Boundaries of groups with isolated flats are path connected
Michael Ben-Zvi

	
	
	A seminal result in geometric group theory is that a 1-ended hyperbolic group has a locally connected visual boundary. As a consequence, a 1-ended hyperbolic group also has a path connected visual boundary. In this paper, we study when this phenomenon occurs for CAT(0) groups. We show if a 1-ended CAT(0) group with isolated flats acts geometrically on a CAT(0) space, then the visual boundary of the space is path connected. As a corollary, we prove all CAT(0) groups with isolated flats are semistable at infinity.
	
	

Importance of the `Higgs' Amplitude Mode in Understanding the “Ideal Glass Transition” and the Kauzmann Entropy Paradox at the Four-Dimensional Crystal/Glass Quantum Critical Point
    David E. Laughlin
    December 30, 2023
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§ INTRODUCTION

	
	We study (0) groups and the boundaries of the spaces on which they act. A group is (0) if it acts geometrically (properly discontinuously, cocompactly, and by isometries) on some (0) space. The motivating question for this article is the following:



 
	If G acts geometrically on a 1-ended (0) space X, when is the visual boundary, X, path connected?


There are two main reasons to investigate this property for boundaries of (0) spaces. The first comes from the theory of hyperbolic groups. All 1-ended hyperbolic groups have locally connected boundaries  <cit.>. Some natural examples of non-hyperbolic (0) groups, such as F_2×, show that this phenomenon does not occur in general. However, a boundary which is connected and locally connected is also globally path connected. In this sense, path connectivity is the `next best thing' one can hope for in the boundary of a (0) group. At the same time, there are known examples of (0) groups which act on spaces with non-path connected visual boundaries (see Example <ref> for one). Therefore, we want to find conditions on G (or on X) which determine when X is path connected.

Another reason to be interested in path connectivity is due to its relationship with semistability. A proper, 1-ended (0) space X is semistable at infinity if any two geodesic rays are properly homotopic. Being semistable at infinity is a quasi-isometry invarient, so showing a group is semistable at infinity comes down to finding an appropriate space on which this group acts. Geoghegan  conjectured that all (0) groups are semistable at infinity (in fact, he has conjectured that all finitely presented groups are semistable at infinity). Geoghegan also shows that given a (0) space X if X is path connected, then X is semistable at infinity <cit.>. Piecing all of this together, knowing that G acts geometrically on a (0) space with a path connected visual boundary shows that G is semistable at infinity.

The majority of this paper is dedicated to answering Question <ref> for a class of (0) groups which exhibit more hyperbolicity than others: (0) groups with isolated flats.  These groups are hyperbolic relative to a collection of flat stabilizers and therefore share many properties with hyperbolic groups. Hruska and Ruane have found necessary and sufficient conditions for a  1-ended (0) group with isolated flats to have a locally connected boundary <cit.>. There are many examples which do not have locally connected boundaries, see Section <ref> for a few. We therefore study the general case. In this paper, we prove the following:

<ref>Let G be a 1-ended (0) group with isolated flats acting geometrically on X. Then X is path connected.
 

As an immediate corollary (via the results in <cit.>) we have the following:


	Let G be a 1-ended (0) group with isolated flats. Then G is semistable at infinity.

Other authors have some results which overlap with this corollary. For example, Mihalik and Swenson  show that many 1-ended relatively hyperbolic groups are semistable at infinity <cit.>. However, they do not cover all cases. They assume that their relatively hyperbolic groups have a trivial maximal peripheral splitting and many (0) groups with isolated flats have non-trivial maximal peripheral splittings. Hruska and Ruane  show that most groups which are hyperbolic relative to polycyclic groups are semistable at infinity <cit.>. They assume that these groups have no non-central elements of order 2. We require no assumptions on the (0) groups with isolated flats to conclude it is semistable at infinity.

Along the way to proving Theorem <ref> we introduce a combination theorem for (0) spaces and their boundaries:

<ref>

		Let X be a (0) space and  a block decomposition. Suppose the following conditions hold:
	
	
		
  * for each B∈, B is path connected,
		
  * for each W∈, W is non-empty, and
		
  * all irrational rays are lonely.
	
	Then X is path connected.


For exact definitions of the terms in Theorem <ref>, see Section <ref>. The example to keep in mind is when we have a (0) group G which is an amalgamated product G=A∗_C B. In many cases of interest, there is a natural `tree-of-spaces' X on which G acts geometrically and then Theorem <ref> essentially says if (1) A and B act on spaces X_A and X_B with path connected boundaries, (2) X is connected,  and (3) a technical condition related to how the boundaries of X_A and X_B interact in X, then X is path connected. Amalgamating (0) groups is the primary method for forming new (0) groups. In order to fully understand what types of spaces can appear as the boundary of a (0) group, we need to understand properties of the boundaries of amalgams.


	In Section <ref>, we describe three key examples that we return to throughout the paper. Sections <ref>, <ref>, and <ref> provide the necessary background information on the visual boundary, relatively hyperbolic groups, and (0) groups with isolated flats. We formally define a block decompositions and prove Theorem <ref> in Section <ref>. 
	
	The remaining sections build up to the proof of Theorem <ref>. In Section <ref>, for a given 1-ended (0) group G with isolated flats, we use the group's maximal peripheral splitting (introduced by Bowditch in <cit.>), Hruska-Ruane's convex splitting theorem <cit.>, and Bridson-Haefliger's Equivariant Gluing Theorem <cit.> to fix a space X on which G acts geometrically. Hruska-Kleiner's work <cit.> shows that if G is (0) with isolated flats, then the boundary is well-defined. Therefore fixing a space on which G acts does not affect the boundary. We then focus on a collection of subsets Y_v⊂ X which cover X. These subsets are defined at the end of Section <ref>. In Sections <ref>, <ref>, and <ref> we show that Y_v is path connected. Applying  Theorem <ref> finishes the proof of Theorem <ref>. 
	
	Of potentially independent interest is the work in Section <ref>. Here, we establish a connection between neighborhoods of points in F, the closure of a flat, and neighborhoods in  Y_v ∖ F. This work was done by Haulmark in the locally connected setting <cit.> and we expand the results to our setting.
	



	I would like to thank Kim Ruane for her help and guidance throughout this project. I would also like to thank Genevieve Walsh, Mike Mihalik, and Rob Kropholler for many interesting discussions and for feedback on my drafts.






§ KEY EXAMPLES
 

In this section, we outline three key examples that capture the difficulty in this work. Two are surface group amalgams and the third is a right-angled Coxeter group which is an amalgamation of groups which are virtually surface groups. All three examples are (0) groups with isolated flats and the boundaries of the spaces on which they act are not locally connected.








 
	Let X̃_̃1̃ be the surface amalgam seen in Figure <ref>, G_1 its fundamental group, and X_1 its universal cover. This group is (0) with isolated flats, with the flats coming from the torus. X_1 consists of Euclidean and hyperbolic planes fitting together in a tree-like way. The boundary, X_1, is made up of circles coming from these two types of planes along with the boundary points of rays which pass through infinitely many such planes. It is clear that the union of the circles is path connected since adjacent planes share a pair of boundary points. Because of the isolated flats, if two based rays pass through the same infinite collection of planes, then the rays are asymptotic. Applying Theorem <ref>, we can conclude that X_1 is path connected.





	Let X̃_̃2̃ be 2 tori with boundary components as identified in Figure <ref> and let G_2 be the fundamental group. By identifying ([a,b])^2 with ([c,d])^2, we create a Klein bottle group in G generated by [a,b] and [c,d]. This subgroup (and its conjugates) are the peripheral subgroups for the relatively hyperbolic structure on G_2. Since the Klein bottle group is virtually ^2, this makes G_2 (0) with isolated flats and with maximal peripheral splitting seen in Figure <ref>.
	
		
A (0) space X_2 on which G_2 acts geometrically can be constructed using Bridson and Haefliger's Equivariant Gluing Theorem <cit.>. This construction captures the isolated flats explicitly since X_2 consists of Euclidean planes coming from the Klein bottle group and truncated hyperbolic planes coming from the free groups. Once again, these spaces fit together in a tree-like way and the boundary X_2 is made up of boundary points from the truncated hyperbolic planes, the Euclidean planes, and the rays which pass through infinitely many such spaces.

To see that X_2 is path connected, consider a truncated hyperbolic plane union every flat which shares a boundary point with it. Call this space Y. This is equivalent to taking a F_2-vertex v of the Bass-Serre tree for the splitting and letting Y be the of the vertex space for v union the vertex spaces for each of the vertices of the Bass-Serre tree adjacent to v. It is clear that X_2 is the union of all such Y. To see that Y is path connected, notice that it is a Cantor set coming from the truncated hyperbolic planes union a collection of circles coming from the Euclidean planes. These circle `fill in' the gaps between points of the Cantor set, essentially replacing the missing intervals with circles. Therefore Y is a circle with a collection of extra arcs attached and is path connected. 

