Right-angled Coxeter groups with totally disconnected Morse boundaries

This paper introduces a new class of right-angled Coxeter groups with totally disconnected Morse boundaries. We construct this class recursively by examining how the Morse boundary of a right-angled Coxeter group changes if we glue a graph to its defining graph. More generally, we present a method to construct amalgamated free products of CAT(0) groups with totally disconnected Morse boundaries that act geometrically on CAT(0) spaces that have a treelike block decomposition. We deduce a new proof for the result of Charney-Cordes-Sisto (Complete topological descriptions of certain Morse boundaries, Groups Geom. Dyn. 17(1),157–184 (2023)) that every right-angled Artin group has totally disconnected Morse boundary, and discuss concrete examples of surface amalgams studied by Ben-Zvi (Boundaries of groups with isolated flats are path connected. arXiv:1909.12360, 2019).


Introduction
This paper presents new examples of right-angled Coxeter groups that have totally disconnected Morse boundaries.These examples arise from a more general construction of CAT(0) spaces with treelike block decompositions that have totally disconnected Morse boundaries.1.1.Motivation.The Morse boundary ∂ M Σ of a proper geodesic metric space Σ is a quasiisometry invariant defined by Cordes [Cor17].It generalizes the contracting boundary introduced by Charney-Sultan [CS15] in the CAT(0) case.If Σ is a proper, geodesic hyperbolic space, its Morse boundary ∂ M Σ coincides with the Gromov boundary.In general, ∂ M Σ is a topological space consisting of equivalence classes of Morse geodesic rays, i.e. geodesic rays that behave similar to geodesic rays in hyperbolic spaces.
The Morse boundary ∂ M G of a finitely generated group G is the Morse boundary of a proper geodesic metric space on which G acts geometrically, ie.properly and cocompactly by isometries.If every geodesic ray bounds a half-flat, e.g. in higher-rank lattices, then ∂ M G is empty.However, there is a large class of non-hyperbolic finitely generated groups with non-empty Morse boundaries.
Interesting examples arise among right-angled Coxeter groups (RACGs) and right-angled Artin groups (RAAGs).Each such group is defined by a finite, simplicial graph, its defining graph and acts geometrically on an associated CAT(0) cube complex.Charney-Cordes-Sisto [CCS] showed: Theorem 1.1 (Charney-Cordes-Sisto).The Morse boundary of every RAAG is totally disconnected.It is empty, a Cantor space, an ω-Cantor space or consists of two points.
If a RACG has totally disconnected Morse boundary, its Morse boundary is also homeomorphic to one of the spaces listed in the theorem above by Theorem 1.4 in [CCS] (see Section 6.1).But in contrast to RAAGs, it is often difficult to determine whether a RACG has totally disconnected Morse boundary or not as many different topological spaces arise as Morse boundaries of RACGs.
1.2.RACGs with totally disconnected Morse boundaries.The right-angled Coxeter group (RACG) associated to a finite, simplicial graph ∆ = (V, E) is the group The group W ∆ acts geometrically on an associated CAT(0) cube complex Σ ∆ , its Davis complex.Hence, the Morse boundary of W ∆ , denoted by ∂ M W ∆ , is the Morse boundary of Σ ∆ .For instance, if ∆ is a 4-cycle, then Σ ∆ is isometric to R 2 and has empty Morse boundary.If ∆ is a 5-cycle, then Σ ∆ is quasi-isometric to the hyperbolic plane and ∂ M Σ ∆ is a circle.If we glue a 4-cycle to a cycle of length at least 5 so that the 4-cycle contains a non-adjacent vertex-pair of the other cycle as in Figure 2, then the corresponding Davis complex has totally disconnected Morse boundary (see Lemma 6.7).On the other hand, if a graph ∆ contains an induced cycle C of length at least 5 without such a glued 4-cycle, then ∂ M Σ ∆ contains a circle [Tra19,Cor 7.12], [Gen20, Prop.4.9].See also [Beh19] and [RST,Thm 7.5].Tran conjectured [Tra19][Conj.1.14] that the nonexistence of such a cycle C implies that the associated Davis complex has totally disconnected Morse boundary.This was disproved in [GKLS21].The problem, which right-angled Coxeter groups have totally disconnected Morse boundaries turns out to be difficult and is still open.
In this paper, we present a new class of right-angled Coxeter groups with totally disconnected Morse boundaries by examining the following question: Question 1.2.Suppose that ∆ is a finite, simplicial graph that can be decomposed into two distinct proper induced subgraphs ∆ 1 and ∆ 2 with the intersection graph Λ = ∆ 1 ∩ ∆ 2 .Are there conditions in terms of ∆ 1 , ∆ 2 and Λ implying that ∂ M Σ ∆ is totally disconnected?Question 1.2 is inspired by an example of Charney-Sultan [CS15, Sec.4.2]: Let ∆ be the graph in Figure 3. Charney-Sultan show that Σ ∆ has totally disconnected Morse boundary.For the proof, they decompose ∆ into two induced subgraphs ∆1 and ∆2 pictured in Figure 3. Since ∆1 and ∆2 are induced subgraphs of ∆, their corresponding Davis complexes Σ ∆1 and Σ ∆2 are isometrically embedded in Σ ∆.Contrary to the case of visual boundaries, this does not imply that the Morse boundaries ∂ M Σ ∆1 and ∂ M Σ ∆2 are topologically embedded in ∂ M Σ ∆.Definition 1.3.Let Σ be a proper geodesic metric space and B ⊆ Σ.We denote by (∂ M B, Σ) the relative Morse boundary of B in Σ, i.e. the subset of ∂ M Σ that consists of all equivalence classes of geodesic rays in B that are Morse in the ambient space Σ.
For instance, if Σ = R 2 and B is the x-axis, then ∂ M B = {−∞, +∞} but (∂ M B, Σ) = ∅.If we endow (∂ M B, Σ) with the subspace topology of ∂ M B and ∂ M Σ, we obtain two topological spaces that might be distinct (see Example 4.17).If B is closed and convex, the second topology is finer than the first one (see Lemma 4.12).Charney-Sultan use this observation implicitly.They show in [CS15,Sec. 4.2, that the relative Morse boundaries (∂ M Σ ∆1 , Σ ∆) and (∂ M Σ ∆2 , Σ ∆) endowed with the subspace topology of ∂ M Σ ∆1 and ∂ M Σ ∆2 are totally disconnected and conclude that ∂ M Σ ∆ is totally disconnected.An essential ingredient of their proof is that ∂ M Σ ∆2 = ∅.
We generalize this approach using the keyobservation that the intersection graph ∆1 ∩ ∆2 lies in a subgraph of ∆ that corresponds to a RACG with empty Morse boundary (namely ∆2 ).RACGs with empty Morse boundary can be characterized in terms of the following definitions.Definition 1.4.A graph is a clique if every pair of vertices is linked by an edge.A graph ∆ is a join of two vertex disjoint graphs ∆ 1 and ∆ 2 if ∆ is obtained from ∆ 1 and ∆ 2 by linking each vertex of ∆ 1 with each vertex of ∆ 2 .If neither ∆ 1 nor ∆ 2 is a clique, then ∆ is a non-trivial join.
For instance, the graph ∆2 is a non-trivial join of two graphs each consisting of three vertices.Corollary B in [CS11] implies Lemma 1.5 (Caprace-Sageev).A RACG has empty Morse boundary if and only if its defining graph is a clique or a non-trivial join.
We are now able to formulate the main result of this paper.
Theorem 1.6.Suppose that ∆ is a finite, simplicial graph that can be decomposed into two distinct proper induced subgraphs ∆ 1 and ∆ 2 with the intersection graph Λ = ∆ 1 ∩ ∆ 2 .Suppose that Λ is a clique or contained in a non-trivial join of two induced subgraphs of ∆.Then every connected component of ∂ M Σ ∆ is either (1) a single point; or (2) homeomorphic to a connected component of (∂ M Σ ∆i , Σ ∆ ) endowed with the subspace topology of ∂ M Σ ∆ where i ∈ {1, 2}.
In Definition 6.8, we will define a large class C of graphs, that can be built iteratively from pieces to which Corollary 1.7 can be applied.
The class C is much larger than the class CF S 0 defined in Definition 6.10 below for which Corollary 1.8 was established by Nguyen-Tran [NT19].For instance, the graphs in Figure 1 are contained in C \ CF S 0 .The left graph was studied by Russell-Spriano-Tran [RST,Ex. 7.7].They asked whether the associated RACG has totally disconnected Morse boundary or not.The other graphs in Figure 1 correspond to RACGs with polynomial divergence of arbitrarily high degree [DT15, Sec.5] (see Lemma 6.13).In contrast, all graphs in CF S 0 have quadratic divergence.
1.3.CAT(0) spaces with a treelike block decomposition that have totally disconnected Morse boundaries.Our results concerning RACGs follow from a more general theorem concerning groups acting geometrically on CAT(0) spaces with treelike block decompositions.Such spaces were studied in [CK00, BZ, BZK21, Moo10] since they arise naturally as spaces on which interesting examples of amalgamated free products of CAT(0) groups act geometrically.We will give a precise definition in Definition 2.1 below.For this introduction, it suffices to know that a block decomposition B of a CAT(0) space Σ is a collection of convex, closed subsets of Σ, called blocks, whose union covers Σ.The non-trivial intersection of a pair of blocks is called a wall.The block decomposition is treelike if the blocks intersect each other so that we obtain a simplicial tree if we add a vertex for every block and an edge for every pair of blocks that intersect non-trivially.Theorem 1.6 is a special case of Theorem 1.9.Indeed, in Proposition 6.5 we will show that a Davis complex of a RACG as in Theorem 1.6 admits a treelike block decomposition whose blocks are isometric to Σ ∆1 or Σ ∆2 and whose walls are isometric to Σ Λ .Because of Lemma 1.5, this decomposition satisfies the the conditions of Theorem 1.9.
1.4.Beyond RACGs.Theorem 1.9 has many applications beyond RACGs.It can be applied to a class of RAAGs partially reproving Theorem 1.1 (see Corollary 7.8), to surface amalgams and to spaces arising from the equivariant gluing theorem of Thm II.11.18].We will finish this paper with a few concrete examples that were studied by Ben-Zvi [BZ].
1.5.Organization of the paper.Section 2 concerns treelike block decompositions of CAT(0) spaces.In Section 3, we will prove a cutset property for visual boundaries of CAT(0) spaces with a fixed treelike block decomposition.In Section 4, we will transfer this property to the Morse boundary and study two further keyproperties of Morse boundaries.In Section 5, we will use these three keyproperties to prove Theorem 1.9.In Section 6, we will apply our insights to RACGs.Finally, we close this paper with applications beyond RACGs in Section 7.

