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Sublinearly Morse Boundary of CAT(0) admissible groups

We show that if G is an admissible group acting geometrically on a CAT(0) space X, then G is a hierarchically hyperbolic space and with mild assumptions the sublinearly-Morse boundary of the group is a topological model for associated Poisson boundaries .


Introduction
Sublinearly Morse boundaries are recently constructed for all finitely generated groups [QRT19,QRT20]. It is a metrizable topological space that is a group invariant. Similar Gromov boundary of hyperbolic spaces, sublinearly Morse boundaries are particularly illuminating in revealing features of groups that contains hyperbolic-like features [IZ21,MQZ20,QZ21,Zal20]. One of these feature centers around asymptotic behavior of random walk on the associated groups. In several important classes of groups, such right-angled Artin groups, relative hyperbolic groups, and the mapping class groups of surfaces of finite type, an appropriately chosen sublinear function yields a sublinearly Morse boundary that serves as a topological model for the Poisson boundaries(with mild assumptions) of the group.
In this paper we continue to prove the connection between a geometric boundary and random walk boundary for a new class of groups: CAT(0) admissible groups.
CAT(0) admissible groups is a particular type of graph of groups that generalizes graph manifolds. A compact, orientable, irreducible 3-manifold with empty or tori boundary is called a graph manifold if it is obtained by gluing finitely many Seifert manifolds with orientable hyperbolic base orbifolds where the gluing maps between the Seifert components do not identify (unoriented) Seifert fibers up to homotopy. CAT(0) admissible group was first introduced by Croke-Kleiner [CK02] as key counterexamples to the nice properties of Gromov boundary when the hyperbolicity assumption is dropped. This property is further studied in [Qin16].
Roughly speaking, a CAT(0) admissible group models the JSJ structure of nonpositively curved graph manifolds where the Seifert fibration is replaced by the following central extension of a general hyperbolic group: In this paper, we also refer to CAT(0) admissible groups as CKA groups (CKA stands for Croke-Kleiner admissible). This class of groups serves as a simple algebraic means to produce interesting groups from an arbitrary finite collection of n CAT(0) hyperbolic groups.
A basic example. Let H 1 and H 2 be two torsion-free hyperbolic groups which act geometrically on CAT(0) spaces X 1 and X 2 respectively. Then G i = H i × t i 1 arXiv:2203.00935v1 [math.GR] 2 Mar 2022 (with i = 1, 2) acts geometrically on the CAT(0) space Y i = X i × R. Any primitive hyperbolic element h i in H i gives a totally geodesic torus T i in the quotient space We rescale Y i so that the lengths of [h 1 ] and of [t 2 ] are the same, and the lengths of [t 1 ] and [h 2 ] are the same. Let f : T 1 → T 2 be an isometry that flips the Z-axis with the axis of an loxodromic element in H i . Let M be the space obtained by gluing Y 1 to Y 2 by the isometry f . One can show that there exists a metric on M such that with respect to this metric, M is a locally CAT(0) space (see e.g. [BH99,Proposition 11.6]). By the Cartan-Hadamard Theorem, then the universal cover M with the induced length metric from M is a CAT(0) space, and thus π 1 (M ) is a CAT(0) admissible group and π 1 (M ) M is a CAT(0) action.
Theorem A. Let G be a CAT(0) admissible group. The following holds.
(1) G is a hierarchically hyperbolic space with respect to a word metric.
(2) Let µ be a finitely supported, non-elementary probability measure on G.
Then for κ(t) = log p (t), where p is the complexity of the HHS. Then almost every sample path (w n ) converges to a point in ∂ κ G; and the κ-Morse boundary (∂ κ G, ν) is a model for the Poisson boundary of (G, µ)where ν is the hitting measure associated to the random walk driven by µ.
The key step to show Theorem A involves proving that CAT(0) admissible groups acts geometrically on hierarchically hyperbolic spaces. Hierarchically hyperbolic spaces are introduced axiomatically, modeling the Masur-Minsky deconstruction of mapping class groups [MM00]. A hierarchically hyperbolic space (HHS) consists of a metric space X together with a partially ordered set of δ-hyperbolic spaces. There are projections maps from X to each of the hyperbolic spaces and from the hyperbolic spaces to each other. In some sense, this set of hyperbolic spaces form a tree-like metric that can be used to measure distances in X. Many interesting groups are quasi-isometric to hierarchically hyperbolic spaces, examples of classes of groups include but are not limited to: • hyperbolic groups and relative hyperbolic groups • Mapping class groups of orientable surfaces of finite type.
• Fundamental groups of compact three-manifolds with no Nil or Sol in their prime decomposition. • Groups that act properly and cocompactly on a special CAT(0) cube complex, and more generally any cubical group which admits a factor system. It is established in [BHS19] that all graph manifolds are a hierarchically hyperbolic space. To show this, we have Kapovich-Leeb [KL98] to show that all graph manifolds are quasi-isometric to flip graph manifolds, and the combination theorem of [BHS19] shows that the fundamental group of a flip graph manifold is HHS. When we lose the manifold structure and instead we look at graph of groups and in the setting of CAT(0) admissible groups, it is unknown whether all fundamental group of graph of hyperbolic-by-Z groups are quasi-isometric to the ones whose gluing map is a flip isometry(here flip roughly means the same as in the basic example). In Theorem A, we drop both the assumption of flip and also the assumption of the space being a manifold.
Lastly, an intermediate step in the proof of Theorem A depends on the following fact: Theorem B. Let µ be a non-elementary, finitely supported measure on G and assume that G acts geometrically on an hierarchically hyperbolic space. Then for κ(r) := log p (r), where p is the complexity of the hierarchy, the κ-Morse boundary is a topological model for the Poisson boundary of (G, µ).
The claim is proven for all mapping class groups and is remarked in [QRT20]. We include the proof for hierarchically hyperbolic spaces in the appendix of this paper.

Preliminaries
In this section, we review definitions of κ-projection, κ-Morse boundary in Section 2.1, random walk on groups and Posisson boundaries in Section 2.2. These two sections (Section 2.1 and Section 2.2) are only used in Section 4, so the reader could skip these two sections until Section 4 . In Section 2.3, we recall the definition of hierachically hyperbolic space (HHS) given in [BHS19]. We then review backgrounds on CAT(0) admissible groups in Section 2.4.
2.1. κ-projection, κ-Morse boundary. We refer to and follow the construction in [QRT20] as a complete reference for κ-Morse boundaries of proper geodesic metric spaces. Here we briefly recall the construction.
Definition 2.1. Let (X, d X ) be a proper geodesic metric space and Z ⊆ X a closed subset, and let κ be a concave sublinear function. A map π Z : X → P(Z) is a κ-projection if there exist constants D 1 , D 2 , depending only on Z and κ, such that for any points x ∈ X and z ∈ Z, A κ-projection differs from a nearest point projection by a uniform multiplicative error and a sublinear additive error [QRT20].
Lemma 2.2. [QRT20] Given a closed set Z, we have for any Definition 2.3. Let Z ⊆ X be a closed set, and let κ be a concave sublinear function. We say that Z is κ-Morse if there exists a proper function m Z : R 2 → R such that for any sublinear function κ and for any r > 0, there exists R such that for any (q, Q)-quasi-geodesic ray β with m Z (q, Q) small compared to r, if The function m Z will be called a Morse gauge of Z.
Definition 2.4. Let X be a proper geodesic space with a base-point o. Two κ-Morse quasi-geodesic rays α and β are equivalent, written as α ∼ β, if they diverge sublinearly. The set of all equivalence classes of κ-Morse geodesic quasi-geodesic rays, together with the fellow traveling quasi-geodesic topology [QRT20].