Since Y_v is path connected for each v in the Bass-Serre tree and X is the union of all such Y_v, then the only points to check are those which come from rays traveling through infinitely many truncated hyperbolic and Euclidean plans. Once again, the isolated flats condition covers this and we can apply Theorem <ref> to conclude X_2 is path connected.





Let G_3 be the Right-angled Coxeter group in Figure <ref> and let X be the Davis complex for G_3. This group is (0) with isolated flats, with the flats coming from the square in the graph. Let H be the RACG whose defining graph is a hexagon, then G_3 has the splitting seen in Figure <ref>.



H is virtually a surface group, therefore X consists of spaces quasi-isometric to ^̋2, which we will call H̃, and Euclidean planes. Applying the same reasoning as in Example <ref>, we can conclude that X is path connected. This example is important for two reasons. First, the way we split the group is vital for applying Theorem <ref>. If we split the group in the `obvious' way:

    H∗__2∗_2 H

then the virtual ^2 is not captured in the splitting (i.e. this is not a peripheral splitting). Because of this, it is no longer true that if two rays go through the same sequence of H̃s, then the rays are asymptotic. Second, this is the type of group for which Hruska-Ruane's results on semistability do not apply <cit.>. However, it is known that RACGs  are semistable at infinity <cit.>.





We will refer to all three of these examples again in Section <ref> to describe the isolated flats structure on each group, and again in Section <ref> to describe the block decompositions of each space.
	


§ THE VISUAL BOUNDARY
 
We refer the reader to <cit.> for an introduction to the theory of (0) spaces and groups. Throughout this section, X is a proper (0) space. 

We describe the topology on X=X∪_x_0 X and then restrict this topology to the visual boundary.


	
	Fix a basepoint x_0∈ X. Let _x_0 X be the set of geodesic rays c [0,∞)→ X based at x_0. Let B(x_0,r) be the closed ball of radius r and π_r be the projection map to this ball. Define sets of the following form:
	
	
    U(c,r,D):= { x∈ : d(x,x_0)>r, d(π_r(x),c(r))<D}

	where c is a geodesic segment or ray based at x_0, r,D>0. A neighborhood basis for the cone topology on  consists of sets of the form U(c,r,D) along with metric balls in X.
	
	The cone topology can be restricted to _x_0 X by taking sets of the following form to be a neighborhood basis:
	
	
    U(c,r,D)= { c'∈_x_0 X: d(c(r),c'(r))<D}

	with r,D>0.
		
	
	Due to the follow result in <cit.>, the topology on _x_0 X is independent of choice of basepoint. Therefore we can denote the visual boundary as X without ambiguity. In practice, however, it often useful to fix a basepoint and work with _x_0 X.
	
	
		For any choice of x_0 and x_1, _x_0 X and _x_1 X are homeomorphic.
	


We use a metric on X which was first introduced by Osajda. We use this metric in Section <ref>.


	Fix x_0∈ X and D>0. The metric d_D on _x_0 X is defined as follows: given two rays c,c'∈_x_0 X, d_D(c,c')=1r where r∈ [0,∞) such that d(c(r),c'(r))=D. Since d(c(t),c'(t)) increases monotonically with t, there is a unique r satisfying d(c(r),c'(r))=D.
	
	This metric is compatible with the cone topology <cit.>. Furthermore, the spaces ( X, d_D) and ( X, d_D') are isometric, so the metric is independent of choice of D.


	


§ RELATIVELY HYPERBOLIC GROUP AND PERIPHERAL SPLITTINGS
 

In this section, we define relatively hyperbolic groups, the Bowditch boundary, and state a theorem of Tran which relates the Bowditch boundary to the (0) boundary. We then introduce the maximal peripheral splitting of a relatively hyperbolic group. We will use the definition of relative hyperbolicity from <cit.>. This definition uses an action of a group on a compact, metrizable topological space M, which will ultimately be the Bowditch boundary.


Let M be a compact metrizable space and G a finitely generated group. An action of G on M is a convergence action if it is properly discontinuous on distinct triples of M.



A convergence group action of G on M is geometrically finite if every point of M is either a bounded parabolic point or a conical limit point. 




	Let the action of G on M be a geometrically finite convergence action and let  be a collection of subgroups of G. Then G is hyperbolic relative to  if  is exactly the collection of maximal parabolic subgroups. 



	The space M is uniquely determined by the pair (G,) and is the Bowditch boundary, denoted (G,).


Since we do not deal with the Bowditch boundary directly, we will not define all the terms above. See <cit.> for precise definitions. We need the Bowditch boundary in order to apply Theorem <ref> and to use a handful results of Bowditch which are summarized in Theorem <ref>. Both theorems can be found below.

There are many other ways to define a relatively hyperbolic group, see for example <cit.>. Each of these ultimately has G acting on a δ-hyperbolic metric space and takes the boundary of this space to be the Bowditch boundary. This is equivalent to our definition of the Bowditch boundary. Furthermore, all of these definitions of relatively hyperbolic are equivalent.


Suppose G is a (0) group acting geometrically on a (0) space X and (G,) is also relatively hyperbolic. We have so far introduced two different boundaries for G: the (0) boundary and the Bowditch boundary. Work of Tran tells us how to relate these two boundaries:


 
	Let (G,) be a relatively hyperbolic group acting geometrically on a δ-hyperbolic or (0) space X. The quotient space formed from X by collapsing the limit sets of each P∈ to a point is G-equivariantly homeomorphic to the Bowditch boundary (G,).


We will now introduce the maximal peripheral splitting of a relatively hyperbolic group. This plays a vital role in establishing the block decomposition of the relatively hyperbolic groups we study.


	Let (G,) be a relatively hyperbolic group. A peripheral splitting of (G,) is a splitting of G as a finite bipartite graph of groups  whose vertices have two type: peripheral and component. Furthermore, the collection of subgroups of G which are conjugate to a peripheral vertex group is the same as , and  does not contain a vertex of degree 1 which is contained in the adjacent peripheral vertex group.


A splitting of this form is called trivial if one of the vertex groups is equal to G and nontrivial otherwise. A splitting ℋ is a refinement of another splitting  if  can be obtained from ℋ by a sequence of foldings over edges which preserves the peripheral/component coloring. A peripheral splitting is maximal if it is not a refinement of any other splitting. 


We combine a number of results of Bowditch about peripheral splittings into one large theorem:
 
	Let (G,) be a 1-ended relatively hyperbolic group. Then the following hold:
	
	
	
	
		
  * if each P∈ is finitely presented, 1- or 2-ended, and does not contain an infinite torsion subgroup, then (G,) is connected <cit.> and locally connected <cit.>,
		
		
  * if G_v is a component vertex group of a peripheral splitting, then G_v is hyperbolic relative to _v:= { P∩ G_v: P∈} <cit.>,
		
	
	
  * if G_v is a component vertex of a maximal peripheral splitting and both  and _v consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, the (G_v,_v) is connected and locally connected <cit.>.
		
	
		
	
		
	
		
		
  * if G_v is a component vertex of a maximal peripheral splitting and both  and _v consist of finitely generated, 1- or 2-ended groups with no infinite torsion subgroups, then (G_v,_v) contains no global cut points <cit.>.
	


In the setting we are dealing with ((0) groups with isolated flats) each component vertex is virtually ^n for n≥ 2, and so 1-ended with no infinite torsion subgroups. Furthermore, if G_v is a component vertex, _v consists of 1- or 2-ended groups.  We use the first point of this theorem in order to accurately restate a result of Hruska-Ruane in Section <ref>. The second point allows us to make sense of the last two. The third point is used at the end of Section <ref> and the final point is used in a proposition stated in Section <ref>.














§ ISOLATED FLATS
 

Here we define (0) isolated flats and outline some results about their large scale geometry. 


	A subset F⊂ X is a flat if it is isometric to ^n for some n≥ 2.


There are many equivalent definitions of isolated flats. Hruska and Kleiner prove the equivalence of these definitions in <cit.>. We will take the following as our definition:



	Let X be a (0) space admitting a geometric group action by G. Let  be a G-invariant collection of flats. Then X has isolated flats if the following two properties hold:
	
	
		
  * Maximality: There is  D<∞ such that for any flat F⊂ X, there is a flat F'∈ with F⊂ N_D(F'). 
		
  * Isolated: For any r>0, there is a ρ=ρ(r)>0 such that for any distinct F,F'∈,	
		
    diam(N_r(F)∩ N_r(F'))<ρ
		
	


We say a group G is (0) with isolated flats if it acts geometrically on a (0) space with isolated flats.


 
	
	
	
		
  * Return to Example <ref>. The universal cover of the surface group amalgam in Figure <ref> is a (0) space with isolated flats. The fundamental group acts geometrically on the universal cover. The flats, coming from the universal cover of the torus, are isolated in the sense that there is a hyperbolic plane separating them.
		
		
  * Consider Example <ref>. The universal cover of the two tori with boundary components with identifications given in Figure <ref> is a (0) space with isolated flats. The ^2's coming from the Klein bottle are separated by truncated ^̋2's.
		