CAT(0) spaces with treelike block decompositions
In Section 2.1, we will fix notation.Section 2.2 concerns basic properties of CAT(0) spaces with treelike block decompositions.Section 2.3 is about itineraries of geodesic rays in such spaces.
2.1.Notation concerning simplicial graphs.For the background of graphs, see [Wes01].For us, a simplicial graph ∆ = (V (∆), E(∆)) consists of a set V (∆) and set E(∆) of 2-element subsets of V (∆).The elements of V are called vertices and the elements of E are called edges.If ∆ 1 and ∆ 2 are two graphs, then ∆ 1 ∪ ∆ 2 is the graph whose vertex set is V (∆ 1 ) ∪ V (∆ 2 ) and whose edge set is the set E(∆ 1 ) ∪ E(∆ 2 ).Analogously, ∆ 1 ∩ ∆ 2 denotes the graph whose vertex set is V (∆ 1 ) ∩ V (∆ 2 ) and whose edge set is the set E(∆ 1 ) ∩ E(∆ 2 ).A subgraph ∆ of a graph ∆ is a graph whose vertex set is contained in V (∆) and whose edge set is contained in E(∆).The subgraph ∆ is a proper subgraph if it does not coincide with ∆.A graph ∆ is an induced subgraph of a graph ∆ if every edge e ∈ E(∆) whose endvertices are contained in V (∆ ) is contained in E(∆ ).We say in this case that ∆ is spanned by the vertex set V .
Two vertices are adjacent if they are contained in an edge.The degree of a vertex v is the number of vertices that are adjacent to v. Let v 1 , . . ., v n ∈ V (∆) and e 1 , . . ., e n−1 ∈ E(∆).The list for all i ∈ {1, . . ., n − 1} and v n ∈ e n−1 .In this case, v 1 and v n are linked by a finite path.If v 1 = v n , P is a closed path.Let (v i ) i∈N and (e i ) i∈N be two sequences of vertices and edges in V (∆) and E(∆) respectively.The infinite list (v 1 , e 1 , v 2 , e 2 , . . .e n−1 v n . . . ) is an infinite path if v i ∈ e i for all i ∈ N. Let (v i ) i∈Z and (e i ) i∈Z be two sequences of vertices and edges in V (∆) and E(∆) respectively.The bi-infinite list (.
If we speak of a path, we mean a finite, infinite or bi-infinite path.
A path P is geodesic, if each vertex occurs at most once in P .A path P is a subpath of a path P if P has two vertices v 1 and v 2 so that P is obtained from P by removing all vertices and edges that occur before the vertex v 1 or after the vertex v 2 in P .The underlying graph of a path P , denoted by P , is the graph whose vertex set consists of all vertices in P and whose edge set consists of all edges in P .If P is a geodesic path, each vertex in P has degree at most two.
A graph is connected if every two vertices are linked by a finite path.A cycle is a graph with an equal number of vertices and edges whose vertices can be placed around a cycle so that two vertices are adjacent if and only if they appear consecutively along the cycle.A graph is a (simplicial) tree if it is connected and does not contain a cycle.An important property of trees is that every geodesic path linking two vertices is unique.In this case, the elements of B are called blocks.A block decomposition is non-trivial if there are at least two blocks.If B is a non-trivial block decomposition, the elements of the set Definition 2.2.The adjacency graph of a block decomposition B is the simplicial graph whose vertex set is B and whose set of edges consists of all pairs of blocks with non-empty intersection.
Recall that a simplicial tree is a connected simplicial graph that does not contain a cycle.
Definition 2.3.A block decomposition is called treelike if (1) the adjacency graph is a simplicial tree; and if (2) there exists d W > 0 so that for all Let Σ be a complete CAT(0) space with treelike block decomposition B and T the corresponding adjacency graph.The following two properties are important for us.
Lemma 2.4.If Σ is proper, then no ball of finite radius in Σ is intersected by infinitely many walls.
Proof.Let B be a ball of finite radius.As Σ is proper, B is compact and we are able to cover B by finitely many balls of radius d W 4 .By the separating property 2.3 (2), each ball of radius d W 4 is intersected by at most one wall.Hence, the number of walls intersecting B is less or equal to the number of the balls that are used for covering B.
Lemma 2.5.Let T be a (possibly infinite) subtree of T with vertex set V .Then the set B∈V (T ) B is closed and convex.
We need the following lemmas for proving Lemma 2.5.
Lemma 2.6.The intersection of more than two blocks is empty.In particular, every two distinct walls are disjoint and every point in Σ is either contaied in exactly one wall or in exactly one block but not both.
We show that distinct walls are disjoint: Indeed, let It remains to show that every x ∈ Σ is either contained in exactly one wall or in exactly one block but not both: Let x ∈ Σ. Suppose that x is not contained in exactly one block.Then x is contained in at least two blocks.The intersection of more than two blocks is empty.Thus, x is contained in exactly two blocks.This means, that x is contained in a wall.As distinct walls are disjoint, there exists exactly one wall containing x. Lemma 2.7.Each wall is closed and convex and the map Proof.The intersection of two closed convex sets is closed and convex.Every wall is the intersection of two blocks and by definition, each block is closed and convex.Thus, each wall is closed and convex.
Next, we show that f is surjective.Let W be a wall.Then there are two blocks B 1 , B 2 , Proof.Let B be a block and Assume for a contradiction that δ = 0. Then for each > 0 there exists a wall W ∈ W such that W ∩ U (x) = ∅.By the separating property 2.3 (2), W = W for all , ∈ (0, d W ). Thus, there exists a wall W such that U (x) ∩ W = ∅ for all > 0. Hence, x is a limit point of W .By Lemma 2.7, W is closed.So, W contains all its limits points.In particular, x ∈ W .But then, x / ∈ B -a contradiction to the choice of x.
Lemma 2.9.Let c : [a, b] → Σ be a curve connecting two distinct blocks B 1 and B 2 .Let Then γ(t 0 ) ∈ W B1 and γ(t , then there exists t ∈ (t 0 , t 1 ) such that γ(t ) is not contained in a wall.
Proof.Suppose that we have already proven that γ(t 0 ) ∈ W B1 and γ(t 1 ) ∈ W B2 .By Lemma 2.6, , then γ(t 0 ) and γ(t 1 ) are contained in two distinct walls.Then γ([t 0 , t 1 ]) is a curve connecting two distinct walls.By the separating property 2.3 (2), the distance of two distinct walls is at least d W . Thus, γ((t 0 , t 1 )) contains a point x that is not contained in a wall.
It remains to prove that γ(t 0 ) lies in W B1 and that γ(t 1 ) lies in W B2 .We focus on proving that γ(t 0 ) lies in W B1 .Therefore, we observe that the topological frontier of B 1 is contained in W B1 .Indeed, Let x be a point in the topological frontier of B 1 .As B 1 is closed, x ∈ B 1 .As the set B1 is open by Lemma 2.8, B1 is contained in the interior of B 1 .Thus, x / ∈ B1 .Hence, x ∈ B 1 \ B1 , i.e. x ∈ W B1 and the topological frontier of B 1 is contained in W B1 .
A similar argumentation shows that γ(t 1 ) lies in W B2 .
Suppose that this would not be the case.Then there exists a block It remains to consider the case that there is no block B in V (T 1 ) such that x ∈ B. In that case, there exists a unique wall Ŵ = W corresponding to an edge in T 1 that contains x by Lemma 2.6.Let Now, we prove the lemma.Let P be a path in T linking two blocks and c : [a, b] → Σ a curve linking two points in these two blocks.We have to show that c passes through every wall that corresponds to an edge of P .Assume for a contradiction that there exists a wall W corresponding to an edge in It remains to study the case where P 1 has length at least one.We assume Without loss of generality that P 1 does not contain the vertex B (otherwise, we switch the roles of B and B ).
We have to prove that c 1 does not intersect W . Assume for a contradiction that there exists j ∈ {0, . . ., k − 1} such that W ∩ ĉj = ∅.By the choice of {w j } j∈{1,...,k} , every pair of two consecutive points w j and w j+1 are contained in B j+1 for all j ∈ {0, . . ., k − 1}.As each block is convex, ĉj ⊆ B j+1 for all j ∈ {0, . . ., k − 1}.Thus, if As B is not contained in P 1 , and as P 1 is a geodesic path, the three blocks B , B 1 and B j+1 are distinct.That contradicts Lemma 2.6.Thus, c 1 does not intersect W .The curve c 2 can be defined analogously.
Proof of Lemma 2.5.At first, we show that As Σ is CAT(0), there exists a unique geodesic segment γ : [a, b] → Σ connecting x and y.We have to show that γ([a, b]) ⊆ M .Let B and B be two blocks in V (T ) containing x and y respectively.As T is connected, there exists a geodesic path P connecting B and B .As T is a tree, this path is unique.As T is a subtree of T , P is a path in T .We show the statement by induction on the length of P .
If P consists of a single vertex, x and y lie in a common block Let P be a path of length k.Let W be a wall corresponding to an edge in P .By Lemma 2.11, there exists such that the geodesic path P connecting B with B does not contain both B 1 and B 2 .As geodesic paths in trees are unique, P is a subpath of P that is shorter than We have to prove that x is an interior point of Σ \ M .First suppose that x is not contained in a wall.Then there exists a unique block B containing x by Lemma 2.6.As x ∈ Σ \ M , B is not contained in V (T ).As x is not contained in a wall, x ∈ B. By Lemma 2.8, B is open.Thus, there exists an open neighborhood U about x that is contained in B. By definition, B consists of points that are not contained in any other block than B. Thus, Remark 2.12.It is possible to prove the lemmas in this section for geodesic metric spaces Σ without assuming that Σ is CAT(0).Furthermore, one can omit the assumption that blocks are convex except for the convexity-statement in Lemma 2.7 and Lemma 2.5.Remark 2.13 (Criterion for treelike block decompositions).If a CAT(0) space satisfies the following three conditions introduced by Mooney [Moo10], then Σ is a CAT(0) space with a treelike block decomposition as in Definition 2.1.
(1) Σ = B∈B B (covering condition); (2) every block has a parity (+) or (−) such that two blocks intersect only if they have opposite parity (parity condition); (3) there is an > 0 such that two blocks intersect if and only if their -neighborhoods intersect ( -condition).
Mooney [Moo10] argues that the adjacency graph of B is a tree T .It remains to show that there exists d W > 0 such that every two distinct walls have distance at least d W .Let We show that d(x, y) ≥ d W = 2 .Indeed otherwise, d(x, y) < 2 .Since x and y are contained in walls, there are two blocks B 1 x and B 2 x containing x such that x and two blocks B 1 y and B 2 y containing y such that , 4}} contains at least three elements.Thus, the underlying graph of C contains a cycle of length at least 3.This is impossible as T is a tree.

