γ , then p induces a continuous map i p : B → ∂ Γ . Proof.
Let p : X → Γ be as in the statement of the lemma. In particular, there exists constants
K, C such that d ( p ( x ) , p ( y )) ≤ Kd ( x, y ) + C. Suppose that B is a subspace as in the statement of the lemma(in particular, it is equipped with the subspace topology of the visual boundary) and let c , c be theconstants so that p ( B ) are all ( c , c )-quasi-geodesics, after reparametrizing. Fix o ∈ X such that everygeodesic ray in B starts at o . Let b n → b ∈ B , and let q n , q denote the infinite ( c , c )-quasi-geodesicswhich are the images of b n , b respectively. Let c n , c denote some geodesic rays with [ c n ] = [ q n ] and[ c ] = [ q ] in ∂ Γ . We need to show that for each t there exists an integer k = k t such that for all n ≥ k, we have d ( c n ( t ) , c ( t )) ≤ D, for some constant D not depending on c n or c. Using the stability of quasi-geodesics in hyperbolic spaces, there exists a constant M = M ( c , c , δ ) ≥ c n and q n is bounded above by M. Similarly, the Haus-dorff distance between c and q is bounded above by M. Let t ∈ R + , notice that using the above,there exists a point q t ∈ q such that d ( c ( t ) , q t ) ≤ M. Let b t be the furthest point in b from o with p ( b t ) = q t . That is, b t is the unique point in b with p ( b t ) = q t and d ( b t , o ) ≥ d ( x, o ) forany x ∈ b with p ( x ) = q t . Let s t ∈ [0 , ∞ ) with b ( s t ) = b t , since b n → b, there exists some k such that d ( b n ( s t ) , b ( s t )) ≤ n ≥ k. Since p is coarsely distance decreasing, we have d ( p ( b n ( s t )) , p ( b ( s t ))) = d ( p ( b n ( s t )) , p ( b t )) = d ( p ( b n ( s t )) , q t ) ≤ Kd ( b n ( s t ) , b t )) + C ≤ K + C. Denote p ( b n ( s t )) ∈ q n by x tn . Notice that d ( x tn , q t ) ≤ K + C . By stability of quasi-geodesics, there exists apoint y tn ∈ c n with d ( y tn , x tn ) ≤ M. Using the above, we get that d ( c ( t ) , y tn ) ≤ M + K + C + M = 2 M + K + C. In particular, since c n , c are geodesics, if s tn is the point in [0 , ∞ ) with c n ( s tn ) = y tn , then s tn ∈ [ t − (2 M + K + C ) , t + (2 M + K + C )]. This implies that d ( c n ( t ) , c ( t )) ≤ D = 2(2 M + K + C ) for all n ≥ k which finishes the proof. (cid:3) Corollary 6.19.
Fix a cocompact
CAT(0) cube complex X . If L is the separation constant, then wehave the following:(1) There exists a coarsely distance decreasing map v : ( X, d ) → (Γ L , d Γ L ) .(2) The map v induces a continuous injection i v : ∂ κ ∞ X → ∂ Γ L . (3) The map v induces a continuous injection i v : ∂ κ X → ∂ Γ L . Proof.
Part (1) of the statement is Corollary 6.15. Part 2 follows using Lemma 6.12, Lemma 6.18,Corollary 6.16 and part (1). Part(3) follows using Lemma 2.12 and part (2). (cid:3) a remark on hierarchically hyperbolic groups The first part of Theorem 1.1 states that for a cocompact CAT(0) cube complex X with a factorsystem, there exists a hyperbolic graph Γ such that infinite κ -Morse geodesic rays projects to infiniteunparamterized quasi-geodesics in Γ defining a map ∂ κ X → ∂ Γ where ∂ κ X and ∂ Γ are the κ -Morseboundary and the Gromov boundary of X and Γ respectively.It is natural to wonder if the same conclusion of Theorem 1.1 holds in the settings of hierarchicallyhyperbolic spaces. Question 1.
Let X be a geodesic hierarchically hyperbolic space and let Y be a hyperbolic space suchthat Morse geodesic rays in X project to infinite unparameterized quasi-geodesics in Y . Do κ -Morsegeodesic rays in X project to infinite unparameterized quasi-geodesics in Y ? We give an example showing that the answer to this question is no. In order to do that, we will usethe following theorem by Rafi and Verberne (Theorem 1.1 in [RV18]). For a surface S, we let Map( S )denote the mapping class group of S and C ( S ) denote the curve graph of S . Theorem 7.1.
Let S = S , be the five-times punctured sphere and let H = M ap ( S ) be the mappingclass group of H . There exists a finite generating set A of H and an infinite sequence of finite geodesics b n ∈ Cay ( H, A ) such that projections of b n to C ( S ) are not unparameterized ( K, C ) -quasi-geodesics forany K, C > . Now, let H and A be as in Theorem 7.1. Define G = H ∗ Z and choose the finite generating set for G by A ′ = A ∪ h a i , where a is a generator for Z . Let Y = C ( S ) ∗ R be the hyperbolic space which is a“tree of curve graphs”. Notice that the action of H on C ( S ) extends naturally to an action of G on Y giving a projection map from G to Y. Also, it’s worth pointing out that the space Y is precisely themaximal hyperbolic element of the HHS structure from [ABD]. Lemma 7.2. (Special case of Theorem 7.9 in [ABO19] and Theorem 6.6 in [ABD]) If X = Cay ( G, A ′ ) ,then X is a hierarchically hyperbolic space such that Morse geodesic ray in X project to infinite unpa-rameterized quasi-geodesics in Y. Corollary 7.3.
Let
G, A ′ , Y be as above and consider the hierarchically hyperbolic space X = Cay ( G, A ′ ) .There exists a √ t -Morse geodesic ray in X whose projection to Y is not an unparameterized ( K, C ) -quasi-geodesic for any K, C > . Proof.
Recall that G = H ∗ h a i , were Z = h a i . Notice that geodesics b n from Theorem 7.1 are allnaturally geodesics in X = Cay ( G, A ′ ), and hence the projection of the sequence b n to Y are notunparameterized ( K, C )-quasi-geodesics for any
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Institut f¨ur Mathematik, ETH Z¨urich, Switzerland
Email address : [email protected] Department of Mathematics, Queen’s University, Kingston, ON
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