		
  * From Example <ref>, the Davis complex of this RACG is (0) with isolated flats admitting a geometric group action by the RACG. Here, the flat comes from the square in the defining graph. As with the example above, pairs of flats are separated by (something quasi-isometric to) hyperbolic planes.
		
		
  * Let K be the figure 8-knot and M=S^3∖ K and let X be the universal cover of M. Then X is isometric to truncated ^̋3 and is a (0) space with isolated flats, with the lifts of the torus-boundary of the manifold making up the flats.
	




One of the major theorems from Hruska and Kleiner's work is that there are a number of equivalent characterizations of (0) spaces with isolated flats. 


	Let X be a (0) space admitting a geometric group action by G. Then the following are equivalent
	
	
		
  * X has isolated flats.
		
  * Each component of the Tits boundary ∂_T X is either an isolated point or a standard Euclidean sphere.
		
  * X is relatively hyperbolic with respect to a family of flats .
		
  * G is relatively hyperbolic with respect to a collection of virtually abelian subgroups of rank at least 2.
	


One key fact we use to establish that (0) groups with isolated flats satisfy condition (3) in Theorem <ref> is a quasiconvexity result from Hruska-Ruane. Recall the definition of quasiconvex:


	Let X be a geodesic metric space. A subset C is κ-quasiconvex if for any pair of points x,y∈ C, any geodesic between x and y is in N_κ(C). 
	
	We say a subset is quasiconvex if it is κ-quasiconvex for some κ. 


 
	
	Let X be a (0) space with isolated flats with respect to . There is a constant κ>0 such that the following holds:
	
		
  * Given two flats F_1,F_2∈ with c the shortest length geodesic from F_1 to F_2, then F_1∪ F_2∪ c is κ-quasiconvex in X.
		
  * Given a point p and a flat F∈ with c the shortest path from p to F, then c∪ F is κ-quasiconvex in X. In particular, if c' is any geodesic joining a point of c to a point of F, then c' intersects the κ-neighborhood of the endpoint in the flat of c.
	





§ BLOCK DECOMPOSITIONS
 

This section is devoted to defining a block decomposition of a space, providing some motivating examples, and concludes with the proof of Theorem <ref>. A block decomposition was first introduced in <cit.> as a generalization of the results from <cit.>. In both papers, the authors focus on spaces which admit geometric group actions by groups of the form G=A∗_C B. The spaces on which these groups act can be viewed as the union of closed, convex pieces, called blocks, and the intersection of these blocks follows the subgroup intersection described in the Bass-Serre tree for the splitting. While the definition of a block decomposition is independent of a group action, this paper's motivating examples of such a decomposition come from the splitting of a group.

We will use the following definition from <cit.>:

 
	A block decomposition of a CAT(0) space X is a collection  of closed, convex subsets (called blocks) such that the following conditions hold:
	
	
		
  * X = ⋃_B∈ B,
		
  * each block intersects at least two other blocks,
		
  * Parity condition: every block has a (+) or (-) parity such that two blocks intersect only if they have opposite parity,
		
  * ϵ-condition: there is an ϵ>0 such that two blocks intersect iff their ϵ-neighborhoods intersect.
	
	


We will mostly use the facts that each B is closed and convex and that X=⋃_B∈ B. Conditions (2)-(4) are there to prevent any degenerate cases (e.g. splitting X into only two pieces, decomposing a plane as a union of lines). 


	A wall is a non-trivial intersection of blocks and we denote the collection of walls . 


Since each block is convex, note that each wall is convex as well. The nerve, denoted (), is a graph which records block intersections. There is a vertex for each block and an edge if two blocks (non-trivially) intersect. We will use the following notation: If v  and w are vertices of (), then B_v, B_w∈ are the blocks corresponding to v and w, respectively. Given an edge e with vertices v and w, then B_v and B_w intersect in a wall by definition and we denote this wall W_e.

Following results from <cit.> and <cit.>, we get that the nerve is always a tree.

	() is a tree.


Since () is a tree, we will denote it _ or  when  is understood. Here are some examples of block decompositions:






 
	
	
	
	In <cit.>, the authors describe what they call a torus complex, see Figure <ref>. Let G be the fundamental group of the torus complex and X the universal cover. The space X is (0) and admits a block decomposition coming from a the following splitting of G:
	
    G = ( F_2×) ∗_^2( F_2×)

	The fundamental group of two tori with a pair of curves identified is F_2× which acts geometrically on T_4×, where T_4 is the valence 4 tree. The blocks come from the lifts of the middle torus and either the left or right torus, meaning each B∈ is isometric to T_4×. The walls are the lifts of the middle torus and are all isometric to ^2. 
	
	 Croke and Kleiner <cit.> use  to show that by changing the angle of intersection between the curves b and c on the middle torus, G can act geometrically on spaces X_1 and X_2 which have non-homeomorphic boundaries. This answered a question of Gromov. Later, in <cit.>, Croke and Kleiner vastly extend their findings on this specific example to a class of groups they call `admissible.'


 We return to Example <ref>. The universal cover of the surface amalgam from this example is (0) and has a block decomposition coming from the splitting of the fundamental group. Here, if G_1 is the fundamental group, then G_1 splits as
	
    G_1 = π_1(S_2)∗_^2

	where S_2 is the surface with genus 2. The blocks of this decomposition come from the lifts of the two surfaces. Therefore there are two types of blocks: those isometric to ^̋2 and those isometric to ^2. The walls, however, are all isometric to . Unlike the torus complex, this group is (0) with isolated flats. 



	Consider Example <ref> once again. Let G_2 be the fundamental group of this surface amalgam and X_2 its universal cover. Recall that this group admits the maximal peripheral splitting as seen in Figure <ref>. X_2 admits a block structure coming from this splitting. The vertex groups which are free act geometrically on truncated hyperbolic space and the vertex group which is the Klein bottle group acts geometrically on a Euclidean plane. These truncated hyperbolic planes and Euclidean planes in X_2 make up the block structure. That is, each block is isometric to either a Euclidean plane or a truncated hyperbolic plane.



	Return now to the RACG in Example <ref>. Let G_3 be the fundamental group and X_3 the universal cover of its Davis complex. The block structure on X_3 once again comes from a splitting of G_3 which we saw in Figure <ref>. The blocks of X_3 come from the subspaces on which the vertex groups act geometrically. The hexagon subgroup H acts geometrically on a space quasi-isometric to ^̋2 while (_2∗_2)^2 acts geometrically on ^2. These two types of spaces form the blocks of X_3. 




	More generally, if X is a (0) space coming from the construction in the Equivariant Gluing Theorem (<cit.> Theorem II.11.18), then X admits a block decomposition coming from the splitting of the group G=G_1∗_H G_2. There is a natural map p  X→, where  is the Bass-Serre tree for the splitting. If v is a vertex of  then the blocks are each of the form  p(N_1/2(v)). We will be using this in Section <ref>.


With these examples in mind, we can say something about the way block boundaries intersect. We say a wall W_e∈ separates blocks B_v and B_u if any path from B_v to B_u must pass through W_e. This is equivalent to e being an edge on any path from v to u in . 

 
	For two blocks B_u,B_v∈ corresponding to vertices u,v∈, then one of the following holds:
	
	
		
  * B_u ∩ B_v = ∅
		
  * The vertices u and v are adjacent and B_u ∩ B_v =  W where W=B_u ∩ B_v.
		
  * The vertices u and v are not adjacent in  but B_u ∩ B_v is non-empty. Then B_u ∩ B_v ⊂ W where W is any wall between B_u and B_v.
	



	We may assume that B_u ∩ B_v ≠∅, so we are in either case (2) or case (3). If u and v are adjacent, then case (2) follows immediately. Suppose then that α∈ B_u ∩ B_v and that u and v are not adjacent in . Let W be any wall separating B_u and B_v. 
	
	Pick basepoints x_0∈ B_u and x_1∈ B_v and let y be a point in W along the geodesic segment from x_0 to x_1. Let c_0 and c_1 be rays representing α based at x_0 and x_1, respectively. For each n∈, let y_n be a point in W along the geodesic segment from c_0(n) to c_1(n). Since W separates the blocks, these y_n exist. Since c_0 and c_1 are asymptotic, there is a constant K such that d(c_0(t),c_1(t))≤ K for all t. Therefore y_n is K-close to both c_0 and c_1. The limit of the geodesic segments from y to y_n, therefore, is asymptotic to c_0 and c_1, so is a representative of α. Since W is convex, then α∈ W.





	The following should be noted from the proof above: it is possible for α∈ B but a ray representing α is disjoint from B entirely. This happens above with the block B_u and the ray c_1.


We use the following terminology and the definition below from <cit.>. A ray c enters a block B if for some time t, c(t)∈ B and c(t) is not in any other block (and therefore is not in a wall). Similarly, a ray c exits a block B if for some t, c(t)∈ B and not in any wall and for some t'>t c(t')∉ B. 