2.3.
Itineraries.Let Σ be a CAT(0) space with treelike block decomposition B. Let T be the adjacency graph of B. From now on, we identify the set of walls W with the edges of T .this is possible because of Lemma 2.7.
be the shortest geodesic path in T linking the unique block containing γ(a) with one of the (at most two) blocks containing γ(b).Then the times t 1 := a, . ., k} and t k+1 := b satisfy the following four properties: (1) γ(t 1 ) ∈ B1 and γ(t i ) Proof.We proof the statement by constructing the path I(γ) as follows. • Step 1: As γ does not start in a wall, there exists a unique block B 1 such that γ(a) ∈ B1 by Lemma 2.6.Let P 1 be the path consisting of the vertex B 1 .
Let P be the path we obtain after the algorithm terminates.First, we will show that P = I(γ), i.e. that P is the shortest geodesic path linking the unique block containing γ(a) with a block containing γ(b).At first, we show by induction that P is a geodesic path: The path P 1 is a geodesic path as it consists of a single vertex.We might assume that P i is a geodesic path and have to show that P i+1 is a geodesic path.If we are in case 1, I(γ) = P i and we are done.Otherwise, P i+1 = (B 1 , . . ., B i+1 ).By induction hypothesis, the path P i = (B 1 , . . ., B i ) is a geodesic path.Hence, it suffices to show that B i+1 = B j for all j ∈ {1, . . ., i}.This is the case as otherwise, there exists j ∈ {1, . . ., i} such that t i+1 > t j and γ(t i+1 ) ∈ B j .As B j is convex, that contradicts the choice of t j .We conclude that P is a geodesic path.
By construction, the path P stars in the unique block containing γ(a).As P is a geodesic path and since the algorithm above terminates in case (1), P is the shortest geodesic path linking the unique block containing γ(a) with a block containing γ(b).Thus P = I(γ).
The following example shows that a geodesic segment γ might intersect a block B that does not occur in I(γ).In such a case, γ ∩ B is contained a wall W that does not appear in I(γ).In the tree T , W is an edge that links B to a block that occurs in I(γ).
We say that two geodesic rays Lemma 2.19.Let γ be a geodesic ray in Σ that does not start in a wall.If γ is asymptotic to a geodesic ray in a wall, then I(γ) is finite.
Proof.Let γ be a geodesic ray that is contained in a wall W .Let γ be a geodesic ray that does not start in a wall so that I(γ ) is infinite.We have to show that γ and γ are not asymptotic, i.e. we have to prove that for each D ∈ R ≥0 there exists t such that d(γ(t), γ(t )) > D. Let D ∈ R ≥0 .As γ ⊆ W , there exists t 0 ∈ R and a block B ∈ B such that γ(t) ∈ B for all t ≥ t 0 .Since I(γ ) is an infinite path, there exists a block B in I(γ ) such that the unique geodesic path in T linking B and B has more than D d W edges.By Lemma 2.14, there exists a time t such that γ(t ) ∈ B .Let γ be the geodesic segment connecting γ(t ) with a point b ∈ B. By Lemma 2.11 and the separating property 2.1 (2), the length of γ is at least D. Since b was chosen arbitrarily and γ ends in B, this implies that d(γ(t), γ(t )) > D.

The visual boundary of every wall behaves like a cutset
In Section 3.1, we will recall the definition of visual boundaries of CAT(0) spaces.In Section 3.2, we will study a cutset property of walls in CAT(0) spaces with a treelike block decomposition.

3.1.
The visual boundary of a CAT(0) space.Let Σ be a complete CAT(0) space and p a chosen basepoint in Σ.Let If α is a geodesic ray starting at p and > 0 r ≥ 0, then the following sets define an open neighborhood basis for the cone topology τ cone on ∂Σ p .
Recall that two geodesic rays Being asymptotic is an equivalence relation and we denote the equivalence class of a geodesic ray γ by γ(∞).We call such an equivalence class a boundary point and denote the set of all boundary points by Every equivalence class in ∂Σ is represented by a unique geodesic ray starting at p ([BH99, Prop.8.2, II]).Hence, the map  [CK00] proved that the visual boundary is not a quasi-isometry invariant.

3.2.
The cutset property.The goal of this section is to prove the following proposition illustrated in Figure 5.I would like to thank Emily Stark for the inspiration to study this property.Proposition 3.2 (cutset property).Let Σ be a complete CAT(0) space with treelike block decomposition B with adjacency graph T and p a chosen basepoint in Σ that does not lie in a wall.Let κ be a connected component of ∂Σ that contains two distinct boundary points γ 1 (∞) and γ 2 (∞).Let γ 1 and γ 2 be the corresponding representatives starting at p.
For every wall W that occurs in I(γ 1 ) or I(γ 2 ) but not in both I(γ 1 ) and I(γ 2 ), there is a geodesic ray γ ⊆ W such that γ(∞) ∈ κ.For the proof, we use the following lemma for general complete CAT(0) spaces, which is similar to Lemma 3.1 in [BZK21] which deals with path-components of visual boundaries.Lemma 3.3 (Ben-Zvi-Kropholler). Let Σ be a complete CAT(0) space and Σ 1 , Σ 2 closed, convex subsets such that the intersection Proof.Assume for a contradiction that there exists a connected component κ in ∂Σ containing γ 1 (∞) and γ 2 (∞) but no element of ∂W .Then κ is a connected component of the topological subspace Y := ∂Σ \ ∂W .We use the following observations.
The space ∂Σ 1 p \ ∂W p is a topological subspace of ∂Σ 1 p and ∂Σ p \ ∂Σ 2 p is a topological subspace of ∂Σ p .We have to show that ∂Σ 1 p \ ∂W p and ∂Σ p \ ∂Σ 2 p are homeomorphic.As Σ 1 is a closed, convex subspace of Σ, the inclusion ι : Σ The geodesic ray γ starts in W and is not contained in W . Since W is convex, there exists a time Proof of Proposition 3.2 .Let γ 1 , γ 2 and κ as in the claim.Suppose that W is a wall that appears in I(γ 1 ) or I(γ 2 ) but not in both.Let T 1 and T 2 be the two subtrees of the adjacency graph T of B we obtain by removing W from T .Let Σ i := B∈V (Ti) B. Then Σ = Σ 1 ∪ Σ 2 and Σ 1 ∩ Σ 2 = W .By Lemma 2.5, the spaces Σ 1 and Σ 2 are closed and convex.The wall W is closed and convex by Lemma 2.7.We will show that one of the two geodesic rays γ 1 and γ 2 ends in Σ 1 and that the other one ends in Σ 2 .Then the claim follows by applying Lemma 3.
We assume without loss of generality that W appears in I(γ 2 ) but not in I(γ 1 ).Let B 1 ∈ V (T 1 ) and B 2 ∈ V (T 2 ) be the blocks so that W = B 1 ∩ B 2 and so that B 2 occurs after B 1 in I(γ 2 ).By Lemma 2.14, there are two times t 1 , t 2 , t 1 < t 2 so that γ 2 (t 1 ) ∈ B1 and Assume for a contradiction that there exists ) is a geodesic segment connecting two points in Σ 1 that contains the point γ 2 (t 2 ) outside of Σ 1a contradiction to the convexity of Σ 1 .Thus, γ 2 (t) ∈ Σ 2 for all t ≥ t 2 , i.e. γ 2 (t) ends in Σ 2 .
It remains to show that γ 1 ends in Σ 1 .Since geodesic paths in trees are unique, each path linking a block in T 1 with a block in T 2 passes through the wall W . Since W does not appear in I(γ 1 ), the geodesic path I(γ 1 ) is either contained in T 1 or in T 2 .As γ 1 and γ 2 start at the same point, I(γ 1 ) and I(γ 2 ) start with the same block B 0 .Recall that W = B 1 ∩ B 2 and that B 2 appears after B 1 in I(γ 2 ).As I(γ 2 ) is a geodesic path, W does not occur twice in I(γ 2 ) and the subpath (B 0 , . . ., B 1 ) ⊆ I(γ 2 ) does not contain W . Thus, as B 1 ∈ T 1 and B 0 ∈ V (T 1 ), the itinerary I(γ 1 ) is a path in T 1 .By Lemma 2.14, γ 1 ([0, ∞)) ⊆ Σ 1 .In particular, γ 1 ends in Σ 1 .