Theorem 2.5. [QRT20] Let X be a proper geodesic metric space and let κ be a sublinear function. The associated κ-Morse boundary ∂ κ X is a metrizable topological space and ∂ κ X is quasi-isometrically invariant.

Random walk and Poisson boundaries.
We now review random walk on groups per the study of this project. Let G be a locally compact, second countable group, with left Haar measure m, and let µ be a Borel probability measure on G, which we assume to spread-out, i.e. such that there exists n for which µ n is not singular w.r.t. m. Given µ, we consider the step space (G N , µ N ), whose elements we denote as (g n ). The random walk driven by µ is the G-valued stochastic process (w n ), where for each n we define the product w n := g 1 g 2 . . . g n .
We denote as (Ω, P) the path space, i.e. the space of sequences (w n ), where P is the measure induced by pushing forward the measure µ N from the step space. Elements of Ω are called sample paths and will be also denoted as ω. Let T : Ω → Ω be the left shift on the path space, and let (B, A) be a measurable space on which G acts by measurable isomorphisms; a measure ν on B is µ-stationary if ν = G g ν dµ(g), and in that case the pair (B, ν) is called a (G, µ)-space. Recall that a µ-boundary is a measurable (G, µ)-space (B, ν) such that there exists a T -invariant, measurable map bnd : (Ω, P) → (B, ν), called the boundary map.
Moreover, a function f : We denote by H ∞ (G, µ) the space of bounded, µ-harmonic functions. One says a µ-boundary is the Poisson boundary of (G, µ) if the map given by Φ(f )(g) := B f dg ν is a bijection. The Poisson boundary (B, ν) is the maximal µ-boundary, in the sense that for any other µ-boundary (B , ν ) there exists a G-equivariant, measurable map p : (B, ν) → (B , ν ). For more details, see [Kai00].
The κ-Morse boundary realizes the Poisson boundary. The following result is due to Ilya Gekhtman and [QRT20].
Theorem 2.6. Let G be a finitely generated group, and let (X, d X ) be a Cayley graph of G. Let µ be a probability measure on G with finite first moment with respect to d X , such that the semigroup generated by the support of µ is a non-amenable group. Let κ be a concave sublinear function, and suppose that for almost every sample path ω = (w n ), there exists a κ-Morse geodesic ray γ ω such that Then almost every sample path converges to a point in ∂ κ X, and moreover the space (∂ κ X, ν), where ν is the hitting measure for the random walk, is a model for the Poisson boundary of (G, µ).

2.3.
Hierarchically hyperbolic spaces. We first recall the definition of a hierarchically hyperbolic space (HHS) as given in [BHS19].
Definition 2.7 (Hierarchically hyperbolic spaces). A q-quasigeodesic space (X, d) (i.e, there exists a constant q such that any two points in X is joining by a (q, q)quasigeodesic segment) is a hierarchically hyperbolic space if there exists δ ≥ 0, an index set Λ, a collection of δ-hyperbolic spaces {CW : W ∈ Λ} such that the following conditions are satisfied: Projections: There exists a uniform constant ξ ≥ 0, there is a set of projections such that diam CW (π W (x)) ≤ ξ for all x ∈ CW . Moreover, there exists K such that for all W ∈ Λ, π W is coarsely Lipschitz and the image π 1 (W )(X) (which is defined to be the union of π W (x)) is a K-quasiconvex set in CW .
Nesting: Λ is equipped with a partial order such that either Λ = ∅ or Λ contains a unique -maximal element. In this paper the maximal element is Also, there is a projection ρ W V from CW to subsets of CV . Orthogonality: Λ has a symmetric and anti-reflexive relation called orthogonality: (we use notation ⊥). Also, whenever V W and W ⊥ U then we require that V ⊥ U . We require that for each S ∈ Λ, for each U ∈ Λ S such that the set For any x ∈ X we have W , and for each x ∈ X we have: Finite complexity: The complexity of X is an integer n ≥ 0 such that any set of pairwise -compatible elements has cardinality at most n.
Large links: There exists λ ≥ 1 and E ≥ ξ, κ 0 such that the following holds. Let W ∈ Λ and let x, x ∈ X. Let N = λd w (π W (x), π W (x )) + λ. Then there exists A 1 , A 2 , · · · , A N ∈ Λ W \{W } such that for all A ∈ Λ W \{W } then either A = T i for some i, or d A (π A (x), π A (x )) < E. Also, d W (π W (x), P Ai W ) ≤ N for each i. Bounded geodesic image: There exists E > 0 such that for all W ∈ Λ, all V ∈ Λ W \{W }, and all geodesic γ of CW , either There exists a constant α with the following property. Let {V j } be a family of pairwise orthogonal elements of Λ, and let p j ∈ π Vj (X). Then there exists an element x ∈ X such that Vj V ) ≤ α and • If W and V j are transverse for some j then d W (x, p Vj W ) ≤ α. Uniqueness: For each k ≥ 0, there exists Θ = Θ(k) such that for any x, y in X, d X (x, y) ≥ Θ then there exists V ∈ Λ such that d V (x, y) ≥ k.
2.4. CAT(0) admissible groups. Admissible groups firstly introduced in [CK02] are a particular class of graph of groups that includes fundamental groups of 3dimensional graph manifolds. In this section, we review admissible groups and their properties that will used throughout in this paper.
We often consider oriented edges from e − to e + and denote by e = [e − , e + ]. Thenē = [e + , e − ] denotes the oriented edge with reversed orientation. Denote by G 0 the set of vertices and by G 1 the set of all oriented edges.
(1) G is a finite graph with at least one edge.
(2) Each vertex group G v has center Z( is a nonelementary hyperbolic group, and every edge subgroup G e is isomorphic to Z 2 . (3) Let e 1 and e 2 be distinct directed edges entering a vertex v, and for i = 1, 2, let K i ⊂ G v be the image of the edge homomorphism G ei → G v . Then for every g ∈ G v , gK 1 g −1 is not commensurable with K 2 , and for every g ∈ G v K i , gK i g −1 is not commensurable with K i . (4) For every edge group G e , if α i : G e → G vi are the edge monomorphism, then the subgroup generated by α −1 1 (Z(G v1 )) and α −1 2 (Z(G v1 )) has finite index in G e . A group G is admissible if it is the fundamental group of an admissible graph of groups.
Definition 2.9. We say that an admissible group G is CAT(0) if it acts properly discontinuously, cocompactly and by isometries on a complete proper CAT(0) space. For the remainder of this paper, the pair (G, X) will always denote a CAT(0) admissible group with the associated CAT(0) space. For examples of admissible groups that are not CAT(0), see [KL96].
Let G be a CAT(0) admissible group, and let G T be the action of G on the associated Bass-Serre tree T of the graph of group G (we refer the reader to Section 2.5 in [CK02] for a brief discussion). Let V (T ) and E(T ) be the vertex and edge sets of T . For each σ ∈ V (T ) ∪ E(T ), we let G σ ≤ G be the stabilizer of σ.