	Let x_0∉ W for all W∈. The itinerary of a ray c based at x_0, denoted _x_0(c), is the sequence of blocks c enters. If x_0 is understood, then the itinerary will be denoted (c). Abusing notation, if the basepoint is understood and c(∞)=α∈ X, then (α) is used to denote the itinerary of the ray representing α. 
	


In order for this definition to be well defined, we need the following lemma:


	If  is a block decomposition for X, then there are points of X which are not in any walls. In fact, for each B∈, there are points of B which do not lie in any walls.



	Let B∈ and let W be a wall of B such that B∩ B_0 = W. Define D=inf_W'⊂ B d(W,W').
	
	If D>0, then all points in N_D(W)∖ W ∩ B are points of B which do not lie in any walls. Since D>0, then this set is non-empty.
	
	Assume, for contradiction, that D=0. Then pick δ< ϵ and consider N_δ(W). Since D=0, there is some wall W'⊂ B which non-trivially intersects N_δ(W). Let B∩ B'=W'. Since δ<ϵ, N_δ(W)∩ W' means that in fact N_ϵ(B_0)∩ W'≠ 0. Since W'⊂ B' and applying the ϵ-condition on the block decomposition, B_0∩ B'≠∅. But then B, B', B_0 all pairwise intersect, contradicting the parity condition. Therefore D>0.
	

 It is important to note that it is possible for α∈ B but for B∉(α). This can happen as in the proof of Lemma <ref>, where c_0 represent α, α∈ B_v, but _x_0(c_0)=_x_0(α)={B_u}. In other words, this can happen when α lies in the boundary of multiple blocks.



The lemma below follows exactly in the same way as <cit.> (Lemma 2), which uses the fact that each B is convex and and B's topological frontier is covered by the collection of blocks corresponding to the link in  of the vertex corresponding to B.

  
	Let X be a (0) space with a block decomposition  and corresponding nerve . The itinerary for any α∈ X consists of blocks which correspond to vertices on a geodesic segment or ray in .  





This lemma gives rise to the following definition:


	
	
	
	Fix a basepoint x_0 which is not in any wall and let c be a geodesic ray based at x_0 representing the point α∈ X. The point α is rational if (c) is finite. The point α is irrational if (c) is infinite.


In order for this definition to be useful, being rational or irrational must be independent of representative for α.


	If c represents α and c is rational, then all representatives of α are rational. 



	Let α∈ X and let c [0,∞)→ X be a ray representing α based at x_0 be a finite itinerary and let B be the final block of this itinerary. So α∈ B. Let c' [0,∞)→ X be a ray representing α based at x_1, and suppose, for contradiction, that _x_1(c') is infinite. By Lemma <ref>, (c') represents a geodesic ray in . 
	
	First, we show that if d(v_B,v_B')≥ 2n for v_B,v_B'∈, then d(B,B')≥ 2nϵ in X. Let B=B_1,B_2,...,B_2n=B' denote the 2n-blocks between B and B'. To see d(B,B')≥ 2ϵ, notice if d(v_B,v_B')=2, then by the ϵ-condition, d(B,B')≥ 2ϵ since the ϵ-neighborhoods of the blocks do not intersect. So let x∈ B and x'∈ B' and let [x,x'] be the geodesic segment between them. For each 0≤ i ≤ n, there are z_i∈ B_2i such that d(z_i,z_i+2)≥ 2ϵ. Therefore d(x,x')≥ 2nϵ. 
	
	Now, for any K>0, there exists some n such that 2nϵ>K. Furthermore, since (c') represent a geodesic ray in , there is a block B' such that d(v_B,v_B')≥ 2n. Thus c' is not in the K-neighborhood of B for any K. But c and c' are asymptotic, a contradiction. Thus (c') must be finite.
	



	If (c) is infinite for some ray c, then all rays representing c(∞) have infinite itineraries.



	Suppose c' represents c(∞) and is finite. Then by the argument above, all rays representing c'(∞) have finite itineraries, which cannot happen since (c) is infinite.

Let ∂_R X and ∂_I X denote the rational and irrational boundary points of X, respectively. Then we have:

	
  * ∂_R X = ⋃_B∈ B
	
  * ∂_I X =  X ∖∂_R X


Once a basepoint is fixed we can use Lemma <ref> to create a map η∂_I X →, mapping a point α to the end of  representing its itinerary at that basepoint. This map is surjective but not necessarily injective. The points in η(α) are the points of the X with the same itinerary as α. We will care when α is the only boundary point with its itinerary.


	If α is irrational and is the only point in X with its itinerary, then we say α is lonely. We will denote the set of lonely rays X.





	Let X be a (0) space and  a block decomposition. Suppose the following conditions holds:
	
	
		
  * for each B∈, B is path connected,
		
  * for each W∈, W is non-empty,
		
  * all irrational rays are lonely
	
	then X is path connected.



	It should be noted that we only assume W is non-empty. We do not need that W is path connected. The walls having non-empty boundary is equivalent to X being connected.


Going forward, we shall refer to condition (3) as the lonely condition.

Before proving Theorem <ref>, it should be noted that conditions (1) and (2), without (3), are insufficient for guaranteeing X being path connected. Returning to Example <ref>, the block decomposition satisfies conditions (1) and (2), but X is not path connected. In <cit.>, the authors explicitly construct a pair of distinct irrational rays with the same itinerary which are in a different path component from the union of the block boundaries. In that example, there is something akin to the Topologist's Sine Curve in the boundary, which obstructs path connectivity. Theorem <ref> shows that the lonely condition is sufficient for avoiding this pathology.

We will prove Theorem <ref> in two parts. First we will show that ∂_R X is path connected. We then show if ∂_R X is path connected, then ∂_R X ∪ X is path connected as well. By the lonely condition, X = _I X, concluding the proof.


	Suppose for each B∈, B is path connected and for each W∈, W is non-empty. Then ∂_R X is path connected.



	Fix x_0∈ X∖.	Let x_0∈ B and B' be an adjacent block. We show that B ∪ B' is path connected then proceed by induction. Since block boundaries are path connected, it suffices to show that there is a path from a point in B to a point in B'. Let α∈ B and α' ∈ B'. Since B and B' are adjacent, there is a W∈ such that B∩ B' = W. Let β∈ W, which exists since W is non-empty. Since W =  B ∩ B' and each block has path connected boundary, there are paths from α to β and from β to α' in B ∪ B'. The concatenation of these paths is a path from α to α'.
	
	Let (α')= {B_1, B_2, ..., B_n} with B_1=B. The union of the boundary of adjacent blocks is path connected, so B_n-1∪ B_n is path connected. By the induction hypothesis, B_1 ∪ B_2 ∪ ... ∪ B_n-1 is path connected, and therefore B_1 ∪ B_2 ∪ ... ∪ B_n is path connected, which concludes the proof. 


Before proving the next proposition, we need the following lemma from <cit.>:

 
	Let X be a locally compact CAT(0) space with a closed, convex subset Y. Let x_0∈ X and α_i∈ X with α_i→α and the rays r_i based at x_0 representing α_i and r the ray based at x_0 representing α. If r_i intersects Y for all i, then either r intersects Y or α∈ Y.


 
	If ∂_R X is path connected, then ∂_R X ∪ X is path connected.



	Fix x_0∈ X∖ and let α∈ X with (α) = {B_1, B_2, ...}. Let c [0,∞) be a path in ∂_R X such that c([n-1,n])⊂ B_n for all n∈. Our goal is to extend c to [0,∞] such that c(∞)=α. 
	
	Let {t_n}⊂ [0,∞) be a monotone sequence tending to infinity. 
	
	
	Since X is compact then, by passing to a subsequence if necessary, {c(t_n)} converges to some α'. It suffices to show that α = α ', which we will do by showing (α) = (α').
	
	For each n, (c(t_n))={B_1, B_2, ..., B_k_n} where k_n = ⌈ t_n ⌉. Let r_n be the ray based at x_0 representing c(t_n) and let r be the ray based at x_0 representing α'. By our construction, r_n→ r.
	
	Fix a block B_m in (α). There is an N such that for all n>N, B_m∈(c(t_n)) and so r_n∩ B_m ≠∅. By Lemma <ref>, either r∩ B_m≠∅ or r∈ B_m. If the former, then B_m∈(α'). If the latter is true, pick points x_n(m)∈ r_n ∩ B_m such that d(x_0, x_n(m))→∞. Such a sequence exists since r∈ B_m and r_n→ r. Then, after possibly passing to a subsequence, the geodesic segments from x_0 to x_n(m) converge to a ray ρ_m with ρ_m(∞) = α'. For each i<m, B_i separates B_m from x_0, so B_i∈(α').
	
	But if this is true for some m, then it is true for all M>m and so α∈ B_M for all M>m. Consider the ray ρ_M by the same construction as the previous paragraph. Using the same argument, B_i∈(ρ_M(∞)) = (α') for all i<M. Since m<M, this means B_m∈(α'). Therefore α' and α have the same itinerary. Since α is lonely, α = α'.