Key properties of the Morse boundary
In Section 4.1, we will recall the definition of Morse boundaries and will transfer the observations of Section 3 to Morse boundaries.In Section 4.2, we will prove that geodesic rays of infinite itinerary are lonely.This is the only point, where we use the Morse-property in this paper.In Section 4.3, we will study relative Morse boundaries.4.1.From the visual boundary to the Morse boundary.In this section, we shortly recap the definition of the contracting boundary and the Morse boundary and obtain as a consequence that the cutset property in Proposition 3.2 holds not only for the visual boundary but also for the Morse boundary.
Let Σ be a complete CAT(0) space and C be a convex subset that is complete in the induced metric.Then there is a well-defined nearest point projection map π C : Σ → C.This projection map is continuous and does not increase distances (See [BH99, Prop.2.4 in II ]).
Definition 4.1 (contracting geodesics).Given a fixed constant D, a geodesic ray or geodesic segment γ in a complete CAT(0) space (Σ, d) is said to be D-contracting if for all x, y ∈ Σ, We say that γ is contracting if it is D-contracting for some D.
Charney-Sultan [CS15] introduced a quasi-iosmetry invariant of complete CAT(0) spaces, called contracting boundary.Let Σ be complete CAT(0) space.The underlying set of the contracting boundary of Σ is the set By definition, ∂ c Σ ⊆ ∂Σ (as sets).Let (∂ c Σ, τ cone ) be the set ∂ c Σ equipped with the subspace topology of ∂Σ.Cashen [Cas16] proved that (∂ c Σ, τ cone ) isn't a quasi-isometry invariant.For obtaining a quasi-isometry invariant-topology, we choose a basepoint p in Σ and define As before in Section 3.1, The contracting boundary ∂ c Σ of Σ is the topological space that we obtain by pushing the topology Cordes [Cor17] generalized the contracting boundary to a quasi-isometry invariant of proper geodesic metric spaces, called the Morse boundary.This generalization is based on the following characterizations of contracting geodesic rays in complete CAT(0) spaces.
Definition 4.3.Let γ : [0, ∞) → Σ be a geodesic ray in a proper geodesic metric space Σ.Given a Morse-gauge M , γ is M-Morse if, for every K ≥ 1, L ≥ 0, every (K, L)-quasi-geodesic σ with endpoints on γ is contained in the M -neighborhood of γ.Definition 4.4.Let Σ be a complete CAT(0) space.A geodesic ray γ is slim if there exists δ > 0 such that for all x ∈ Σ, y ∈ γ, the distance of π γ (x) and the geodesic segment connecting x and y is less than δ.
Sultan [Sul14] and Charney-Sultan [CS15] showed: Lemma 4.5 (Charney-Sultan).Let γ be a geodesic ray in a complete CAT(0) space.The following are equivalent: The Morse boundary ∂ M Σ of a proper geodesic metric space Σ is a topological space with underlying set The topology of ∂ M Σ is a direct-limit topology so that in the CAT(0)-case, ∂ M Σ and ∂ c Σ are homeomorphic: Suppose that p is a basepoint in the proper geodesic metric space Σ and N is a Morse gauge.Let

Now, let
with the induced direct limit topology, ie.
p for all N ∈ M. Cordes proves that this is well-defined, i.e. he shows independence of the basepoint.
Finally, we will use the following consequence of the Theorem of Arzelà-Ascoli.It is Corollary 1.4 in [Cor17] and conform with [Mun00].
The topology of the Morse boundary is finer than the subspace topology of the visual boundary, i.e. if a set is open in the subspace topology of the visual boundary, then it is also open in the Morse boundary.Thus, Proposition 3.2 implies Corollary 4.9 (cutset property).Let Σ be a complete CAT(0) space with treelike block decomposition B with adjacency graph T and p a chosen basepoint in Σ that does not lie in a wall.Let κ be a connected component of ∂ M Σ that contains two distinct boundary points γ 1 (∞) and γ 2 (∞).Let γ 1 and γ 2 be the corresponding representatives starting at p.
For every wall W that occurs in I(γ 1 ) or I(γ 2 ) but not in both I(γ 1 ) and I(γ 2 ), there is a geodesic ray γ ⊆ W such that γ(∞) ∈ κ.

4.2.
Loneliness of Morse geodesic rays with infinite itinerary.Let Σ be a proper CAT(0) space with treelike block decomposition B with adjacency graph T and p a chosen basepoint in Σ that does not lie in a wall.The goal of this subsection is to prove that geodesic rays starting at p of infinite itinerary are lonely.I would like to thank Tobias Hartnick for his help to simplify the proof for this property.This property is quite remarkable as it is the only point in this paper where we use the Morseproperty.The Loneliness property does not hold for non-Morse geodesic rays in general.The following example shows that visual boundaries might contain infinitely many geodesic rays that are pairwise non-asymptotic and have all the same infinite itinerary.Let p := ( 1 2 , 0).A geodesic ray γ starting at p can be of three different kinds: If γ is parallel to the Y -axes, γ is contained in the block B 0 , i.e. its itinerary is the path that consists of the block B 0 .If γ intersects the Y -axis R × 0, the itinerary of γ is the infinite path (B 0 , B −1 , . . .).In the remaining case, γ is not parallel to the Y -axis and does not intersect the Y -axis.In this situation, the itinerary of γ is the infinite path (B 0 , B 1 , . . .).
The following proof is inspired by the example of Charney-Sultan discussed in Section 1.2 and uses methods of the proof of Proposition 3.7 in [CS15].

Let
• (W i ) i∈N be the sequence of consecutive walls that are contained in I; • (γ i ) i∈N the sequence of geodesic segments γ i connecting a i with b i .
• (a * i ) i∈N be the sequence of projection points a * i := π β (a i ).
By proposition 3.7 (2) in [CS15], there exists R > 0 such that {a * i | i ∈ N} ⊆ B R (p), where B R (p) denotes the closed R-ball about p.For i ∈ N, let ∆ i := ∆(a i , b i , a * i ) be the geodesic triangle in Σ with corners a i , b i and a * i .As β is Morse, β is slim by Lemma 4.5.Hence, there exists δ > 0 such that U δ (a * i ) Recall that for each i ∈ N, a i and b i are contained in a wall.As each wall is convex, γ i ⊆ W i .Thus, W i ∩ B R+δ(p) = ∅ for all i ∈ N. We conclude that infinitely many walls intersects the ball B R+δ (x 0 ).But this is impossible because of Lemma 2.4.4.3.Relative Morse boundaries of convex subspaces.Let Σ be a proper geodesic metric space and B a convex, complete subspace whose Morse boundary is σ-compact, i.e. a union of countably many compact subspaces.Recall that (∂ M B, Σ) denotes the relative Morse boundary of B in Σ, i.e. the subset of ∂ M Σ that consists of all equivalence classes of geodesic rays in B that are Morse in the ambient space Σ.In this section, we study the relation of the two topological spaces that are obtained by endowing (∂ M B, Σ) with the subspace topology of ∂ M B and ∂ M Σ.Our goal is to proof Lemma 4.12 and Corollary 4.13.I would like to thank Nir Lazarovich for his help to improve this section.Moreover, I would like to thank Elia Fioravanti for sending me an example showing that the inverse of the embedding in the following lemma need not be continuous (see Example 4.17).
Lemma 4.12.Let Σ be a proper geodesic metric space with σ-compact Morse boundary.Let B be a complete, convex subspace of Σ that contains all geodesic rays in Σ that start in B and are asymptotic to a geodesic ray in B. If we endow (∂ M B, Σ) with the subspace topology of ∂ M Σ, then the map Proof of Corollary 4.13.If the assumptions of Lemma 4.12 are satisfied, then the continuity of ι * implies Corollary 4.13.Hence it remains to verify the assumptions of Lemma 4.12: The Main theorem in [CS15] implies that ∂ M Σ is σ-compact.Moreover, B contains all geodesic rays in Σ that start in B and are asymptotic to a geodesic ray in B because Σ is CAT(0) and B is closed and convex.
For proving Lemma 4.12, we need the following facts about direct limits.The following lemma is Lemma 3.10 in [CS15].For completeness, we cite their proof as well.
Lemma 4.14 (Charney-Sultan).Let X = lim In the following lemma, we study the direct limit of countably many topological spaces.
Lemma 4.15.Let X = lim −→ i∈N X i be a direct limit of topological spaces X i with associated topological embeddings ι i,j : X i → X j where i, j ∈ N, i ≤ j.Let B ⊆ X be a closed subspace of X.We equip B ∩ X i with the supspace topology of X i for all i.Then lim equipped with the subspace topology of X.

Proof.
Let O be an open set in B equipped with the subspace topology of X.By Lemma 4.14, O is open in the direct limit lim We have to find a set Õ that is open in X and satisfies Õ ∩ B = O ∩ B. Since each ι i,j : X i → X j is a topological embedding for each i, j ∈ N, we may assume that X i ⊆ X j for all i ≤ j.For every i ∈ N, we will define a set Õi Then the set Õ := i∈N O i is the set, we are looking for.Indeed, It remains to define the sets Õi , i ∈ N. Let i ∈ N. In step 1, we will prepare the definition of Õi .In step 2, we will define Õi .In step 3, we will show that Õi satisfies (1), ( 2) and (3).
Step 1: Induction base: Let O i i := O i .Induction step: Suppose that O j i is a set in X j with the properties listed above.Since ι j,j+1 : X j → X j+1 is a topological embedding, O j i is an open set in X j equipped with the subspace topology of X j+1 .Thus, there exists a set Step 3: Let i ∈ N. It remains to show that Õi satisfies (1), ( 2) and ( 3). ( It remains to show that O i j and X i \ A are open in X i for all j ≤ i.By definition, For applying Lemma 4.15 to the relative Morse boundary of B in Σ, we need the following variant of Example 8.11 (4) in Chapter II of [BH99] (see Lemma 3.1) for Morse boundaries.
Lemma 4.16.If B is a complete, convex subspace of a proper geodesic metric space Σ, then Proof of Lemma 4.16.Let p a basepoint in B and M be the set of all Morse gauges.We have to show that ) be a sequence of equivalence classes of geodesic rays in (∂ M B, Σ) ∩ ∂ N M Σ p and (β n ) n∈N be corresponding representatives that start at p and lie in B. By the Theorem of Arzelà-Ascoli 4.8, the sequence (β n ) n∈N has a convergent subsequence (β nm ) m∈N .As B is complete, β := lim m→∞ β nm is a geodesic ray in B.Moreover, β is N -Morse.Indeed, Let q be a point on a quasi-geodesic with endpoints on β and t ∈ R ≥0 such that d(β, q) = d(β(t), q).The continuity of the distance function d and the N -Morseness of β nm , m ∈ N implies that d(q, β(t)) = d(q, lim m→∞ β nm (t)) = lim m→∞ d(q, β nm ) ≤ N .
Proof of Lemma 4.12: Let p be a basepoint in B ⊆ Σ and M be the set of all Morse gauges.We equip (∂ M B, Σ) with the subspace topology of ∂ M Σ = lim We have to show that the map is continuous.We prove the statement in two steps.For any Morse gauge N ∈ M, let We endow ∂ N M Σ p ∩ ∂ M B with the supspace topology of ∂ N M Σ p and study the direct limit lim with the induced direct limit topology.
We will show in two steps that the following inclusions are continuous: Since ι * = ι • ῑ, this will imply that ι * is continuous as composition of continuous maps.
Step 1: For proving that the inclusion ῑ is continuous, it is sufficient to show that ∂ M Σ and its subspace (∂ M B, Σ) satisfy the assumptions of Lemma 4.15.
(a) Since ∂ M Σ p is σ-compact, there exists an ascending sequence of natural numbers Step 2: For proving that the inclusion ι is continuous, it is sufficient to show all assumptions of Lemma 4.14 are satisfied.
Let N be a Morse gauge.Since B is a convex subspace of Σ and because B contains all geodesic rays in Σ that start in B and are asymptotic to a geodesic ray in B, M B p as in Lemma 4.6.Note that ∂ N M Σ p contains a neighborhood about γ(∞) of the form V n (γ) as well.Since B is a convex subset of Σ and because B contains all geodesic rays in Σ that start in B and are asymptotic to a geodesic ray in B we have Hence, ι N is continuous.Thus, all assumptions of Lemma 4.14 are satisfied.
We finish this section with the following example of Elia Fioravanti showing that the inverse of ι * in Lemma 4.12 need not be continuous.Example 4.17 (See Figure 8).Let B be the subspace of R 2 that consists of [0, ∞) and all vertical geodesic rays of the form {(n, y) ∈ R 2 | y ∈ R ≥0 } where n ∈ N. We choose p := (0, 0) as basepoint.Let γ : R → B, γ(t) := (t, 0) and γ n : R → B, γ n (t) := γ(t) for all t ≤ n and γ n (t) := (n, t − n) for all t ≥ n.Then γ and γ n , n ∈ N, are Morse geodesic rays and lim n→∞ γ n (∞) = γ(∞) in ∂ M B. Now, we attach to each geodesic ray γ n a filled square along the geodesic segment connecting (n, 1) and (n, n + 1) as in the figure.Let Σ be the space we obtain this way.The geodesic rays γ n , n ∈ N, and γ are still Morse geodesic rays in Σ.But now, the Morse gauge of γ n grows in n.Hence, the sequence (γ n (∞)) n∈N does not converge to γ(∞) if we endow ∂ M B = (∂ M B, Σ) with the subspace topology of ∂ M Σ.Indeed, for each N , the set of N -Morse geodesic rays in Σ is finite.Hence, {γ n (∞)} ∩ ∂ N M Σ p is finite for each Morse gauge and thus, {γ n (∞)} is a closed subspace of ∂ M Σ.In particular, no limit point of {γ n (∞)} lies outside of this set.