In the rest of this section, we review facts from Section 3.2 in [CK02] that will be used thoroughly this paper. We refer the reader to [CK02] for further explanation. Given (G, X) pair, (1) for every vertex v ∈ V (T ), the set B v := ∩ g∈Z(Gv) minset(g) splits as metric product H v × R where Z(G v ) acts by translation on the R-factor and the quotient G v /Z(G v ) acts geometrically on the Hadamard space H v . (2) for every edge e ∈ E(T ), the minimal set B e := ∩ g∈Ge minset(g) splits as B e × R 2 ⊂ B v where B e is a compact CAT(0) space and G e = Z 2 acts co-compactly on the Euclidean plane R 2 . We note that the assignments v → B v and e → B e are G-equivariant with respect to the natural G actions in the sense that gB v = B gv and gB e = B ge for any g ∈ G.
Strips in X: We reviews strips in X that was discussed in the second paragraph in Section 4.1 in [CK02]). We first choose, in a G-equivariant, a plane F e ⊂ B e for each edge e in T (i.e, gF e = F ge for each g ∈ G). We will call F e is a boundary plane of B e± . For every pair of adjacent edges e 1 , e 2 . we choose, again equivariantly, a shortest geodesic α e1e2 from F e1 to F e2 (equivariantly here we mean α (ge)(ge ) = gα ee for adjacent edges e, e ). By the convexity of B v = H v × R, v := e 1 ∩ e 2 , this geodesic α e1e2 determines a Euclidean strip S e1e2 := γ e1e2 × R (possibly of width zero) for some geodesic segment γ e1e2 ⊂ H v .
Remark 2.10. There exists a G-equivariant coarse L-Lipschitz map π : X → V (T ) such that x ∈ X ρ(x) for all x ∈ X. The map π is called a index map. We refer the reader to Section 3.3 in [CK02] for the existence of such a map ρ.
Lemma 2.11. [CK02, Lemma 2.9] There exists a constant > 0 such that the following hold. Let F e and F e be two boundary planes in a vertex space Definition 2.12 (Boundary lines). Let {F e } be the collection of boundary planes of the block B v . We note that the intersection of a boundary plane F e of B v with the hyperbolic space H v is a line. We define the collection of lines L v , which we call boundary lines, in the hyperbolic space H v as the following: Remark 2.13. There exists a constant r > 0 such that for any vertex v ∈ T , for any The following lemmas are needed in Section 3 to check the HHS Axioms.
Proof. Let K e be the image of an edge group G e into G v andK e be its projection is virtually cyclic. The almost malnormality follows from non-commensurability of K e in G v . To see this, assume thatK e ∩ hK e h −1 contains an infinite order element by the hyperbolicity of Q v . If g ∈ G v is sent to h, then K e ∩gK e g −1 is sent toK e ∩hK e h −1 . Thus, K e ∩gK e g −1 contains an abelian group of rank 2. The non-commensurability of K e in G v implies that e = e and g ∈ K e . This shows that P is almost malnormal. It is well-known that a hyperbolic group is hyperbolic relative to any almost malnormal collection of quasi-convex subgroups ( [Bow12]). Thus Q v is hyperbolic relative to P.

CAT(0) admissible groups are HHS
In this section, we are going to prove the following proposition (that is also part (1) in Theorem A).
Proposition 3.1. Let G be a CAT(0) admissible group. Then G is a hierarchically hyperbolic space with respect to a word metric.
First at all, we will construct an index set Λ and a set {CW : W ∈ Λ } of δ-hyperbolic spaces that satisfies HHS's axioms (see Definition 2.7). We note that if metric spaces X and Y are quasi-isometric, then X is a hierarchically hyperbolic space whenever Y is a hierarchically hyperbolic space (see Proposition 1.10 in [BHS19]). Hence to see that G is a hierarchically hyperbolic space with respect to word metrics, we only need to work on its appropriate finite index subgroups.
Define V (T ) := V 1 ∪ V 2 where V i consists of vertices in T with pairwise even distances. Passing to a finite index subgroup if necessary, we could assume that G preserves each V i (see Lemma 4.6 in [NY20]). We emphasize here that each vertex group G v in a CAT(0) admissible groups maybe not split trivially as a product of a hyperbolic space with Z. We need to pass to a further finite index subgroup to make our group is in a simpler form. By Lemma 7.2 in [HNY21], there exists a finite index subgroup of the CAT(0) admissible group G such that this subgroup of G preserves V 1 , V 2 , and each vertex group of the subgroup is a product of a nonelementary hyperbolic group with Z. By abuse of notations, we still denote this subgroup by G (since HHS is preserved under quasi-isometry [BHS19]).

Constructing an index set.
3.1.1. Elements in index set Λ: A collection of coned-off hyperbolic spaces. Recall from Section 2.4 that for each vertex v in the Bass-Serre tree T , there is associated block For a line and a constant r > 0, a hyperbolic r-cone, denoted by cone r ( ) is the quotient space of ×[0, r] by collapsing ×0 as a point called apex. A metric is endowed on cone r ( ) so that it is isometric to the metric completion of the universal covering of a closed disk of radius r in the real hyperbolic plane H 2 punctured at the center (see [BH99,Part I be the set of boundary lines in the hyperbolic space H v (see Definition 2.12). We attach a hyperbolic r-cone to each boundary line e . The resulted space is denoted byĤ v .
Since L v has the bounded intersection property, it follows from Corollary 5.39 in [DGO] that for a sufficiently large constant r, the space H v is a hyperbolic space. Since there are finitely many vertices up to G-action, the constant r can be chosen large enough so that H v is a δ(r)-hyperbolic space for all vertex v in T .
The collection of δ(r)-hyperbolic spaces H v v ∈ V(T ) is a part in the index set Λ (in Definition 2.7) that we will define later.
3.1.2. Elements in the index set Λ: a collection of quasi-lines. We need to add another collection of hyperbolic spaces into the collection above as the following. Let e be an oriented edge in T such that e + = v. Let f e be a line in the boundary plane F e parallel to the R-direction in Recall that two lines f e = fē in the plane F e must be intersect. The action of G e on the set of lines orthogonal to f e induces an action of G e on f e . Moreover, this action of G e on f e extends to the boundary plane F e so that it commutes with the orthogonal projection to f e . Note that this new action of G e on F e preserves f e so does not coincide with the original action on F e .
We choose a Z(G v )-invariant set Kē of fiber lines of B w in F e orthogonal to the boundary line H w ∩ F e of H w in F e . A hyperbolic cone of radius r = 1 is then attached to F e along each line in Kē. The resulted coned-off space denoted by F e . We note that F e is quasi-isometric to a line and thus it is a hyperbolic space.
Let Lk(v) and St(v) be the link and the star of v in T . Then F l(v) := St(v) × R is the union of flat strips S e 's (of width 1) corresponding to the set of oriented edges e = [w, v] towards v. Let R v be the space obtained from the disjoint union F l(v) e∈St(v) F e with the boundary lines of every strip S e in F l(v) are identified with the marked lines f e in F e . Endowed with length metric, Remark 3.2. By our construction of R v 's, there exists a uniform constant r > 0 such that the following holds.
(1) R v is a r-hyperbolic space for every vertex v in T .
(2) Let v be an arbitrary vertex in T , and let w be a vertex that is adjacent to v. Denote the edge joining two vertices v and w by e. Let F e be the boundary plane in B v associated to the edge e. Let be an arbitrary line in the plane F e that is parallel to the R-factor of the adjace pice is no more than r.
(3) Let and be two lines in the plane F e that is parallel to the R-factor of the adjacent piece B w . These two lines intersect the boundary line w := F e ∩ H w at two points x and y respectively.