	Proving path connectivity of the boundary does not require each block to have a path connected boundary. The proposition only assumes that ∂_R X is path connected. This distinction will come up later when we show all 1-ended (0) groups with isolated flats have path connected visual boundaries.




	While the lonely condition is sufficient, it is not necessary for path connectivity. This comes up for the following group
	
    G=F_2× = ^2∗_^2

	G acts the universal cover of two tori with a curve on each identified. The block decomposition comes from viewing G as the amalgamated product of two ^2s over a . The boundary is path connected since it is a suspension of a Cantor set, but irrational rays are not lonely.







§ FIXING A SPACE


Normally when dealing with (0) groups and their boundaries, one needs to be particularly careful about the space they are working with. This is because Croke and Kleiner <cit.> showed  that, unlike in the hyperbolic setting, (0) groups do not have a well defined visual boundary. When it comes to (0) groups with isolated flats, however, work of Hruska and Kleiner <cit.> shows that the visual boundary of these groups is well defined . In this section, we will fix the space on which a 1-ended (0) group with isolated flats will act on and define the block decomposition for this space.

Throughout this section, let G be a (0) group which is hyperbolic relative to , the collection of flat-stabilizers so that (G,) is the relatively hyperbolic pair. Let G act geometrically on some (0) space X. 

We want more control over the space on which G acts, so we use the peripheral splitting theorem of Bowditch introduced in Section <ref>, a convex splitting theorem of Hruska and Ruane <cit.>, and a combination theorem of Bridson and Haefliger <cit.> to fix a different space which admits a geometric group action by G. We quote the two remaining theorems below. But first, a definition.



	Let G be a CAT(0) group acting geometrically on a (0) space X. A subgroup H≤ G is convex if H stabilizes a closed, convex subspace Y of X, and H acts cocompactly on Y. Therefore H act geometrically on Y, a (0) space.



	Let G act geometrically on a (0) space X. Suppose G splits as the fundamental group of a graph of groups  such that each edge group of  is convex. Then each vertex group is convex as well. In particular, each vertex group is a (0) group.



Let  be the maximal peripheral splitting for our group G. Since G has isolated flats, all peripheral subgroups in the graph of groups  are virtually abelian and therefore so are all edge groups. By the Flat Torus Theorem <cit.>, all virtually abelian subgroup of a (0) group are convex and therefore we can apply the Convex Splitting Theorem to the maximal peripheral splitting of G.

Let  be the collection of component vertices and  the set of peripheral vertices of . For each v∈ with vertex group G_v, let X_v be the convex subspace of X on which G_v acts geometrically. For w∈, the vertex group G_w is virtually abelian and therefore acts geometrically on some Euclidean space F_w. 

Each edge e of  is incident to a peripheral vertex w and a component vertex v. Let F_e be the flat subspace of F_w on which P_e acts geometrically. Each edge also comes equipped with a monomorphism ϕ_e P_e→ G_v. The Flat Torus Theorem gives a ϕ_e-equivariant isometric embedding of F_e into X_v. 

With all of this set up, we can use the following theorem to build our desired X:


	Let Γ_0, Γ_1, and H be groups acting geometrically on complete CAT(0) spaces X_0, X_1, and Y, respectively. Suppose for j=0,1, there exists monomorphisms ϕ_j H ↪ X_j and ϕ_j-equivariant isometric embeddings f_j  Y→ X_j, then Γ=Γ_0∗_H Γ_1 acts geometrically on a CAT(0) space X.


The space uses the Bass-Serre tree  for the splitting  of G. For each component vertex v of , take a copy of X_v and for each peripheral vertex w of , take a copy of P_w. We shall refer to these spaces as vertex spaces. Then for each edge e incident to v and w, take a copy of F_e× [0,1], which we shall refer to as edge spaces. Identify the F_e×{0} part with the image of F_e in X_v and F_e×{1} is identified with the image of F_e in F_w. Following the Bass-Serre tree, all the copies of the different vertex and edge spaces are glued up to form X. 

With this X fixed, we can also describe the block decomposition of X. For each vertex space X_v or P_w, take the closed 1/2-tubular neighborhood to be a block B. Then each block intersects in F_e×{1/2}. This defines a block decomposition of X.


	Throughout the remaining sections, when referring to a (0) group with isolated flats acting geometrically on X, we shall be referring to the X from this construction. 


Our goal for the next few sections is to show that X is path connected. In Lemma <ref>, we will show that with this block decomposition, all irrational rays are lonely. Therefore we just need to show that ∂_R X is path connected.

In general, each component vertex is not 1-ended. When this happens, X_v is cannot be path connected and therefore showing ∂_R X is path connected requires more work. Therefore we introduce Y_v:


	For each component vertex v, let Star(v) be the star of v in . There is a continuous map p X→ which maps vertex spaces to vertices and edge spaces to edges in the obvious way. Let Y_v= p  ( Star(v)).
	
	 Put another way, let ℰ_v be the collection of edges incident to v and _v be the collection of vertices of  adjacent to v. Since  is the Bass-Serre tree for the maximal peripheral splitting, each w∈_v is a peripheral vertex. Then Y_v is
	
	
    Y_v := X_v ∪(⋃_e∈ℰ_v F_e×[0,1]) ∪(⋃_w∈_w F_w )

	Put a third way, Y_v consists of X_v and every flat of X which shares a boundary point with Y_v.
	
	Ultimately, we want to show that Y_v is path connected for each v∈. When dealing with a generic Y_v, we will drop the subscript and denote it Y.


The motivating example for needing to this is Example <ref>. The maximal peripheral splitting of this group is in Figure <ref>. Since the component vertices are free groups, they have Cantor sets for boundaries. Additionally, there is no obvious intermediate peripheral splitting which has component vertices with path connected visual boundaries. Therefore, one needs to look at one of the Cantor sets along with all the circles attached to it. We will show this space is path connected.





§ DECOMPOSITION SPACES

In this section we will repeat some results from <cit.> about decompositions of spaces and their relationship to CAT(0) groups with isolated flats. In particular, we establish that the map π X → (G,) from Theorem <ref> can be viewed through the lens of decomposition theory. We then use this to establish a decomposition of Y and ultimately conclude that boundary points in Y which are not in the boundary of a flat are locally connected. 

Much of the results here follow from elementary decomposition theory, which can be found in <cit.>. For a topological space M, a decomposition, , is a partition of M. The decomposition map π M→ M/ is the quotient map where each d∈ is collapsed to a point and the resulting space is given the quotient topology. In this way, decompositions of M are equivalent to quotients of M. Any decomposition can be made into a quotient map and any quotient map can be made into a decomposition by taking point pre-images as elements of the decomposition.



	A decomposition  of M is upper semicontinuous if each d∈ is compact and for each d∈ and each open U⊂ M containing d, there is an open V⊂ M containing d such that every d∈ which intersects V is contained in U. A quotient is upper semicontinuous if the associated decomposition is upper semicontinuous.



	Let  be a decomposition of a space M with each d∈ compact. The following are equivalent:
	
		
  *  is upper semicontinuous.
		
  * For an open set U⊂ M, let U^* be the union of d∈ such that d⊂ U. Then U^* is open.
		
  * The decomposition map π  M→ M/ is closed.
	
	


 
	
	(1)⇒ (2): Let U open in M. If there are no d∈ such that d⊂ U, then U^* is empty and therefore open. For each d⊂ U, let V_d be the open subset of U from the definition of upper semicontinuous. Let V=∪_d⊂ UV_d, then V is open and we claim that V=U^*. The inclusion U^*⊂ V is obvious. Suppose x∈ V, then x∈ V_d for some d⊂ U. Since  is a decomposition, x∈ d' for some d'∈. Since d'∩ V_d≠∅, then d'⊂  V_d ⊂ U and therefore d'⊂ U^*. This shows the other inclusion.
	
	(2)⇒ (3): Let C⊂ M be closed and let U be C's complement in M. To show π M→ M/ is closed, we must show that π (π(C)) is closed. Call this set C', then we have 
	
    C'= {x∈ M: π(x)∈π(C)} = ∪{ d∈: π(d) ∈π(C) }

	Let U'= M∖ C'. We claim U^*=U'. First, U^*⊂ U' since if d⊂ U, then π(d)∉π(C) and therefore d∩ C'=∅. For the other inclusion, suppose x∉ C', then x∈ d for some d∈ such that π(d)∉π(C). Since π(d)∉π(C), then d∩ C = ∅, so d⊂ U and thus d⊂ U^*. This shows the other inclusion.
	
	(3)⇒ (1): Let d∈ and U open in M such that d⊂ U. Let C=M∖ U. Since π is a closed map, then π(C) is closed and therefore π(π(C)) is closed as well. Let U'=M∖π(π(C)). Then U'⊂ U and if d∩ U' ≠∅, then d⊂ U. If d⊂ U, then π(d)∉π(C) and therefore d⊂ U'. Suppose d'∩ U' ≠∅, then π(d') ∉π(C), so d'⊂ U. 