Proof of Theorem 1.9
Let Σ be a proper CAT(0) space with treelike block decomposition B. Let T be the adjacency graph of B and p ∈ Σ be a basepoint that is not contained in a wall.Now, we apply the cutset property (Corollary 4.9) and the loneliness property (Proposition 4.10) to study the connected components of ∂ M Σ.This will lead to a proof of Theorem 1.9.Definition 5.1 (Itineraries of boundary points).Let γ(∞) ∈ ∂Σ be a boundary point of Σ.The itinerary I(γ(∞)) of γ(∞) is the itinerary of the representative of γ(∞) that starts at p. Definition 5.2.We say that a connected component κ of ∂ M Σ is of (1) If κ is of type A, then there exists a block B so that the representative of every point in κ starting at p ends in a block B.Moreover, κ is homeomorphic to a connected component of (∂ M B, Σ) endowed with the subspace topology of ∂ M Σ. (2) If κ is of type B, then |κ| = 1.
(3) If κ is of type C, then it contains an equivalence class of a geodesic ray in a wall.(1) Suppose that κ is of type A. Then there exists a finite path I in T such that all elements in κ have itinerary I; Then there exists a block B such that each geodesic ray starting at p and representing an element in κ ends in B. In particular, every equivalence class in κ has a representative that is contained in B, i.e. κ ⊆ (∂ M B, Σ).Thus, κ is homeomorphic to a connected component of (∂ M B, Σ) equipped with the subspace topology of ∂ M Σ. (2) Suppose that κ is of type B. Then there exists an infinite path I in T such that I(γ) = I for each geodesic ray γ starting at p such that γ(∞) ∈ κ.By the loneliness property (Proposition 4.10), all geodesic rays with itinerary I are asymptotic.Hence, κ contains only one element.(3) Suppose that κ is of type C. Then κ contains two points with different itineraries I 1 and I 2 .According to the cutset property (Corollary 4.9), κ contains the equivalence class of a geodesic ray that is contained in a wall.
Remark 5.5.The content of this section is with respect to the direct-limit topology of the Morse boundary.This does not imply the validity of the statements above for the visual boundary.However, the methods of the proofs can be transferred to (path) components of the visual boundary.The arguments used above can be repeated to confirm the following statements.or there is a time t such that c(t) has finite itinerary.