Definition 3.3 (The index set Λ). We define the index set Λ to be the union of the Bass-Serre tree T , the coned-off spaces H v , and the thickened fiber lines R v , i.e, 3.2. Verifying the HHS Axioms. To prove Proposition 3.1, we need to associate to each element in this index set Λ a hyperbolic space, and verify that this index set together with these hyperbolic spaces satisfy HHS's axioms in Definition 2.7. Recall that G X is a CAT(0) admissible action. Since HHS preserves under quasi-isometries (see Proposition 1.10 in [BHS19]), it suffices to show that the orbit space G(x 0 ) ⊂ X is a HHS for some x 0 ∈ X. Thus we assume, without loss of generality that x 0 belongs to a plane F e0 for some edge e 0 in T . By abusing notation, we still denote G(x 0 ) by X.
First at all, each element W in the index set Λ is associated to a hyperbolic space as the following: If W = T , we let CW denote the Bass-Serre tree T equipped with the metric d T . If W = R v for some vertex v, we let CW denote the space R v equipped with the metric d Rv described early. We recall that R v , d Rv is quasiisometric to a line, and thus R v , d Rv is a hyperbolic space. If W = H v for some vertex v, we let CW denote the space H v equipped with the coned-off metric d Hv described early in previous paragraphs. Note that these spaces are all hyperbolic spaces, and there are finitely many vertices up to G action, thus there exists a constant δ > 0 such that CW is a δ-hyperbolic space for all W in the index set Λ.
3.2.1. Projections. In this section we are going to verify the Projection Axiom of Definition 2.7. Recall that x 0 belongs to a plane F e0 for some edge e 0 in T . Let W be an element in the index set Λ, we are going to define projections π W : X → 2 CW as follows.
Assume that W = T , then we define the π W = π T to be the index map π : X → T given by Remark 2.10. Now, assume that W = H v for some vertex v ∈ T . then we define the projection π W : X → 2 CW to be the nearest point projection from We are going to define the projection π X : X → 2 CW = 2 Rv as the following: Let x be an arbitrary element in X, then π(x) is a vertex in the tree T . We note that x ∈ Y π(x) . If π(x) = v then we define π W (x) := x. If π(x) and v are two adjacent vertices then we let x be the projection of x into be base We remind here that the intersection S e k−1 ,e k ∩F e k is a line in the plane F e k , but this line in the coned-off space R v is no longer a line, and it is a bounded set in R v (see Remark 3.2).
To verify the HHS Axioms in Definition 2.7, we will use the following lemma. . Let e , e1 , e2 be boundary lines in H v associated to the edges e, e 1 , and e 2 respectively. Suppose that x and y are two points in X such that π(x) = u and π(y) = v. Then with π e ( ei ) is the projection from ei into e in the CAT(0) hyperbolic space H o .
Proof. If u = v then the result is vacuously true since both π Rw (x) and π Rw (y) are the same. We thus assume that u = v. By our definition of projections, we have π Rw (x) = S e1e ∩ F e and π Rw (y) = S e2e ∩ F e . Note that π Rw (x) and π Rw (y) are parallel lines in F e . These two lines intersect the boundary line e at two points denoted byx andȳ. By Remark 3.2, we have that We note thatx andȳ are within a uniformed bounded distance from π e ( e1 ) (projection from e1 into e in the CAT(0) hyperbolic space H o ) and π e ( e2 ) respectively. It follows that there exists a uniform constant A > 0 such that the conclusion of the lemma holds. Proof. By our definition of projection maps in Section 3.2.1, it is obvious that diam CW π W (x) ≤ ξ for some uniform constant ξ > 0. Note that each π W is coarsely Lipschitz map. Indeed, this fact is clear when W ∈ Λ\{T }. In the case W = T then the coarse Lipschitz property of π W is followed from Remark 2.10. Also, it is clear from our definition of projection maps that π W (X) is K-quasiconvex in CW for some constant K large enough. Hence Projection Axiom is verified.
3.2.2. Nesting. We equip the index set Λ with a partial order as follows: We declare W W (i.e, W is nested in itself) for all W in the index set Λ. For each W ∈ Λ, we declare that W T (i.e, every element in Λ is nested in T ). Let V, W be distinct elements in Λ\{T }. We say V is properly nested in W if CW = H v for some v ∈ V (T ) and CV = R w for some vertex w ∈ T such that v and w are adjacent vertices.
Proof. By our definition of , every element in the index set is nested in T , hence the element T ∈ Λ is the unique -maximal element. For each W in the index set Λ, we denote by Λ W the set of V ∈ Λ such that V W . Assume that V, W ∈ Λ so that V is properly nested in W . We are going to define a specific subset ρ V W ⊂ CW such that diam(ρ V W ) ≤ ξ. Also, there is a projection ρ W V from CW to subsets of CV . Note that in the Definition 2.7, there is no restriction Thus, we only need to consider the following cases: Case 1: Assume that W is the Bass-Serre tree T . In this case, we have that CW = (T, d T ). Since V is properly nested in W , it follows that V ∈ Λ\{T }. Hence CV is either the coned-off space H v , d Hv or the thickened fiber line R v , d Rv for some vertex v in T . We thus define ρ V W to be the vertex v ∈ T = CW . Obviously, diam(ρ V W ) = 0 < ξ. A projection map ρ W V : CW = (T, d T ) → 2 CV from CW to subsets of CV is defined as follows. Let x be an arbitrary vertex in the Bass-Serre tree T . If d T (x, v) ≤ 1, then we define ρ W V (x) to be the entire space CV . Now we assume that d T (x, v) ≥ 2. Let e 1 and e 2 be the last two consecutive oriented edges in the geodesic [x, v] in T such that v = (e 2 ) + . Let u be the common vertex of e 1 and e 2 , and let S e1e2 be the strip in the space B u (see Subsection 2.4).
Let r > 0 be the constant given by Remark 3.2. Applying Remark 3.2 to := S e1e2 ∩F e2 , we have that the diameter of ρ W Case 2: The last case we should consider is that W = H w . Since V is properly nested in W , it follows that V = R v for some vertex w where v and w are two adjacent vertices in T . Let e denote the edge [w, v]. We define ρ V W to be the apex-point c e in H w , and thus diam(ρ V W ) = 0 < ξ. Now, we are going to define ρ W V : CW → 2 CV as follows. Let x be a point in CW = H w , d Hw . If x is the apex-point c e then we define ρ W V (x) to be R v . Now we assume that x is not the apex-point c e . Letx the projection of x to the subspace H v of H w . Let Sx be a strip in B w such that its projection into H w is a shortest path fromx to the boundary line = F e ∩ H w . We define ρ W V (x) to be the Sx ∩ F e ⊂ R v . 3.2.3. Orthogonality. We define orthogonality ⊥ as the following: • We declare R v ⊥ R w whenever v and w are adjacent vertices.
Proof. Let V , W , and U be elements in the index set Λ such that V W and W ⊥ U . Note that W = T since every element in Λ is nested in T . If V = W then it is obvious that V ⊥ U . We thus assume that V is properly nested in W . It For the rest of the proof, we are going to verify the statement in Orthogonality Axiom (see Definition 2.7): "for each S ∈ Λ, for each U ∈ Λ S such that the set S then we have V W ". We observe that the element S must be T . Indeed, suppose that S = T . In this case then S is either Now we consider U = R w . Since V ∈ Λ S , it follows that V = H v or V = R a where a and v are adjacent. Note that H v is not orthogonal to U = R w , and V = R a is not orthogonal to U = R w (since a and w are not adjacent vertices). In other words, the set {V ∈ Λ S : V ⊥ U } = ∅.