	If  is an upper semicontinuous decomposition of a Hausdorff space, then M/ is Hausdorff. If  is an upper semicontinuous decomposition of a metric space, then M/ is metrizable.


Boundaries of proper CAT(0) spaces are metrizable, and therefore so are their quotients coming from upper semicontinuous decompositions.


	If  is an upper semicontinuous decomposition of M, then π M→ M/ is a proper map.



	A decomposition  of M is monotone if each d∈ is compact and connected.


 
	Let  be an upper semicontinuous decomposition of a space M. Then  is monotone if and only if π(C) is connected whenever C is a connected subset of M/.



	(⇒) Let C be a connected subset of M/ and suppose, for contradiction, that π(C) is disconnected. Then there are disjoint closed sets U and V such that π(C)⊂ U ∪ V. Since each d∈ is connected, if d⊂π(C), then d⊂ U or d⊂ V. Furthermore, if x∈π(C) and x∈ d for some d∈, then d⊂π(C). But then π(U)∪π(V) are disjoint, cover C and are closed since the decomposition induces a closed map. This is a contradiction, so π(C) is connected.
	
	(⇐) Singletons are connected in M/, therefore π(x) is connected for each x∈ M/ and π(x)=d for some d∈ by definition of the decomposition map. Thus each d∈ is connected so the decomposition is monotone.



 
	Let  be an upper semicontinuous monotone decomposition of M. Let {x} be a singleton member of . If M/ is locally connected at π(x), then M is locally connected at x.



	Let U be a neighborhood of x in M. Then U^* is an open neighborhood of x contained in U. Since M/ is locally connected at x, then π(U^*) contains a connected neighborhood V containing π(x). By Proposition <ref> π(V) is connected and π(V)⊂ U^*⊂ U. 


The following definition allows us to construct subspace decompositions:


	Let  be a decomposition of a Hausdorff space M. Let W⊂ M such that if d ∩ W is non-empty for any d∈, then d⊂ W. The induced subspace decomposition of W is the decomposition consisting of all members of  which are contained in W. If  is upper semi-continuous, then the induced subspace decomposition is as well.

This definition will become vital for understanding the decomposition of Y⊂ X.


	A collection of subsets  of a metric space is a null family if for each ε>0, only finitely many A∈ have diameter greater than ε.


 
	Let  be a null family of compact sets in a metric space M. Suppose q∈ M and q is not in any of the members of . Then each neighborhood U of q contains a smaller neighborhood V of q such that for each A∈, if A∩ V ≠∅, then A⊂ U.



	Let U a neighborhood of q and suppose B(q,ε)⊂ U. Then choose δ>0 such that δ<ε/2 and such that d(q,A)>δ for each of the finitely many A with diameter greater than ε/2. If V=B(q,δ), then the result follows. 




		In view of how the metric works on X, the notion of a null family can be reformulated topologically as follows: Fix x_0 as the basepoint for the cone topology. A collection  of subspaces is a null family in X if there is some D>0 such that for each r<∞, only finitely many members of  are not contained in a set of the form U(·, r, D). Under the metric d_D on ∂_x_0 X, only those members of  which lie in U(·, r, D) have diameter at most 1/r.
	
	A similar condition can be put on the cone topology of X. We do not need to deal with this directly, however. Work of Bestvina says that X is metrizable and this metric is compatible with the cone topology, which is sufficient for our needs <cit.>.



 
	Let X be a CAT(0) space with isolated flats with respect to the family . Let  be the collection of spheres { F: F∈}. Then  is a null family of disjoint compact subsets of X. 





	Let  be a decomposition of a metric space M such that each d∈ is compact. Let  be the collection of d∈ such that d is not a singleton. Then if  is a null family,  is an upper semicontinuous decomposition. 



	Let U be an open set in M and d⊂ U. For each x∈ d, let ε_x be such that B(x,ε_x)⊂ U. For each x, let 0<δ_x<ε_x/2 such that d(x,A)>δ for each A∈ with diameter at least ε_x/2. Then let V=∪_x∈ d B(x,δ_x). It is clear that V⊂ U by our choices of δ_x. Suppose y∈ d'∩ V for some d'∈. If d'={y}, then d'⊂ U. If not, then y∈ B(x,δ_x) for some x∈ d. By our choice of δ_x, it must be that the diameter of d' is smaller than ε_x/2, so d'⊂ B(x,ε)⊂ U. Therefore  defines an upper semicontinuous decomposition of M.



	Let G be a CAT(0) group with isolated flats relative to  acting geometrically on a CAT(0) space X. Let  be the collection of spheres which bound maximal dimensional flats in X. Then the quotient map X → X/ ≅∂ (G,) given in Theorem <ref> is upper semicontinuous and monotone.


   
	Let G be 1-ended and CAT(0) with isolated flats relative to . Then X is locally connected at each x which is not in the boundary of a flat. 



	From Theorem <ref>, ∂(G,) is locally connected. Since the decomposition is upper semicontinuous and monotone, the singletons in the decomposition are locally connected as well. In this decomposition, the singletons are exactly the points which are not in the boundary of any flat.



Recall the definition of Y(Definition <ref>) at the end of Section <ref>: Let p X→ be the projection map onto the Bass-Serre tree . Then for a component vertex v∈, Y_v=p(Star(v)). Since we will being working with generic Y_v, we shall just denote it as Y.
 
	The restriction of π X → (G,) to Y defines a monotone upper semicontinuous decomposition and the image of Y is ∂ (G_v, _v), the Bowditch boundary for the corresponding component vertex. Here _v : = { P∩ G_v: P∈}.



	First, see that restricting π to Y is a subspace decomposition for if d∩ Y, then d⊂ Y. This is clear since for each d∈, either d is a singleton or d is the boundary of a maximal dimensional flat and in either of those cases, d∩ Y means d⊂ Y. Additionally this subspace decomposition is monotone. Therefore we have an upper semicontinuous decomposition π Y →π( Y). 
	
	Next, consider the relatively hyperbolic pair (G_v,_v). By the arguments above, the map π' X_v → (G_v, _v) is an upper semicontinuous decomposition. Since the peripheral structure of G_v is inherited from G, then π( X_v) = π'( X_v).
	
	Lastly, we need to show that π( Y) = π (X_v). To see this, note that if d⊂ Y, then either d is a singleton or d is the boundary of a flat. In the former case, then d∈ X_v. In the latter case, d∩ X_v ≠∅. The map π collapses all points of d to a single point, therefore π(d) = π (d∩ X_v), and therefore π( Y) =  (G_v,_v).
	
	
	



	If x∈ Y and x∉ F for any flat F, then Y is locally connected at x.



	This follows from the fact that we have a monotone, upper semicontinuous decomposition to a locally connected space (Theorem <ref>) and that x is a singleton in this decomposition.






§ PROPER AND COCOMPACT ACTION
 

Our goal of this section is to show that the stabilizer of a flat of X_v also acts properly and cocompactly on Y minus the higher-dimensional flat. Recall X_v is the vertex-space for a component vertex v of the maximal peripheral splitting of G. Since we need to distinguish between flats of X_v and flats of Y, we need to establish some notation. Throughout this section, fix a flat F⊂ X_v and let F^* be the maximal dimensional flat of X which contains F. By our construction of Y, F^*⊂ Y but F^* is not necessarily contained in X_v. Let P and P^* stabilize F and F^* respectively. Lastly, define Ω:=  Y ∖ F^*.

The ideas from this section come from <cit.>, which proves the results in the case where X is locally connected. In that setting, Y and X_v are the same and F=F^*.

The stabilizer P acts properly and cocompactly on F. With some work, one can show that P also acts properly and cocompactly on Ω. Furthermore, the fundamental domains for the action of P on both can be picked in a way which allows us to relate the two spaces. The following lemma of Bowditch  establishes the proper and cocompact action of P on Ω <cit.>:


	Let P be a group acting on topological spaces C and D. Define the action of P on C× D as the diagonal action and let ⊂ C× D. If  is P-invariant and the projection maps π_C and π_D from  to C and D are both proper and surjective, then the following are equivalent:
	
		
  * P acts properly and cocompactly on C
		
  * P acts properly and cocompactly on D
		
  * P acts properly and cocompactly on 
	


Define (F) to be the set of geodesic rays orthogonal to F. Recall a geodesic ray r [0,∞) → X is orthogonal to a convex set C⊂ X if r(0)∈ C and for every t>0 and any y∈ C, the Alexandrov angle, ∠_r(0)(r(t),y), is greater than or equal to π/2. 

We will apply this lemma where F and Ω are C and D respectively.  Let := { (x,q): there exists  q∈(F)  with  d(x,q(0))≤ A}, where A is the diameter of the fundamental domain for the action of P on F. Throughout, we will assume our basepoint for the topology on Y is in F. Since P acts properly and cocompactly on F, we need to show the P-invariance of  and the projection maps to F and Ω are both proper and surjective. 



	 is P-invariant.



	P acts by isometries on Y, so if (x,q)∈ and p∈ P, then d(p· x, p· q(0))≤ A. It is clear that p· q is a ray, but we need to show that it is orthogonal to F.
	