Applications to RACGs
In Section 6.1, we will determine which totally disconnected topological spaces occur as Morse boundaries of RACGs by applying Theorem 1.4 in [CCS].In Section 6.2, we will complete the proof of Theorem 1.6 by showing that each Davis complex of infinite diameter has a non-trivial treelike block decomposition.In Section 6.3, we will introduce a new graph class C consisting of graphs that correspond to RACGs with totally disconnected Morse boundaries.We will investigate this graph class by studying some examples of the literature.6.1.Totally disconnected spaces arising as Morse boundaries of RACGs.In this subsection, we study which totally disconnected topological spaces arise as Morse boundaries of RACGs.We will use Theorem 1.4 in [CCS] and I would like to thank Matthew Cordes for his comment how to apply this theorem to RACGs.
An ω-Cantor space is the direct limit of a sequence of Cantor spaces C 1 ⊂ C 2 ⊂ C 3 . . .such that C i has empty interior in C i+1 for all i ∈ N. By [CCS, Thm 3.3], any two ω-Cantor spaces are homeomorphic.We will apply the following Theorem [CCS, Thm 1.4].Theorem 6.1 (Charney-Cordes-Sisto).Let G be a finitely generated group.Suppose that ∂ M G is totally disconnected, σ-compact, and contains a Cantor subspace.Then ∂ M G is either a Cantor space or an ω-Cantor space.It is a Cantor space if and only if G is hyperbolic, in which case G is virtually free.
A suspension of a graph ∆ is a join of a graph consisting of two vertices and ∆.Corollary 6.2.Let W ∆ be a RACG with defining graph ∆ whose Morse boundary is totally disconnected.If ∆ is a clique or a non-trivial join, then ∂ M W ∆ is empty.If ∆ consists of two non-adjacent vertices or is a suspension of a clique, then ∂ M W ∆ is virtually cyclic and ∂ M W ∆ consists of two points.Otherwise, if ∆ does not contain any induced 4-cycle then ∂ M W ∆ is a Cantor space.In the remaining case, ∂ M W ∆ is an ω-Cantor space.
Proof.If the graph ∆ is a clique or a non-trivial join, then ∂ M W ∆ is empty by Lemma 1.5.Otherwise, W ∆ contains a rank-one isometry by Corollary B in [CS11].If ∆ consists of two nonadjacent vertices, then W ∆ is isomorphic to the infinite Dihedral group.If ∆ is a suspension of a clique, then W ∆ is the direct product of the infinite Dihedral group with a finite right-angled Coxeter group.In both cases, W ∆ is quasi-isometric to Z and the Morse boundary of W ∆ consists of two points.
In the remaining case, an induction on the number of vertices shows that ∆ contains either the graph consisting of three pairwise non-adjacent vertices as induced subgraph or the graph consisting of an edge and a further single vertex.These graphs correspond to special subgroups of W ∆ that are quasi-isometric to a tree whose vertices have a degree of at least three.Thus, in the remaining case, W ∆ is not quasi-isometric to Z.In particular, W ∆ is not virtually cyclic.
It remains the case where W ∆ is not virtually cyclic and contains a rank-one isometry.We will show that the assumptions of Theorem 6.1 are satisfied.If ∆ does not contain an induced 4-cycle, W ∆ is hyperbolic (See[Dav08, Thm 12.2.1,Cor 12.6.3],[Gro87]).Then Theorem 6.1 will imply that ∂ M W ∆ is a Cantor space.Otherwise, if ∆ contains an induced 4-cycle, Theorem 6.1 will imply that ∂ M W ∆ is an ω-Cantor space.
For proving the assumptions of Theorem 6.1, we have to show that ∂ M W ∆ is σ-compact and contains a Cantor space as subspace.The σ-compactness follows from Proposition 3.6 in [CS15].The existence of the Cantor subspace can be proven similarly as in Lemma 4.5 in [CCS]: Osin [Osi16] concluded from [Sis18] that if a group acts properly on a proper CAT(0) space and contains a rank-one isometry, then the group is either virtually cyclic or acylindrically hyperbolic.As W ∆ is not virtually cyclic, W ∆ is acylindrically hyperbolic.Thus, Theorem 6.8 and Theorem 6.14 of [DGO17] imply the existence of a hyperbolically embedded free group.By [Sis16], this free subgroup is quasi-convex and thus stable.The Morse boundary of this stable subgroup is an embedded Cantor space in the Morse boundary of the ambient group.6.2.Block decompositions of Davis complexes.In this subsection, we prove that each Davis complex of infinite diameter has a non-trivial treelike block decomposition.This will complete the proof of Theorem 1.6.
Recall that the RACG associated to a finite, simplicial graph ∆ = (V, E) is the group Let Cay(W ∆ , V ) be the Cayley graph of W ∆ with respect to the generating set V .The Davis complex Σ ∆ of W ∆ is the unique CAT(0) cube complex with Cay(W ∆ , V (∆)) as 1-skeleton so that • each 1-skeleton of a cube in Σ ∆ is an induced subgraph of Cay(W ∆ , V (∆)) and • each set of vertices in Cay(W ∆ , V (∆)) that spans an Euclidean cube is the 1-skeleton of an Euclidean cube in Σ ∆ .
Definition 6.3.Let ∆ be a finite simplicial graph.A subgroup of a RACG W ∆ is special if it has an induced subgraph of ∆ as defining graph.
(1) The trivial graph (∅, ∅) is an induced subgraph of ∆.The Davis complex of (∅, ∅) consists of a vertex corresponding to the identity of the trivial group.
Proposition 6.5.Let ∆ be a finite, simplicial graph that can be decomposed into two distinct proper induced subgraphs ∆ 1 and ∆ 2 with the intersection graph Λ = ∆ 1 ∩ ∆ 2 .Let Σ ∆1 , Σ ∆2 , Σ Λ be the canonically embedded Davis complexes of ∆ 1 , ∆ 2 and Λ in Σ ∆ .The collection is a treelike block decomposition of Σ ∆ .The collection of walls W is given by Proof.By Remark 2.13, it suffices to show that (1) Σ = B∈B B (covering condition); (2) every block has a parity (+) or (−) such that two blocks intersect only if they have opposite parity (parity condition); (3) there is an > 0 such that two blocks intersect if and only if their -neighborhoods intersect ( -condition).
(2) Parity condition: We give parity (-) to each block of the form gΣ ∆1 and parity (+) to each block of the form gΣ ∆2 .Let i ∈ {1, 2} and B 1 := g 1 Σ ∆i and B 2 := g 2 Σ ∆i two distinct blocks of the same parity.By definition of the Davis complex, the 0-skeletons of B 1 and B 2 are the coset g 1 W ∆i and g 2 W ∆i .As B 1 and B 2 are two distinct blocks, g 1 W ∆i = g 2 W ∆i .Hence, g 1 W ∆i ∩ g 2 W ∆j = ∅.Thus, the 0-skeletons of B 1 and B 2 don't intersect.This implies that B 1 ∩ B 2 = ∅.Indeed, by Remark 6.4, the blocks B 1 and B 2 are isometric to the Davis complex of W ∆i .Hence, if a set of vertices in Cay(W ∆i , V (∆ i )) spans an Euclidean cube, then it spans a cube in B j , j ∈ {1, 2} and on the other hand, each 1-skeleton of a cube in B j is an induced subgraph of Cay(W ∆i , V (∆ i )).Thus, B 1 and B 2 intersect if and only if their 0-skeletons intersect.
We observe that there is no hyperplane H that separates B 1 and B 2 .In other words: If we delete a hyperplane H outside of B 1 ∪ B 2 , then B 1 and B 2 lie in a common connected component of the resulting space.Indeed otherwise, each geodesic segment connecting B 1 and B 2 has to pass through H.As hyperplanes are equivalence classes consisting of midcubes, d(x, y) ≥ 1  2 for all x ∈ B 1 ∪ B 2 , y ∈ H.As < 1 2 , this implies that the -neighborhoods of B 1 and B 2 don't intersect -a contradiction.We conclude that there is no hyperplane H outside of B 1 ∪ B 2 that separates B 1 and B 2 .Thus, the distance of the 1-skeletons of B 1 and B 2 is zero by Theorem 4.13 in [Sag95].In particular, B 1 and B 2 intersect.
It remains to show that each wall is of the form gΣ Λ , g ∈ W ∆ .Indeed, let B 1 := g 1 Σ ∆1 and B 2 := g 2 Σ ∆2 be two distinct blocks of distinct parity that have non-empty intersection.We see as in (2) that the (0)-skeletons of B 1 and B 2 are the left-cosets g 1 W ∆1 and g 2 W ∆1 .Since we can write W ∆ as an amalgamated free product W ∆1 * WΛ W ∆2 , the intersection g 1 W ∆1 ∩ g 2 W ∆1 is a left coset of W Λ , and the cube complex C spanned by the vertices in this left coset is contained in B 1 ∩ B 2 .Now, B 1 ∩ B 2 does not contain any other point in B 1 ∪ B 2 because each cube in B j , j ∈ {1, 2} is spanned by vertices that are contained in g j W j .Thus, every point x ∈ B 1 ∪ B 2 \ B 1 ∩ B 2 lies in a cube that is contained in at most one of the two blocks B 1 and B 2 .
As a result, Theorem 1.6 is a direct consequence of Theorem 1.9, Proposition 6.5 and Lemma 1.5.Definition 6.6.A finite, connected graph ∆ is called a Charney-Sultan-graph if ∆ is the union of two distinct proper induced subgraphs C and J so that C is a cycle of length at least 5 and J is a non-trivial join of two induced subgraphs of ∆.Lemma 6.7.If ∆ is a Charney-Sultan-graph, then ∂ M W ∆ is totally disconnected.
Proof.Suppose that ∆ is a Charney-Sultan graph.Then ∆ is the union of a cycle C and a nontrivial join J.If J contains all the vertices of C, ∆ coincides with J. Then W ∆ is the direct product of two RACGs and ∂ M W ∆ is empty.
If J does not contain all vertices of C, C contains a path P of length at least two that connects two vertices in J so that no inner vertex of P is contained in J. Let ∆ be the graph that we obtain by removing the inner vertices of P from ∆. Then ∆ = ∆ ∪ P and ∆ ∩ P consists of the two non-adjacent end vertices of P .If Σ ∆ has totally disconnected Morse boundary, then W ∆ has totally disconnected Morse boundary by Corollary 1.7.
Let C be the subgraph of C that we obtain by removing the inner vertices of P .Then ∆ = C ∪ J.If C ∩ J = J, ∆ is a non-trivial join.Then W ∆ is the direct product of two infinite CAT(0) groups and thus, ∂ M Σ ∆ is empty.Otherwise, C ∩ J is a proper subgraph of J. Since the graph C is a path, W C is quasi-isometric to a free group.Thus, ∂ M Σ C is totally disconnected.In particular, (∂ M Σ C , Σ ∆ ) endowed with the subspace topology of ∂ M Σ C is totally disconnected.As J is a non-trivial join, W J is the direct product of two infinite CAT(0) groups and thus, ∂ M Σ J is empty.By Corollary 1.7, Σ ∆ has totally disconnected Morse boundary.
We enlarge the class of Carney-Sultan-graphs in the following manner.Definition 6.8 (The class C of clique-square-decomposable graphs).asdgtagdsaadfa Let C be the smallest class of finite graphs such that (1) each finite graph without edges is contained in C; (2) each finite tree is contained in C; (3) each Charney-Sultan graph is contained in C; (4) each clique is contained in C; (5) each non-trivial join of two graphs is contained in C; (6) the union of two graphs Λ 1 , Λ 2 ∈ C is contained in C if Λ 1 ∩ Λ 2 is an induced subgraph of Λ 1 ∪ Λ 2 so that one of the following three conditions is satisfied: Let ∆ be a graph in C. If ∆ satisfies (1) or (2) then W ∆ is quasi-isometric to Z/2Z, Z or to a free group of rank at least two and ∂ M W ∆ is totally disconnected.If ∆ satisfies (3), ∂ M W ∆ is totally disconnected by Lemma 6.7.If ∆ satisfies (4), W ∆ is a finite group and ∂ M W ∆ = ∅.If ∆ satisfies (5), W ∆ is the direct product of two infinite RACGs.In this case, each geodesic ray in Σ ∆ is contained in an Euclidean flat and thus, ∂ M W ∆ = ∅.If ∆ satisfies (6), we can decompose ∆ into two graphs Λ 1 , Λ 2 ∈ C satisfying the properties listed in the definition above.If Λ 1 and Λ 2 satisfy (1), ( 2), ( 3), (4) or (5), then Corollary 1.7 implies that ∂ M W ∆ is totally disconnected.Otherwise, we repeat the same argumentation for Λ 1 and Λ 2 .As ∆ is a finite graph, this algorithm ends after finitely many step.Thus, by applying Corollary 1.7 several times we obtain Next, we examine the graph class C by studying so-called CF S graphs.The four-cycle graph ∆ 4 of a graph ∆ is a graph whose vertices are the induced cycles of length four.Two vertices of ∆ 4 are connected by an edge if the corresponding 4-cycles have a pair of vertices in common that are not adjacent in ∆.The support of a subgraph K of ∆ 4 is the set of vertices of ∆ that are contained in a 4-cycle corresponding to a vertex of K.The following is a generalization in [BFRHS18] of the original definition of Dani-Thomas [DT15].Definition 6.9 (CF S).A graph ∆ is CF S if it is a join of two graphs ∆ and K where ∆ is a non-trivial subgraph of ∆ and K is a clique (it is allowed that this clique is trivial, i.e., (∅, ∅)) so that ∆ 4 has a connected component whose support coincides with the vertex set V (∆) of ∆.
Intuitively, a CF S graph ∆ contains a lot of induced 4-cycles.One could expect that W ∆ has totally disconnected Morse boundary.However, an example of Behrstock [Beh19] shows that this is wrong in general.On the other hand, Nguyen-Tran [NT19] used an approach similar to this paper for proving that the Morse boundary of W ∆ is totally disconnected if ∆ is contained in the following graph class CF S 0 ⊆ CFS.Definition 6.10.Let CF S 0 be the class of all graphs that are CF S, connected, triangle-free, planar, having at least 5 vertices and no separating vertices or edges.
On page 3 in [Tra19], Nguyen-Tran conclude the following result from their considerations: It turns out that our theorem includes their result in view of the following proposition: Proposition 6.12.CF S 0 ⊆ C.
Proof.In Proposition 3.11 of [NT19], Nguyen-Tran prove that each ∆ ∈ CFS 0 decomposes as a tree of graphs G so that each vertex of G corresponds to a non-trivial join of two graphs consisting of two and three vertices respectively and so that each edge of G corresponds to an induced 4cycle in ∆.Thus, ∆ can be obtained by starting with a non-trivial join and adding finitely many non-trivial joins to it.Unlike Nguyen-Tran, we allow to add not only non-trivial joins and cliques but also other more sophisticated graphs.Hence, CF S 0 ⊆ C.
The class C is substantially larger than the class CF S 0 .The following lemma shows that C contains graphs corresponding to RACGs with polynomial divergence of arbitrary high degree.In contrast, each graph in CF S 0 ⊆ CF S corresponds to a RACG of quadratic divergence.Indeed, Dani-Thomas [DT15] proved that a triangle-free graph ∆ is CF S if and only if W ∆ has quadratic divergence, and Levcovitz [Lev18, Thm 7.4] proved this statement for general graphs .Remark 6.14.The graphs ∆ i , i ≥ 3, in Figure 10 correspond to RACGs with totally disconnected Morse boundaries that are not quasi-isometric to any RAAG.Indeed, every right-angled Artin group has either linear or quadratic divergence, as remarked by Behrstock [Beh19].Hence, if a graph is not CF S then the corresponding RACG is not quasi-isometric to a RAAG.
Another interesting example is the graph pictured in Figure 12 that was studied by Ben-Zvi.
Lemma 6.15.The graph ∆ pictured in Figure 12 is contained in C \ CF S 0 .Remark 6.16.Ben-Zvi argues that W ∆ is a CAT(0) group with isolated flats and proves that the visual boundary of the corresponding Davis complex is path-connected.Among other things, the graph ∆ was studied by Ben-Zvi for the following reason: Let ∆ 1 and ∆ 2 the two graphs pictured in Figure 14 and Λ = ∆ 1 ∩ ∆ 2 .The graph ∆ consists of two vertices.See Figure 14.Ben-Zvi observes that the virtual Z 2 corresponding to the four-cycle in the middle is hidden if we write W ∆ as W ∆ = W ∆1 * WΛ W ∆2 .In the block decomposition corresponding to this splitting, two geodesic rays might go through the same sequence of hyperbolic planes corresponding to W ∆1 and W ∆2 but their rays are not asymptotic.In other words, there might be pairs of distinct points in the visual boundary having the same infinite itinerary.We have proven in Proposition 4.10 that such an unpleasant situation does not occur among Morse geodesic rays of infinite itinerary.
For completing this section, we show that the graph ∆ in the left upper corner in Figure 15 (mentioned in the introduction, there Figure 1) is contained in C. For the proof, we consider the successive decomposition pictured in Figure 15.The graph ∆ is seen in the left upper corner.We decompose ∆ from left to right and from above to bottom.In each second step, we decompose the graph into a green and a black graph.
The intersection of these two graphs always consists of single vertices marked by the thick red points.These red vertices are contained either in a non-trivial join or in a clique.In every second step, we delete the green subgraph and continue to decompose the resulting graph in the next step.Finally, we end up with a 4-cycle.As a 4-cycle is contained in C, we conclude that ∆ ∈ C. Remark 6.17.The graph ∆ in the left upper corner in Figure 15 is not planar.Thus, it is not contained in CF S 0 .Remark 6.18.Behrstock [Beh19] investigates a graph ∆ similar to ∆ and shows that ∂ M Σ ∆ contains a circle.This circle corresponds to an induced 5-cycle C in ∆ .No pair of non-adjacent vertices of this cycle C is contained in an induced 4-cycle.In particular, no induced subgraph of C is contained in a non-trivial join (since in a non-trivial join, any pair of non-adjacent vertices is contained in an induced 4-cycle).Hence, no matter how often and in which way we decompose the graph ∆ along graphs that are contained in non-trivial joins or cliques, the remaining graph will always contain C. Interestingly, it is possible to decompose ∆ similarly to ∆ so that only the 5-cycle C remains.