Therefore, the element S = T , and hence Λ S = Λ. Since U ∈ Λ S , it follows that U is either T , or R v or H v for some vertex v ∈ T . Since no element in Λ is orthogonal to T , it follows that U could not be T , otherwise the set {V ∈ Λ S : V ⊥ U } = ∅. When U = R v then we define W to be H v . Note that whenever V ⊥ U = R v and V S = T then by our definition of ⊥, the element V is either H v = W W or V = R w for some vertex w that is adjacent to v. By definition of in Section 3.2.2, the element V is nested in W . When U = H v then we choose W to be the R v in the index set Λ. It is obvious that if V ⊥ U = H v and V S = T then V must be 3.2.4. Transversality and consistency. If V and W are not orthogonal and neither is nested in the other, then we say that V and W are transverse, denoted V W . We have the following possibilities for V and W .
Case 1: V = R v and W = R w for some vertices v and w in T . We note that the distance between v and w in T is at least 2. Otherwise, V ⊥ W if v and w are adjacent and V = W if v = w.
Case 2: V = R v and W = H w for some vertices v, w ∈ T . Again, we note that the distance between v and w is at least 2. Indeed, if v and w are adjacent vertices then R v is nested in H w by our definition which contradicts to the fact V and W are transverse. If v = w then V = W that contradicts to the fact V and W are transverse.
Case 3: V = H v and W = H w for some distinct vertices v, w ∈ T . Regarding Case 1: Let e 1 e 2 . . . e k−1 e k (k ≥ 2) be the geodesic edge path in the Bass-Serre tree T connecting v to w with v = (e 1 ) − and (e k ) + = w. Let F e k and F e1 be the boundary planes (see Subsection 2.4) of the blocks B w and B v respectively. Let S e k−1 e k and S e1e2 be the given strips (see Subsection 2.4) in the blocks B (e k )− and B (e1)+ respectively.
We define P V W := S e k−1 e k ∩ F e k and P W V := S e1e2 ∩ F e1 Lemma 3.8. V, W, P V W , P W V defined in Case 1 above satisfying (♣), in Definition 2.7.
Proof. Let r > 0 be the constant given by Remark 3.2. We have that diam P V W ≤ r ≤ ξ and diam P W V ≤ r ≤ ξ. As we have shown early, the distance between v and w is at least 2. Let u = π(x) ∈ V (T ) where π : X → T be the index map given by Remark 2.10.
Suppose that u does not belong to the geodesic [v, w]. If v lies between u and w then by the definition of π W (see Subsection 3.2.1) and P V W , we see that π W (x) and P V W are the same subset in CW . Similarly, if w lies in [v, u] then π V (x) and P W V are the same subset in CV . By Remark 3.2, we have that (♣) holds. Now, we assume that u belongs to [v, w]. Suppose that d(u, v) ≥ 2 (resp. d(u, w) ≥ 2) then we have that π V (x) and P W V are the same subset in CV (resp, π W (x) and P V W are the same subset in CW ). It follows from Remark 3.2 that (♣) holds whenever d(u, v) ≥ 2 or d(u, w) ≥ 2.
The final case we should consider is that d(v, w) = 2 and u lies between [v, w]. Let S := S [v,u], [u,w] be the strip in B u . Since x ∈ X = G(x 0 ), and x 0 is in the boundary plane F e0 , it follows that x must lie in a boundary plane of B u . We denote this boundary plane by F e . If e = [v, u] (the case e = [u, w] is similar) then π W (x) and S ∩ F [u,w] are identical and their diameter is no more than r in CW by Remark 3.2. Now we assume that the edges e, [v, u], and [u.w] are distinct.
We endow each H v with the HHS structure originating from the fact that H v is hyperbolic relative to the collection of boundary lines L v (see Lemma 2.14). As a result, The Transversality Axiom and the Consistency Axiom is applied to this HHS structure. In particular, min π 1 ( , 2 ), π 2 ( 1 , ) ≤ λ for any boundary line ∈ L u . It implies that (♣) holds.
Regarding Case 2: Assume that V = R v and W = H w for some vertices v, w ∈ T . Since V and W are transverse, it follows that v = w (otherwise they are orthogonal). Note that d(v, w) ≥ 2. Indeed, if v and w are adjacent vertices then R v is nested in H w by our definition which contradicts to the fact V and W are transverse. Now, let e 1 e 2 . . . e k−1 e k be the geodesic edge path in the Bass-Serre tree connecting v to w with v = (e 1 ) − and (e k ) + = w. We define and P V W := c e k the apex-point in H w . Hence, subsets P W V and P V W are bounded sets in R v (see Remark 3.2) and H w respectively. Therefore diam(ρ V W ) ≤ ξ, and diam(P W V ) ≤ ξ. Lemma 3.9. V, W, P V W , P W V defined in Case 2 above satisfying (♣), in Definition 2.7.
Proof. Let u = π(x) ∈ V (T ) where π : X → T be the index map given by Remark 2.10. If v lies between u and w then by the definition of π W in Subsection 3.2.1, we have that π W (x) is a point in the boundary line H w ∩ F e where e is the last edge in the geodesic [v, w]. Since P V W is the apex-point c e , it follows that the distance between π W (x) and P V W in H w is bounded above by r. Thus (♣) holds. If w lies in [v, u], then similar arguments as above show that π W (x) and P W V are the same subset in CV , and thus (♣) holds. Now we assume that u belongs to [v, w]. Let e be the last edge in the geodesic path [v, w]. Then P V W is the apex-point c e in H w and π W (x) is a point in the boundary line H w ∩ F e . Since the distance between π W (x) and c e in H w is bounded above by r. It follows that (♣) holds.
Regarding Case 3: Finally, suppose that V = H v and W = H w for some vertices v, w ∈ T . Since V and W are transverse so v = w. Let e 1 . . . e k−1 e k be the geodesic edge path in the Bass-Serre tree connecting v to w with v = (e 1 ) − and (e k ) + = w. We then define P V W := c e k and P W V := c e1 to be apex-points in H w and Lemma 3.11. V, W, P V W , P W V satisfy (♠) in Definition 2.7 Proof. Let u = π(x) where π : X → T be the index map given by Remark 2.10.
Case 1: Now we assume that d(u, v) > 2. Let e and e be two consecutive edges in the geodesic [u, v] such that (e ) + = v, (e) + = (e ) − . If V = R v then by our definition of π V (see Subsection 3.2.1), we have π V (x) = S e,e ∩ F e is a bounded subset in CV by Remark 3.2, and which is a bounded subset in CV by Remark 3.2; thus (♠) holds. If V = H v then π V (x) is a point in the boundary line H v ∩ F e and ρ W V (π W (x)) is the apex-point c e in H v . Since the distance between π V (x) and c e is bounded above by r in H v , it follows that (♠) holds.
When v ∈ [u, w] then π W (x) is a point in the boundary line H w ∩ F e of H w . In H w , this point and the apex-point c e have a distance that is bounded above by r, thus (♠) holds.
When w ∈ [v, u), let e be the edge in the geodesic [u, w] with (e ) + = w. We have π V (x) = π Rv (x) = S e ,e ∩ F e ⊂ R v . By the definition of π W , we have that π W (x) is a point in a boundary line of H w . Note that this line is a subset of F e . By Lemma 2.11, ρ W V (π W (x)) and S e ,e ∩ F e are within an uniform finite Hausdorff distance in the plane F e , and thus their distance is uniformly bounded with respect to the metric in R v . Thus (♠) holds. For the case u = w, then since x lies in the orbit G(x 0 ), so x ∈ F f for some edge f with (f ) + = v. The proof in this case is identical as the previous case.