	For all t, π_F(q(t))=q(0), so q(0) is the closest point of F to each point along the ray. Assume, for contradiction, that π_F( p· q(t)) = y and y≠ p· q(0). Then	
	
    d(p· q(t), y) < d(p· q(t), p· q(0))

	This is a strict inequality because of the uniquness of projection points. Applying p to all the points gives us:
	
    d(q(t),p·  y) < d(q(t),q(0))

	Since p∈ P and P fixes F, then p y ∈ F. But q(0) is the closest point of F to q(t), so y=p· q(0) and therefore p· q∈(F).



	Let α∈Ω. If r is a ray representing α, then there is a ray q∈(F) asymptotic to r.


To prove this lemma, we make use of the following lemma from <cit.>, which follows closely to Lemma I.5.31 in <cit.>.

 
	If (Y,ρ) is a separable metric space, (X,d) a proper metric space, y_0∈ Y, and K a compact subset of X, then any sequence of isometric embeddings, c_n Y → X, with c_n(y_0)∈ K has a subsequence which converges point-wise to an isometric embedding c Y→ X.


Throughout the remaining of the section, let [x,y] denote the geodesic segment connecting points x,y∈ X. 


	Fix a basepoint x_0, let  x_n:=r(n) and let y_n:= π_F(x_n) for all n∈. First, assume for contradiction that y_n is unbounded. Then y_n→η∈ F. By <cit.>, there is a constant M>0 such that for all n d(x_0,[x_n,y_n])<M, where [x_n,y_n] is the geodesic segment in Y from x_n to y_n. Let m be the point of this geodesic closest to x_0. Since the geodesic [x_n,y_n] is orthogonal to F and x_0,y_n∈ F, then the largest side of the triangle with vertices x_0,y_n,m is the edge [x_0,m]. Therefore d(x_0,y_n)<M, contradicting y_n tending to infinity.
	
	For each n∈, let f_n [0,n]→ Y be the geodesic [y_n,x_n], which is orthogonal to F. For each k and all n≥ k, consider the sequence (f_n|_[0,k]). By Lemma <ref>, these converge to a map q_k [0,k]→ Y. Notice for each ℓ < k, q_ℓ = q_k|_[0,ℓ], since the tails of the sequences defining q_ℓ and q_k are identical. Therefore, q [0,∞)→ Y is a geodesic ray which is the limit of the sequence (q_n). 
	
	It suffices to show that q∈(F) and q is asymptotic to r. For the former, fix y∈ F and notice that since [y_n,x_n] is a geodesic segment orthogonal to F,  ∠_y_n(y,x_n) ≥π/2 for all n∈. Then, applying Proposition II.3.3(1) in <cit.>, see that the map (x,y,p)→∠_p(x,y) is upper semicontinuous. Therefore the limiting segments q_n are orthogonal to F so q∈(F). To see that q and r are asymptotic, notice that [y_n,x_n]⊂ B_M(r), and so q is M-close to r, thus q and r are asymptotic. 
	
	
	



	The projection maps π_F and π_Ω are surjective. 



	The lemma above shows that π_Ω is surjective. Using the P-action and the fact that A is the diameter of the fundamental domain, then π_F is surjective as well.


Part of showing the projection maps are proper requires an understanding of what happens to the boundary points when a sequence of rays in (F) converge. 



	If (r_n) is a sequence of rays orthogonal to F which converge to an element r∈(F), then r_n(∞) converges to r(∞) in the cone topology.



	Fix x_0∈ F a basepoint for the cone topology on Y. For each r_n, let c_n be the ray based at x_0 asymptotic to r_n. Since r_n→ r, then there is a constant D such that d(r_n(t),c_n(t))≤ D for all t. Applying Lemma <ref>, the rays c_n converge to a ray c which is asymptotic to r. 
	
	Fix ϵ>0 and let s>0. U(c,s,ϵ) is a basic neighborhood of c in Y. Since c_n→ c pointwise, then there is an N∈ such that d(c_m(s),c(s))<ϵ for all m>N and therefore c_n(∞)→ c(∞).



What remains is to show that the projection maps are both proper. Recall from Theorem <ref> there is a universal constant κ>0 which makes certain subsets of a (0) space with isolated flats κ-quasiconvex. Haulmark extends this result to include geodesic rays which are orthogonal to some flat F.

 
	Let F∈ and q∈(F), then there exists a constant κ such that q∪ F is κ-quasiconvex in X.



	The constants κ in Theorem <ref> and Lemma <ref> above are the same κ.



	Let F^* a flat of X, F a flat subset of F^* and q∈(F), then q∪ F is κ-quasiconvex in Y.



	In Y, (F)=(F^*) since F separates F^* from the rest of Y. Y inherits the metric from X, so q∪ F^* is quasiconvex in Y. Any geodesic in Y from points of F to points of q are κ-close to F^* or q. If they are κ-close to F^*, then they are κ-close to F because F separates F^* from Y. Lastly, since Y is convex, all geodesics from between points of q∪ F^* are in Y to begin with. Thus, q∪ F is κ-quasiconvex.


The two lemmas below relate how close together a ray q∈(F) and c can be, where c is the ray based at x_0 asymptotic to r.


	There is a constant M=M(κ)>0 such that for any ray q∈(F), d(q(0),c(t))< M where c is the ray asymptotic to q based at x_0 and t=d(x_0,q(0)).



	Let β [0,t]→ F be the geodesic from x_0 to q(0). Using the quasiconvexity result above, there is a κ>0 such that c is contained in the κ-neighborhood of q∪ F. Therefore, there are points s∈[0,∞), x∈ q, y∈ F such that d(c(s),x)≤κ and d(c(s),y)≤κ. Using the triangle inequality, d(x,y)≤ 2κ. Since x∈ q, q is orthogonal F and y∈ F, then d(x,q(0))≤ 2κ as well and so d(c(s),q(0))≤ 3κ. 
	
	The triangle (x_0,c(s),q(0)) has edges with side lengths t, s, and at most 3κ, so t∈ [s-3κ,s+3κ]. Thus |t-s|≤ 3κ and so 
    d(c(t),q(0)) ≤ d(c(t),c(s))+ d(c(s),q(0)) ≤ 3κ+3κ.
 Setting M=6κ is the desired constant.


The next lemma relates rays in (F) and their geodesic representatives, based on how close the former ray's starting point is to a ray in F.

 
	Fix ϵ>0 and let M=M(κ) from above. Then there is a constant δ=δ(ϵ,M) such that for any n>0 and η∈ F the following holds: if q∈(F) and q(0)∈ U(η, n, ϵ), then c(n)∈ U(η,n,δ), where c is the ray asymptotic to q.



	Let β  [0,t]→ F be the geodesic from x_0 to q(0). The lemma above shows that if t=n, then d(c(t),q(0))≤ M so d(c(t),η(n))≤ M+ϵ.
	
	If n<t, notice that d(c(t),η(n)) ≤ d(c(t),β(t))+ d(β(t),η(n)) ≤ M+ϵ. Consider the triangle (x_0,c(t),η(n)). This has side lengths t, n, and at most M+ϵ, so t-n≤ M+ϵ. With the triangle inequality we have:	
	
    d(c(n),η(n)) ≤ d(c(t),c(n)) + d(c(t), η(n)) ≤ M+ϵ + M + ϵ

	Lastly, if t<n, then d(β(t),η(n))≤ϵ and d(β(t),c(t)) ≤ M so d(η(n),c(t))≤ M+ϵ. Consider the triangle (x_0,η(n),β(t), which has sidelengths t, n, and at most ϵ. Therefore n-t< ϵ. Applying the triangle inequalty we get:	
	
    d(c(n),η(n)) ≤ d(c(t),c(n))+ d(c(t),η(n)) ≤ϵ + M+ϵ

	Therefore, letting δ = 2(M+ϵ) covers all three cases.


With these two lemmas, we can now prove that the projection maps are proper:


	The projection map π_Ω→Ω is a proper map.



	Let K⊂Ω be a compact set and consider 
	
    π_Ω(K) = { (x,q): q∈(F),  q(∞)∈ K,  d(x,q(0))≤ A}⊂

	Let C be the following set
	
    C := { x∈ F: ∃ q∈(F),  q(∞)∈ K,  d(q(0),x)≤ A}

	We claim C is compact and we will show this by showing that it is closed and bounded. 
	
	First, suppose it is unbounded. Then there is a sequence x_n tending to infinity and so there is a sequence of rays q_n with q_n(0) tending to a point η∈ F. Let c_n be the geodesic ray based at x_0 asymptotic to q_n. Fix ϵ>0 and consider U(η,n,ϵ). Such a neighborhood contains all but finitely many of q_i(0). By Lemma <ref>, U(η,n,δ) contains all but finitely many c_i and therefore c_i(∞)→η. But c_i(∞)∈ K and K is compact, so η∈ K, a contradiction. Therefore C is bounded.
	