Beyond RACGs
In this section, we study applications of Theorem 1.9 that are not RACGs.7.1.Right-angled Artin groups (RAAG).The right-angled Artin group (RAAG) associated to a finite, simplicial graph ∆ = (V, E) is the group The group A ∆ acts geometrically on an associated CAT(0) cube complex Σ A ∆ , its Salvetti complex.Hence, the Morse boundary of A ∆ is the Morse boundary of Σ A ∆ .This splitting corresponds to a treelike block decomposition of the Salvetti complex Σ A ∆ on which A ∆ acts geometrically.The walls in this block decomposition are Euclidean flats.Thus, we can apply Theorem 1.9.The Morse boundary of F 2 × Z is empty as F 2 × Z is a direct product of two infinite CAT(0) groups.We apply Theorem 1.9 and conclude that ∂ M A ∆ is totally disconnected.Though the factors in the splitting of A ∆ have empty Morse boundary, the Morse boundary ∂ M A ∆ is not empty.It consists of Morse geodesic rays that don't end in a block.Accordingly, each connected component of ∂ M A ∆ is of type B, i.e. each connected component consists of an equivalence class of a geodesic ray with infinite itinerary.
More generally, we can transfer the line of argumentation in Section 6 to RAAGs by studying Salvetti-complexes of RAAGs instead of Davis complexes of RACGs.We conclude similarly to the case of RACGs: If ∆ is a join of two graphs, then A ∆ is a direct product of two RAAGs.As each RAAG is an infinite group, each such RAAG is a direct product of two infinite CAT(0) groups.In such a case, each geodesic ray in Σ A ∆ is bounded by a Euclidean half-plane, and A ∆ has empty Morse boundary.Corollary B in [CS11] implies the following lemma: Lemma 7.2 (Sageev-Caprace).The Morse boundary of a RAAG A ∆ is empty if and only if ∆ is the join of two non-empty graphs.Now, let ∆ be a finite, simplicial graph that can be decomposed into two distinct proper induced subgraphs ∆ 1 and ∆ 2 with the intersection graph Λ = ∆ 1 ∩ ∆ 2 .Repeating the arguments in the proof of Proposition 6.5 in the setting of RAAGs yields We need the following lemma for studying Charney-Sultan graphs in the setting of RAAGs.
Lemma 7.6.If ∆ is a finite tree, then A ∆ has totally disconnected Morse boundary.
Proof.Suppose that ∆ is a finite tree.We show by induction on the numbers of edges in ∆ that A ∆ has totally disconnected Morse boundary.If ∆ consists of an edge, A ∆ is isomorphic to Z 2 and ∂ M A ∆ is empty.Now suppose that ∆ is a tree with n edges.Then there exists an edge e such that ∆ is obtained by gluing one endvertex of e to a subree T of ∆ which contains n − 1 vertices.Example 7.9 (Example 2.1 in [BZ]).Let G 1 be the fundamental group of the surface amalgam pictured in Figure 17.Its universal cover Σ 1 admits a treelike block decomposition in blocks that are Euclidean and hyperbolic planes corresponding to the two-torus T 2 on the left and the torus T 1 on the right.As in the example of Charney-Sultan pictured in Figure 3, one can argue that the relative Morse boundary (∂ M T2 , Σ 1 ) of the universal cover of the two-torus T2 on the left endowed with the subspace topology ∂ M T2 is totally disconnected.The relative Morse boundary (∂ M T1 , Σ 1 ) of the universal cover of the torus T1 on the right is empty.By Corollary 1.10, the Morse boundary of G 1 is totally disconnected.
Remark 7.10.Ben-Zvi shows that the group G 1 is a CAT(0) group with isolated flats and that ∂Σ 1 is path-connected.The isolated flat property implies that two rays passing through the same infinite collection of hyperbolic or Euclidean planes are asymptotic.So, in this example, Proposition 4.10 is also true for non-Morse geodesic rays.
Example 7.11 (Example 2.2 in [BZ]).Let G 2 be the fundamental group of two tori with boundary components identified as shown in Figure 18.By applying Theorem 1.9 twice, we prove that Now, we construct the space Σ 2 on which the group G 2 acts geometrically.For that purpose, we glue copies of Σ 2 along bi-infinite geodesic rays corresponding to Z = [c, d] to copies of trunked hyperbolic planes corresponding to the free group F 2 = c, d as in the equivariant gluing theorem of Bridson-Haefliger.This way, we obtain a CAT(0) space with a treelike block decomposition where each wall is a bi-infinite geodesic ray that is contained in a subspace corresponding to the Kleinian group K. Hence, no wall contains a Morse geodesic ray and the Morse boundary of G 2 is totally disconnected by Corollary 1.10.Remark 7.12.Ben-Zvi shows that G 2 is a CAT(0) group with isolated flats and that the visual boundary of Σ 2 is path-connected.
Example 7.13 (Examples arising from the equivariant gluing theorem of Bridson-Haefliger).The spaces arising from the equivariant gluing theorem [BH99, Theorem II.11.18] of Bridson-Haefliger are CAT(0) spaces with a treelike block decompositions on which amalgamated free products of CAT(0) groups act geometrically as observed by Ben-Zvi in Example 6.8 in [BZ].Thus, many other examples can be constructed to which Theorem 1.9 and Corollary 1.10 can be applied.

2. 2 .
Definitions and basic properties.In this subsection, we study CAT(0) spaces that have a treelike block decomposition.The following considerations are variants of definitions and lemmas in [CK00, BZ, BZK21, Moo10].For the background about CAT(0) spaces, see [BH99, Ch.II].Definition 2.1.Let Σ be a CAT(0) space.A collection B of closed convex subsets of Σ is a block decomposition of Σ if it satisfies the covering condition Σ = B∈B B.
It remains to prove that f is injective.Let e 1 = {B 1 , B 2 } and e 2 = {B 3 , B 4 } be two edges in T such that f (e 1 ) = f (e 2 ).Then B 1 ∩ B 2 = ∅ and B 3 ∩ B 4 = ∅ and B 1 ∩ B 2 = B 3 ∩ B 4 .By Lemma 2.6, each point in Σ lies in at most two blocks.Hence, {B 1 , B 2 } ={B 3 , B 4 }, i.e. e 1 = e 2 .For avoiding the use of the term "boundary" in two different meanings, we define the topological frontier of a set S to be the closure of S minus the interior of S. If x is a point in Σ and > 0, we denote by U (x) the open -neighborhood about x.If B is a block in a block decomposition of a CAT(0) space with wall-set W, then W B denotes the set {W ∈ W | W ⊆ B} and B := B \ W B .Lemma 2.8.For every B ∈ B, the set B = B \ W B is open in Σ.
It suffices to find two curves c 1 and c 2 that don't intersect W so that c 1 links a point in B \ W with c(a) and c 2 links c(b) with a point in B \ W . Then the concatenation of c 1 , c and c 2 is a curve in Σ \ W that connects a point in B \ W with a pint in B \ W .That contradicts Claim 1.It remains to find the curves c 1 and c 2 .Let B 1 = B and P 1 = (B 1 , W 1 , B 2 , W 2 , . . ., B k ) be the unique geodesic path in T connecting B with the first block of P .If P 1 consists of a single vertex, then B k = B and c(a) ∈ B. Because we assume that c does not intersect W , c(a) ∈ B \ W . Then the trivial constant curve with value c(a) is the curve c 1 we are looking for.

Figure 4 .
Figure 4.A CAT(0) space with a bock decomposition.Each line is a block.The intersection points of every two lines is a wall.