So far, we have the notation ρ W V (a map from CW to subsets of CV ) whenever V is properly nested in W , and the notation P W V (a bounded subset of CV ) whenever V W . For simplicity, we will write P W V to mean either ρ W V or P W V when it is clear from the context.
Proof. We will assume that U is properly nested in V since if U = V then there is nothing to show. Assume that V = T , then U is either R v or H v for some vertex v ∈ T . The set of W ∈ Λ satisfies either V is properly nested in W or V W and W is not orthogonal to U is empty. The claim is vacuously true. Assume that V = R v for some vertex v, then the set of U ∈ Λ that is properly nested in V is empty. The claim is vacuously true. Assume that V = H v for some vertex v, then since U is properly nested in V , it follows that U = R w for some vertex w that is adjacent to v.
When V is properly nested in W then W = T . In this case, we note that W and W is not orthogonal to U = R w . Then there are two cases we should consider: u]. Let γ be the geodesic in T containing [v, w] with the endpoint u. Let e, e be the two edges in γ such that (e) + = (e ) − and (e ) + = u. We have d W (P U W , P V W ) ≤ κ 0 since both P U W and P V W are S e,e ∩ f e that is a bounded subset in CW . Hence (♦) holds.
Case 2: W = H u where u = v, w and d(u, w) ≥ 2. Let γ be the geodesic in T containing [v, w] with the endpoint u. Let e be the edge in γ with (e) + = u. Then d W (P U W , ρ V W ) ≤ κ 0 since P U W and P V W both are the apex-point c e and the lemma is proven.
3.2.5. Finite complexity. It follows from the construction that the complexity of X is 3. Indeed, we have that V W T whenever V = R v , W = H w for some adjacent vertices v and w in T .
Proof. We endow each H v with the HHS structure originating from the fact that H v is hyperbolic relative to the collection of boundary lines L v (see Lemma 2.14). As a result, Large Link Axiom is applied to this HHS structure. Since there are finitely many vertices up to G-action, there are uniform constants λ ≥ 1, L ≥ 1 such that for each vertex v ∈ T the following holds: Let x and y be two points in H v , and let N = λd Hv (π (x), π (x )) + λ. Then there exists 1 , 2 , · · · , N such that for all boundary lines ∈ L v then either = i for some i, or d (π (x), π (x )) < L. Also, d Hv (π Hv (x), P i Hv ) ≤ N for each i. Let r > 0 be the constant given by Remark 3.2. We could enlarge λ if necessary and assume that λ > 2r. We are going to verify the following statement in the Large Links Axiom of Definition 2.7. " Let W ∈ Λ and let x, x ∈ X. Let N = λd w (π W (x), π W (x ))+λ. Then there exists Recall that x 0 is in the boundary plane F e0 for some edge e 0 . Since X is the orbit G(x 0 ) and the collection of boundary planes is G-equivariant, it follows that x and x belong to boundary planes F e and F e for some edges e and e .
We first observe that if W = R v for some vertex v ∈ V (T ) then there is nothing to show since the set Λ W \{W } = ∅. Hence, in the rest of the proof, we only need to consider the cases W = H v and W = T .
Case 1: Suppose that W = H v for some vertex v ∈ T . Let A be an arbitrary element in Λ W \{W }. As A is nested in W = H v and A = W , it follows that A = R w for some vertex w which is adjacent to v.
Case 1.1: v is a vertex of both edges e and e . Without loss of generality, we assume that (e) + v and (e ) + = v. In this case, we note that F e and F e are two boundary planes in the vertex space B v = H v × R. Let e and e be the boundary lines in H v corresponding to edges e and e respectively. Letx andx be the projection point of x and x into the boundary line e and e respectively.
Each boundary line j with j ∈ {1, . . . , k} is associated to an edge e j with (e j ) + = v. We denote v j by (e j ) − . Let A j := R vj with j ∈ {1, . . . , k}. If the vertex w is equal to v j for some j ∈ {1, . . . , k} then A = A j . Now we assume that w / ∈ {v 1 , . . . , v k }. Let be a boundary line of H v associated to the edge [v, w]. We have that d (x, x ) ≤ L, otherwise w = v j for some j. Let A > 0 be the constant given by Lemma 3.4. We have It follows from the second paragraph above that with each i = 1, . . . , k, we have d W (π W (x), ρ i W ) = d Hv π Hv (x), ρ Ai Hv ≤ N ≤ N Case 1.2: Assume that (e) + = v and v is not a vertex of e (the case (e ) + = v and v is not a vertex of e is proved similarly).
When v lies between (e) − and e then the proof is identical as Case 1.1.
We note that d W π W (x), c e = d Hv π H V (x), c e ≤ r, and d Hv P

Hv
, c e ≤ r, thus v,w] are close within a uniform Hausdorff distance in the plane F [v,w] , it follows that the distance of two bounded subsets (with respect to the metric of R w ) π A (x) and π A (x ) no more than E. Case 1.3: Suppose that F e and F e are not boundary planes of B v . Denote the last edge in the geodesic from e to v is f , and the last edge in the geodesic from e to v is f . Note that both (f ) + = (f ) + = v. We will consider the collection For each i, since π W (x) is a point in the boundary line f of H v , and P Ai W is the apex-point c f , and thus d W (π W (x), P Ai W ) ≤ 2r ≤ N . Case 2: W is the Bass-Serre tree T . Recall that x ∈ F e and x ∈ F e . Let α be the geodesic in T joining the edge e to e . Let Since A is properly nested in W = T , it follows that A = R w or H w for some vertex w in T .
If A / ∈ {A 1 , A 2 , · · · , A N } then w / ∈ α. Using the same argumenrts as in the previous subcase, we have that d A (π A (x), π A (x )) < E. For each i ∈ {1, · · · , N } then A i is either R v or H v for some vertex v ∈ α. Let π : X → T be the index map given by Remark 2.10. Recall that we define the projection π T given by π T (x) = π(x) for any vertex x ∈ T . Denote α(0) and α(1) be the initial and terminal vertices of the geodesic α. Since α is a geodesic joining e to e and x ∈ F e , x ∈ F e , it follows that d(π(x), α(0)) ≤ 2 and d(π(x ), α(1)) ≤ 2. As ρ Ai The lemma is proved.
Proof. We endow each H v with the HHS structure originating from the fact that H v is hyperbolic relative to the collection of boundary lines L v (see Lemma 2.14). As a result, Bounded Geodesic Image Axiom is applied to this HHS structure. Since there are finitely many vertices up to G-action, there exists a constant E > 0 such that properties in Bounded Geodesic Image Axiom hold for every vertices v ∈ T . Let r > 0 be the constant given by Remark 3.2. We could enlarge E so that E > r.
We are going to verify (⊕) in the Definition 2.7. Let γ be a geodesic in CW . Let γ − and γ + be the initial and terminal points of γ. We consider the following cases: Case 1: Suppose that W = T . Since V is properly nested in W , it follows that V is either R v or H v for some vertex v in T . There are two subcases we will consider: share the last common two consecutive edges, denoted by e 1 , e 2 . When V = R v then it follows from the definition of Then there is nothing to show since there is no element V ∈ Λ such that V is properly nested in W .
Case 3: Suppose that W = H v . Since V is properly nested in W , it follows that V = R w where w and v are adjacent vertices. Let e denote the edge [v, w], and let be the boundary line in H v associated to the edge e. Since (⊕) holds for hierarchically hyperbolic structure of H v , we have that Note that diam (ρ Hv (γ)) and diam Rw (ρ Hv (γ)) are within a uniform bounded distance, and ρ Hv and (ρ R W Hv are the apex point c e . Thus it follows that there exists a constant E which does not depend on v, , and γ such that diam Rw (ρ Hv (γ)) ≤ E or γ ∩ N E (ρ R W Hv ) = ∅ Replacing E by max{E, E } if necessary, we have that (⊕) holds.