Figure 5 .
Figure 5. Illustration of the cutset property (Proposition 3.2): The orange marked edge e in T to the left corresponds to a wall W in Σ which is marked orange to the right.The wall W appears in I(γ 2 ) but not in I(γ 1 ).If we delete the edge e from T , I(γ 1 ) and I(γ 2 ) end in different components of the resulting graph.If we delete W form Σ, the rays γ 1 and γ 2 end in different components of Σ \ W and γ 1 (∞) and γ 2 (∞) lie in different components of ∂Σ \ ∂W .
that is an N -Morse geodesic ray with β(0) = p} endowed with the compact-open topology.The following is Lemma 3.1 in[Cor17]: Lemma 4.6.Let Σ be a proper geodesic metric space and p ∈ Σ.For N = N (K, L) a Morse gauge, let δ N := max{4N (1, 2N (5, 0)) + 2N (5, 0), 8N (3, 0)}.Let α : [0, ∞) → Σ be a N -Morse geodesic ray with α(0) = p and for each positive integer n let V n (α) be the set of geodesic rays γ such that γ(0) = p and d(α(t), γ(t)) < δ N for all t < n.Then{V n (α) | n ∈ N} (2) is a fundamental system of (not necessarily open) neighborhoods of α(∞) in ∂ N M Σ p .By Proposition 3.12 in[Cor17], ∂ N M Σ p is compact for each Morse gauge N ∈ M. Let M be the set of all Morse gauges.If N and N are two Morse gauges, we say that N ≤ N if and only if N (λ, ) ≤ N (λ, ).This defines a partial ordering on M. Corollary 3.2 in[Cor17] and the proof of Proposition 4.2 in [Cor17] implies Lemma 4.7.Let N and N two Morse gauges such that N ≤ N and Σ a proper geodesic metric space with basepoint p. Then the associated inclusion map ι N,N : ∂ Proposition 4.10 (Loneliness property).Let α and β be two distinct geodesic rays in Σ starting at p that have infinite itinerary.If at least one of both is Morse, then I(α) = I(β).

Figure 6 .
Figure 6.A block decomposition of the Euclidean plane in which the itinerary of each geodesic ray starting in the interior of a block is either trivial (i.e. a path consisting of one vertex) or an infinite path.The dashed lines denote three geodesic rays with different itineraries.Example 4.11.Let Σ = R 2 and B := {[i, i + 1] × R | i ∈ Z} as pictured in Figure 6.For i ∈ Z, let B i := [i, i+1]×R.The adjacency graph of B is a bi-infinite path of the form (. . ., B −1 , B 0 , B 1 , . . .).

Figure 7 .
Figure 7. Illustration of the proof of the Loneliness property.
Corollary 4.13.Let B be a closed convex subspace of a proper CAT(0) space Σ.If (∂ M B, Σ) endowed with the subspace topology of ∂ M B is totally disconnected, then (∂ M B, Σ) endowed with the subspace topology of ∂ M Σ is totally disconnected.

2 :
where the last equality follows by induction hypothesis.Step For each i ∈ I, we defineÕi := j≤i O i j \ A, where A := B \ O.
): We have to show that i∈N Õi ∩ B = O ∩ B. Since B = O A is the disjoint union of O and A and B ⊆ O, we have that Õ ∩ B = i∈N Õi ∩ B = i∈N j≤i by Lemma 2.6 in[CCS].Hence, ∂ M Σ is a direct limit of countably many topological spaces.(b) By Lemma 4.16, (∂ M B, Σ) is closed in ∂ M Σ, (c) By Lemma 4.7, the inclusion maps ι N,N : ∂ N M Σ p → ∂ N M Σ p are topological embeddings for all Morse gauges N , N such that N ≤ N .

Corollary 5. 6 .
If κ is a connected component of ∂Σ of type B and contains a boundary point that is Morse, then |κ| = 1.The next corollary is related to Lemma 5.3 and Lemma 7 in section 1.7 of [CK00] Corollary 5.7.If a curve c : [a, b] → ∂Σ starts at a point c(0) of infinite itinerary I, then I(c(t)) = I for all t ∈ [a, b]

6. 3 .
Examples.Theorem 1.6 can be used to construct new examples of RACGs with totally disconnected Morse boundaries.First, we generalize the example of Charney-Sultan studied in Section 4.2 of [CS15].Afterwards, we introduce a class C of graphs corresponding to RACGs with totally disconnected Morse boundaries.Finally, we study interesting examples that lie in C. I would like to thank Ivan Levcovitz, Jacob Russell and Hung Cong Tran for their comments on these examples.

Figure 11 .
Figure 11.Decomposition of the graphs in Figure 10 showing that each graph Λ i is contained in C.

Figure 12 .
Figure 12.The defining graph of a RACG studied by Ben-Zvi in [BZ, Ex. 2.3].This graph is contained in C \ CF S 0 .

Figure 14 .
Figure 14.A decomposition of the graph pictured in Figure 12.The common intersection graph consists of the two vertices which are drawn bold in both pictured graphs.

Figure 15 .
Figure 15.Decomposition of the graph ∆ in the left upper corner that was studied in [RST, Example 7.7] (see Figure 1).The Decomposition shows that ∆ is contained in C.

Proposition 7. 3 .
The collectionB := {gΣ A ∆1 | g ∈ A ∆ } ∪ {gΣ A ∆2 | g ∈ A ∆ } is a treelike block decomposition of Σ A ∆ .The collection of walls W is given by W = {gΣ A Λ | g ∈ A ∆ }.Theorem 1.9 combined with Proposition 7.3 and Lemma 7.2 directly implies Theorem 7.4.Suppose that Λ is contained in a join of two induced subgraphs of ∆.Then every connected component of ∂ M Σ A ∆ is either (1) a single point; or (2) homeomorphic to a connected component of (∂ M Σ A ∆i , Σ A ∆ ) equipped with the subspace topology of ∂ M Σ A ∆ where i ∈ {1, 2}.By means of Corollary 4.13 we obtain Corollary 7.5.Suppose that the assumptions of Theorem 7.4 are satisfied.If (∂ M Σ ∆1 , Σ A ∆ ) and (∂ M Σ ∆2 , Σ A ∆ ) equipped with the subspace topology of ∂ M Σ A ∆1 and ∂ M Σ A ∆2 are totally disconnected then ∂ M Σ A ∆ is totally disconnected.

Figure 17 .
Figure 17.Example 2.1 in [BZ]: A torus and a genus 2 surface identified along the curves x and y.
Theorem 1.9.Let Σ be a proper CAT(0) space with treelike block decomposition B. If no wall in Σ contains a geodesic ray that is Morse in Σ, then every connected component of ∂ M Σ is either (1) a single point; or (2) homeomorphic to a connected component of (∂ M B, Σ), where B is a block in B and (∂ M B, Σ) is endowed with the subspace topology of ∂ M Σ.By means of Corollary 4.13 in Section 4.3 we conclude Corollary 1.10.Let Σ be a proper CAT(0) space with a treelike block decomposition.If no wall contains a geodesic ray that is Morse in Σ and (∂ M B, Σ) equipped with the subspace topology of ∂ M B is totally disconnected for every block B, then ∂ M Σ is totally disconnected.
two distinct walls where B 1 , B 2 , B 3 , B 4 ∈ B. As W 1 and W 2 are distinct, there exists i ∈ {1, 2} such that B i / ∈ {B 3 , B 4 }.By definition of walls, B 1 = B 2 and B 3 = B 4 .As the intersection of more than two blocks is empty and and γ be the geodesic segment connecting x and y.By Lemma 2.9, γ has to pass a wall W that is contained in B 3 .As B 3 / ∈ {B 1 , B 2 }, the wall W does not coincide with B 1 ∩ B 2 .By the separating property 2.3 (2), inf y∈W d(x, y) ≥ d W .But this is impossible because y ∈ U d W (x). If P is a path in T linking two blocks B and B and W is a wall corresponding to an edge of P , then each curve in Σ linking a point in B with a point in B passes through W .Proof.We will prove the statement in two steps.Claim 1 If B 1 and B 2 are two distinct blocks with non-empty intersection W , then Σ \ W decomposes so that each pair of pointsx 1 ∈ B 1 \ W , x 2 ∈ B 2 \ W lie in different connected components of Σ \ W . Proof: Let B 1and B 2 be two blocks with non-empty intersection W .Let x 1 ∈ B 1 \ W and x 2 ∈ B 2 \ W .Let e be the edge in T corresponding to W .If we delete the edge e, then T decomposes into two trees T 1 and T 2 .Let T 1 be the tree containing B 1 and T 2 be the tree containing B 2 .Let O 1 := B∈V(T1) B \ W and O 2 := B∈V(T1) B \ W .By Lemma 2.6, each point in Σ is contained in exactly one block or exactly one wall.Hence, O 1 ∩ O 2 = ∅ and Σ \ W = O 1 .∪ O 2 such that x 1 ∈ O 1 and x 2 ∈ O 2 .If O 1 and O 2 are open, this implies that x 1 and x 2 lie in distinct connected components of Σ \ W . Thus, it remains to show that O 1 and O 2 are open.By symmetry reasons it suffices to prove that O 1 is open.Let x ∈ O 1 .If there exists B ∈ V (T 1 ) such that x ∈ B, then there exists > 0 such that U (x) ∈ B as B is open by Lemma 2.8 and x is an interior point of O 1 .
is a bijection.The visual boundary of Σ is the topological space that we obtain by pushing the topology of ∂Σ p to ∂Σ, i.e.A ⊆ ∂Σ is open if and only if f −1 (A) is open in ∂Σ p .For more details see [BH99, Def.8.6 in part II].If Σ is Gromov-hyperbolic, then the visual boundary of Σ coincides with the Gromov boundary of Σ.While the Gromov boundary is a quasi-isometry invariant, Croke-Kleiner 1 p → Σ p induces a topological embedding ι * : ∂Σ 1 p → ∂Σ p by Lemma 3.1.It remains to show that ι * (∂Σ 1 p \ ∂W p ) = ∂Σ p \ ∂Σ 2 p .Let γ ∈ ∂Σ 1 p \ ∂W p .The geodesic ray γ starts in W and is not contained in W . Since W is convex, there exists a time t 0 By the induction hypothesis, A T has totally disconnected Morse boundary.As A e has empty Morse boundary, it follows from Corollary 7.5 that A ∆ has totally disconnected Morse boundary.Adapting the arguments in Lemma 6.7 by dint of Lemma 7.6 yields Lemma 7.7.If ∆ is a Charney-Sultan graph, then A ∆ has totally disconnected Morse boundary.Let C be the graph class obtained by varying the Definition 6.8 in the following manner: • replace C by C ; • replace the condition 6.8 (5) in Definition 6.8 by the condition that each (not necessarily non-trivial) join of two non-empty graphs is contained in C .Applying Lemma 7.2, Corollary 7.5, Lemma 7.6 and Lemma 7.7 similarly to the argumentation after Definition 6.8 leads to the following corollary which gives an alternative prove of Theorem 1.1 for RAAGs whose defining graphs are contained in C .Corollary 7.8.If ∆ ∈ C , then A ∆ has totally disconnected Morse boundary.7.2.Surface amalgams.In this section we study examples of surface amalgams that were examined by Ben-Zvi in [BZ].