Proof. Since {V i } be a family of pairwise orthogonal elements of Λ. It follows from the definition of orthogonality (see Section 3.2.3) that the collection {V i } is either Let p 1 and p 2 be two arbitrary elements in π V1 (X) and π V2 (X). We are going to define a point x ∈ X so that it satisfies the conditions in Partial Realization Axiom as the following.
Case 1: Assume that V 1 = R v and V 2 = H v . Let r > 0 be a sufficiently large constant such that it applies to both Remark 3.2 and Remark 2.13. We note that p 2 is a point in H v . By Remark 3.2, there exists a pointx in a boundary line of H v such that d Hv (p 2 ,x) ≤ r. Let x be a point in B v = H v × R such that its projection into H v isx and its R-coordinate is the same as p 1 .
Case 2: Assume that V 1 = R v and V 2 = H w where v and w are adjacent vertices. Let e denote the edge [v, w]. Since p 1 ∈ π Rv (X) and p 2 ∈ π Rw (X), it follows that they belong to boundary planes of B v and B w (this follows from the definition of π Rv and π Rw ). Choose q 1 and q 2 to be points in the plane F e such that the R-coordinate of q 1 in B v = H v × R is the same as of p 1 and the R-coordinate of q 2 in B w = H w × R is the same as of p 2 . Let be the line in the plane F e that is parallel to R-factor of B v and passes through q 1 . Let be the line in the plane F e that is parallel to R-factor of B w and passes through q 2 . These two lines and intersect at a point, and we will denote this point by x.
Verifying the point x satisfies the conditions in Partial Realization uses similar arguments as in Section 3.2.4 and is hence omitted to avoid redundancy.
3.2.9. Uniqueness. Proof. We are going to verify the following statement. For each k ≥ 0, there exists Θ = Θ(k) such that for any x, y in X, d(x, y) ≥ Θ then there exists V ∈ Λ such that d V (x, y) ≥ k.
We endow H v with the HHS structure originating from the fact that H v is hyperbolic relative to the collection of boundary lines L v (see Lemma 2.14). As a result, Uniqueness Axiom is applied to this HHS structure. Since there are finitely many vertices up to G-action, there exists a constant ξ(k) > 0 such that for each vertex v, for any two pointsx andȳ in Let µ > 1 be the constant given by Lemma 4.6 in [NY20]. Let Θ be a sufficiently large constant such that Let π : X → T be the index map given by Remark 2.10. Case 1: Two vertices π(x) and π(y) has distance at least k in T . Recall that we define the projection π T = π. In this case, we will choose V to be T . It follows that d V (x, y) = d T (x, y) := d T (π T (x), π T (y)) = d T (π(x), π(y)) ≥ k.
Case 2: Two vertices π(x) and π(y) has distance at most k in T . Suppose that π(x) = π(y). Let v be the vertex π(x) = π(y). Then we have that x ∈ Y π(x) = Y v and y ∈ Y π(y) = Y v . Let γ be a geodesic in B v = H v × R connecting x to y. Let γ 1 and γ 2 be the projections of γ into factors H v and R of B v respectively. Since d(x, y) ≥ Θ, it follows that Hence either Len(γ 1 ) or Len(γ 2 ) is greater than or equal to Θ i.e, d Hv ((γ 1 ) − , (γ 1 ) + ) ≥ ξ(k), then either d Hv (x,ȳ) ≥ k or d (x,ȳ) ≥ k for some boundary line ∈ L v . In the first case, we will let V = H v . In the later case, let e be the edge in T with (e) − = v such that is associated to e. We then let V = R w where w = (e) + . Now we assume that π(x) = π(y). Let e 1 · · · e n be the geodesic edge path connecting π(x) to π(y) and let p i = S ei−1ei ∩ S eiei+1 be the intersection point of adjacent strips, where e 0 := x and e n+1 := y. Hence we have the following path which is a concatenation of geodesics: where p 0 := x and p n+1 := y. Since we assume that d T (π(x), π(y)) ≤ k, it follows that n ≤ k + 2.
Proof of Proposition 2.14. The proof is the combination of the previous lemmas in this section.

Appendix: Poisson boundaries of hierarchically hype rbolic spaces
Let G be a group that acts geometrically on a hierarchically hyperbolic space (Definition 2.7). Such a group we refer to as an HHS group, and note that conjecturally it is not equivalent to being a hierarchically hyperbolic group. In this section, we show that for an appropriate choice of κ that only depends on the HHS group G, the κ-Morse boundary of G serves as a topological model for the Poisson boundaries of the pair (G, µ), where µ is a finitely-supported , non-elementary, generating measure.
We need to show that a generic sample path of such a random walk sublinearly tracks a κ-Morse quasi-geodesic ray. We will do so by showing that, in fact, the limiting quasi-geodesic ray is κ-weakly contracting((Definition 5.3 in [QRT20]). This proof follows Section 8 of [QRT20] where the same result is claimed for mapping class groups.
Without loss of generality, equip X with a base-point o and we use o S to denote a point in π S (o). Fix o S once and for all.
The hierarchy of geodesics. Let S denote the maximal element in Λ and recall CY denoted the associated hyperbolic spaces for each Y ∈ Λ. To every pair of points x, y ∈ G one can associate a hierarchy of geodesics, which are a set of geodesic segments in each CY that connects π Y (x) to π Y (y). Hence we also write H(x, y) = {[x, y] Y } for specific given Y ∈ Λ. Given H(x, y) = {[x, y] Y |Y ∈ Λ}, a realization of a hierarchy H(x, y) is a uniform quasi-geodesic segment G(x, y) in G connecting x to y where, for any element of Y ∈ Λ, the projection of G(x, y) to CY is contained in a uniformly bounded neighborhood of the geodesic segment [x, y] Y . The set of all realizations we denote G(x, y).
We can also replace x or y with elements of G whose projection to CS are infinite quasi-geodesic rays.
[We start with a (tight) geodesic [x, ξ) S in CS and build H(x, ξ), the same as before replacing, for every element Y ∈ Λ, π Y (y) with π Y (ξ). The realization G(x, ξ) of H(x, ξ) is then a uniform quasi-geodesic in G starting from x such that the shadow of G(x, ξ) in CS converges to ξ. ] Definition 4.1 (Centers). Since elements of Λ are uniformly hyperbolic spaces, let them all be D-hyperbolic. There exists a constant δ(D) such that for any three points x, y, z, the intersection of the δ(D)-neighbourhood of the geodesic segments [x, y], [y, z] and [x, z] are non-empty. Given any y ∈ Λ, we use ctr Y (x, y, z) to denote any point in the said intersection. It follows from the construction of HHS that there exists a constant D such that: Given x, y, z ∈ G, for any Y there exists a point η ∈ G such that, for any Y ⊆ S, we have We call η the center of x, y and z and we denote it by ctr(x, y, z).
We use the hierarchy paths to show: Proposition 4.2. Let p be the complexity of the hierarchy of G. For any x, y ∈ X, assume that d Y (x, y) ≤ E for all Y = S, Y ∈ G and some E > 1. Then we have Proof. In view of the Distance Formula, we need to show The restriction of H(x, y) to a element of Λ is again a hierarchy which we denote with H Y (x, y). Let α denotes minimal elements in the poset Λ, and let We check the statement of Proposition 4.2 inductively. When p = 1 the statement is vacuously true. Now let p ≥ 2. By induction, that for every element of Λ, the hierarchy Note that, once again we can replace each of x, y, z with an element ξ ∈ ∂CS. That is, ctr(x, y, ξ) is a well-defined element of G. From now on, we will denote as o the identity element in G, which will function as base point.
Definition 4.3. (Projections in HHS) Let D be given from 3, and let ξ be an element of G whose shadow in CS is an infinite diameter, quasi-geodesic ray in ∂CS. We define a D-cloud of a ray in the direction of ξ to be By construction, the realization G(o, ξ) of the hierarchy H(o, ξ) is contained in Z(o, ξ). Fixing ξ ∈ ∂CS, we define a projection map to the cloud: We now check that Π ξ is a κ-projection according to Definition 2.1.
Lemma 4.4. Let ξ ∈ G be an element whose shadow in CS is an infinite diameter, quasi-geodesic ray in ∂CS. , the map Π ξ is coarsely Lipschitz with respect to d G . Furthermore, if x ∈ Z(o, ξ), then d G (x, Π ξ (x)) is uniformly bounded. As a consequence, Π ξ is a κ-projection.
Proof. Consider points x, x ∈ G where d w (x, x ) ≤ 1. Then x(θ) and x (θ) have a uniformly bounded intersection number, which implies that there exists a uniform constant C 1 > 0 such that Let η := ctr(o, x, ξ) and η := ctr(o, x , ξ). Since CY is hyperbolic, the dependence of η Y on x Y is Lipschitz, that is, there exists a uniform constant C 2 > 0 such that Now, Proposition 4.2 implies that Lemma 4.5. There exists L > 0 such that the following holds. Let x, y ∈ G, and let γ ∈ G be an infinite geodesic ray based at o, and let x γ := c γ,y (x). For any Proof. Let y γ := π γ (y). Since Y is hyperbolic, there exists L 1 , R 0 such that, if d Y (π γ (x), π γ (y)) ≥ L 1 , the geodesic γ 1 := [x, y] in G and the broken geodesic lie in a R 0 -neighborhood of each other for the metric d Y . Let p 1 , p 2 be nearest point projections (in G), respectively, of x γ and y γ onto [x, y]. This implies which proves the claim, as long as L ≥ R 0 .
Proof. We now fix L as given by Lemma 4.5, and we start by contradiction, by assuming that d S (π γ (x), π γ (y)) ≥ L. By Lemma 4.5, d S (x, x γ ) ≺ d S (x, y), and d Y (x, x γ ) ≺ d Y (x, y) whenever d Y (x, x γ ) is large enough. Now, applying the Distance Formula (Theorem 4.5 [BHS19]) to the pair of points (x, y) we have: by Eq. (4) That is to say, there exists D 1 = D 1 (L, δ) such that which is a contradiction since x γ ∈ γ. Therefore, setting D 2 = L yields Logarithmic projections. We now consider the set of elements in G whose projection to CS are quasi-geodesic rays in ∂CS that have logarithmically bounded projection to all elements of Λ. Recall in Definition 2.7, When V W then there is a specific subset ρ V W ⊂ CW such that diam(ρ V W ) ≤ ξ. Given a non-maximal element Y ∈ Λ, let Y S := d S (o S , ρ Y S ). In comparison, for x ∈ G, define x S := d S o S , π S (x)) .
Definition 4.7. For a constant c > 0, let L be the set of elements in G whose shadows to an infinite quasi-geodesic ray ξ ∈ ∂CS such that for every non-maximal element Y ∈ Λ.
Proof. In this proof, we use the notations ≺ c and O c to mean that the implicit constants additionally depend on c. Let Z = Z(o, ξ). Given x, x ∈ X where let y = Π ξ (x), y = Π ξ (x ). We claim that, for every proper element of Λ, Since CS is hyperbolic, nearest point projection in CS is coarsely distance decreasing, hence y S ≺ x S . Also, by Theorem 4.6 (6) d S (y, y ) ≤ D 2 therefore, y S ≺ x S . Which means, for every curve α in the geodesic segment [y, y ] S in CS we have d S (o, α) ≺ x S . Axiom (7) (Bounded Geodesic Image) implies that if d Y (y, y ) is large then d S ([y, y ] S , ∂Y ) ≺ 1, hence By the definition of Π ξ , y Y and y Y are D-close to the geodesic segment [o, ξ] Y in CY and, by assumption, the length of this segment is at most a uniform multiple of log Y S . Therefore, In view of Equation (6) and Proposition 4.2, we get d G (y, y ) ≺ c log p x S . Now, by Theorem A.1 in [QRT20], Z(o, ξ) is κ-Morse. Let m Z be the associated Morse gauge for Z(o, ξ). Now we show G(o, ξ) is also κ-Morse. Assume κ and r > 0 be given (see Definition 2.3) and, using the fact that Z(o, ξ) is κ-Morse, let R be a radius such that, for any (q, Q)-quasi-geodesic ray β in G with m Z (q, Q) small compared to r, we have d G (β R , Z(o, ξ)) ≤ κ (R) =⇒ β| r ⊂ N κ (Z(o, ξ), m Z (q, Q)).
We need to show that every x ∈ β| r is close to G(o, ξ).
In fact y can be taken to be Π ξ (x) and hence y S ≺ x S . Let z be a point in G(o, ξ) where d S (z S , y S ) ≤ D (such a point exists since the shadow of G(o, ξ) to CS is the geodesic ray [o, ξ) S ). Since y, z ∈ Z(o, ξ), we have for every element of Λ that d Y (y, z) ≺ c log max( y S , z S ) ≺ log( x S + D) ≺ log x S .
Therefore, by Proposition 4.2, we have d G (y, z) ≺ c log p x S ≺ κ(x).
That is, G(o, ξ) is κ-Morse with a Morse gauge m G = O c (m Z ).

4.1.
Convergence to the κ-Morse boundary. Let µ be a probability measure on G. We say that µ is non-elementary if the semigroup generated by its support contains two loxodromic elements with disjoint fixed sets in ∂CS. Let us recall some useful facts on random walks on any HHS group.
Theorem 4.9. Let µ be a finitely supported, non-elementary probability measure on an HHS group G. Then: (1) For almost every sample path ω = (w n ), the sequence (w n ) S converges to a point ξ ω in the Gromov boundary of CS.
Claim (2) is proven by Maher ([Mah10], [Mah12]), while (1) and (3) are proven by Maher-Tiozzo in [MT18]. Also, exactly the same proof as in [QRT19, Theorem A.17] yields for HHS groups that for finitely supported, non-elementary probability measure on G. Then for any k > 0 there exists C > 0 such that for all n we have where the supremum is taken over all (proper) elements of Λ of S. As a consequence, for almost every sample path there exists C > 0 such that for all n sup Y d Y (o, w n ) ≤ C log n.
It follows that almost every sample path converges to a point in the κ-Morse boundary of the HHS group, where κ(r) = log p (r). We now complete the proof of Theorem B by identifying the κ-Morse boundary with the Poisson boundary.
Theorem 4.10. Let µ be a non-elementary, finitely supported measure on an HHS group G. Then for κ(r) := log p (r), where p is the complexity of the hierarchy, the κ-Morse boundary is a topological model for the Poisson boundary of (G, µ).
Proof. Since almost every sample path sublinearly tracks a κ-Morse geodesic ray, with κ(r) = log p (r), so we can apply Theorem 2.6.