	Next, let x_i→ x be a sequence of points in C. For each x_i, there is a a q_i∈(F) with d(q_i(0),x_i)≤ A. So for sufficiently large i, q_i(0)∈B_A(x). Applying Lemma <ref>, q_i converge to q∈(F) pointwise. Therefore q(0)∈B_A(x). Since q_i(∞)∈ K for all i and K is compact, then q(∞)∈ K and so q(0)∈ C and so is x.
	
	Now let (x_n,q_n) be a sequence in π_Ω(K). Since q_n(0)∈ C and C is compact, by Lemma <ref>, after possibly passing to a subsequence, q_n→ q pointwise. Similarly, x_n∈ C, so after passing to a subsequence, x_n converge as well and so (x_n,q_n) can be made to converge to (x,q).
	
	Lastly, to see that q∈(F), take any y∈ F. Let p_n=q_n(0) and x_n=q_n(1). Since each q_n is orthogonal to F, ∠_p_n(x_,y)≥π/2. The map (x,y,p)→∠_p(x,y) is upper semicontinuous (see <cit.> Proposition II.3.3(1))), so q∈(F). This concludes the proof.
	


	The projection map π_F→ F is a proper map.




	Let C⊂ F be compact. Then we have
	
    π_F (C) = { (x,q): q∈(F),  x∈ C,  d(q(0),x)≤ A}

	Let (x_,q_n) be a sequence in π_F (C). Since C is compact, so is B_A(C) and q_i(0)∈B_A(C) for each i. By Lemma <ref>, after possibly passing to a subsequence, q_n→ q pointwise. Furthermore, x_n→ x since x_n∈ C. Therefore, (x_n,q_n)→ (x,q).
	
		To see that q∈(F), take any y∈ F. Let p_n=q_n(0) and x_n=q_n(1). Since each q_n is orthogonal to F, ∠_p_n(x_,y)≥π/2. The map (x,y,p)→∠_p(x,y) is upper semicontinuous (see <cit.> Proposition II.3.3(1))), so q∈(F) and thus (x,q)∈π_F(C) making this set compact.

Combining everything we get the following theorem:


	Let P be the stabilizer of a flat F of X_v. Then P acts properly and cocompactly on Ω:=  Y ∖ F^*, where F^* is the maximal dimensional flat in Y which contains F.


More importantly, the work above gives the following association between compact sets of F and those of Ω:


	
	Given a compact set K⊂Ω, there exists a compact set C⊂ F ⊂ F^* associated to K  such that for each η∈ K, there is a ray c [0,∞) → Y such that c∈(F)=(F^*), c(0)∈ C and c(∞)=η. Furthermore, the set C can be chosen P-equivariantly in the sense that if C is associated to K, then pC is associated to pK.



	This follows immediately from the proofs of the projection maps being proper. For any K⊂Ω, take C to be π_F(π_Ω(K)), which is compact as seen above and this association is P-equivariant since  is.


This corollary is vital for relating neighborhoods of points in F to neighborhoods of points in Ω, which will ultimately allow us to show that Y is locally connected.



§ LOCAL CONNECTIVITY IN FLATS
 
The goal of this section to show that Y is locally connected at each of the remaining points. Once that is done, we can combine everything we have done to prove Theorem <ref>. The decomposition theory gives that for each ξ∈ Y, if ξ∉ F^* for any flat F^* of Y, then Y is locally connected at ξ. Additionally, if β∈ F^* and β∉ X_v, then Y is locally connected at β since F^* is a sphere. Therefore all that is left to show is that if α∈ X_v ∩ F^*=  F, then Y is locally connected at α.

Instead of showing local connectivity, we show that if α∈ F, then Y is weakly locally connected at α. The distinction between locally connected and weakly locally connected is toward the end of the section, Definition <ref>.

Many of the ideas in this section come from <cit.>. The authors work in the setting where G is a (0) group with isolated flats with a trivial maximal peripheral splitting and ultimately conclude that G is  locally connected at every point. We show here that the ideas work in our setting as well.

The approach is to use the association established in Corollary <ref> to relate neighborhoods in F to neighborhoods of Ω. Recall F is the subflat of F^* such that F ⊂ X_v. Before establishing this connection, we need to understand Ω.


	Ω=Ω∪ F.



	First, we will show that Ω∪ F is closed. We know that F^* and F are both closed in Y. Therefore F^* ∖ F is open. We can also see that 
	
    Y ∖ (Ω∪ F) =  F^* ∖ F

	Therefore Ω∪ F is closed since its complement in Y is open and so Ω⊂Ω∪ F.
	
	Now suppose α∈ F, then we want to show that α∈Ω. We will do this by creating a sequence of points in Ω which limit to α. Fix x_0∈ F and let c [0,∞)→ Y be the ray representing α and let C be a fundamental domain for the action of P on F. Using the P-action, let p_i· C cover the ray c. For each p_i, let q_i∈(F) such that q_i(0)∈ p_i· C. Let r_i be the geodesic ray based at x_0 asymptotic to q_i.
	
	Fix ϵ>0 and notice that since the translates of C cover c, then for all but finitely many i, q_i(0)∈ U(c,n,ϵ). Applying Lemma <ref>, we have  
    r_i(n)∈ U(c,n,ϵ+δ)
for all but finitely many i. This means that r_i(∞)∈ U(c,n,ϵ+δ) for all but finitely i and thus r_i(∞)→α as n→∞, as desired.		
	
	


Next we restate a result of Haulmark's to emphasize the connection between rays orthogonal to the flat F and their corresponding endpoints:

 
	Let κ be the CAT(0) with isolated flats constant as in Theorem <ref>. Suppose c'∈(F) and c is some ray contained in F. If c'(0)∈ U(c,r,D) for some constants r,D, then c'(∞)∈ U(c,r,D+κ). Conversely, if c'(∞)∈ U(c,r,D), then c'(0)∈ U(c,r,D+κ).



We make use of the following result of Bestvina:


	Let H be a group acting geometrically on a (0) space Z. Let C⊂ Z be a compact set. The H-translates of C form a null family in the compact space Z.




	If K⊂Ω is compect, then the collection of P-translates of K is a null family in Ω, where P is the stabilizer of the flat F⊂ X_v.


	
	Let K⊂Ω be compact and let C⊂ F be the compact set associated to K from Corollary <ref>. Fix a positive constant D>0 and let κ be the constant from Theorem <ref>. From the Remark <ref>, it suffices to show that for each r<∞, only finitely many P-translates of K are not contain in any set of the form U(·, r, D). 
	
	Using the Bestvina result above, P^*-translates of C form a null family in F^*. Since P⊂ P^* and F⊂F^*, then the P-translates of C form a null family in F. So only finitely many of the P-translates of C are not in set of the form U(·, r,D+κ). For any p∈ P, if pK lies in a set of the form U(·, r, D), then pC lies in a set of the form U(·, r, D+κ) by Lemma <ref>. Therefore the P-translates of K form a null family in Ω.




The association between compact sets of F and those of Ω established in Corollary <ref> can be improved even further: the compact sets chosen can be connected fundamental domains for the action of P. To see this is connected, first observe the following:

 
	Let (G,) be a CAT(0) group with isolated flats and peripheral structure . Let (G_v,_v) be a component vertex of G's maximal peripheral splitting with the peripheral structure coming from G. Let ρ∈∂(G_v, _v) be a parabolic point stabilized by P. Then there exists a connected, compact fundamental domain C for the action of P on ∂ (G_v, _v) ∖{ρ}. 



	
	Since (G_v,_v) is a component vertex of a 1-ended CAT(0) group with isolated flats, we know that ∂ (G_v, _v) is connected and locally connected and therefore path connected. Notice that π(F)=π(F^*)=ρ. We also know that since G_v is a component vertex of a maximal peripheral splitting, then G_v does not split further relative to _v. Therefore ρ is not a global cut point of ∂ (G_v, _v). These facts follow from Theorem <ref>.
	
	Since ρ is not a global cut point and ∂ (G_v, _v) is path connected, then ∂ (G_v, _v)∖{ρ} is path connected as well. We know that P acts cocompactly on ∂ (G_v, _v)∖{ρ}. Let C_0 be a compact fundamental domain for this action. Our goal is to show that C_0 is contained in a connected fundamental domain C. 
	
	Let d= d(C_0, ρ) in ∂ (G_v, _v). We can cover C_0 with finitely many open connected sets of diameter at most d/2. The union of the closures of these sets form a compact set C_1⊂∂ (G_v, _v) containing C_0 and having only finitely many connected components. By our choice of d, C_1 ⊂∂ (G_v, _v) ∖{ρ}. We can then join the finitely many connected components of C_1 with finitely many paths in ∂ (G_v, _v) ∖{ρ}, giving us a compact connected fundamental domain C.
	


Using this, we can find a connected fundamental domain for P's action on Ω.


	There is a compact, connected fundamental domain K⊂Ω for the action of P on Ω. Furthermore, if S is any finite generating set for P, then K can be chosen such that sK∩ K ≠∅ for each s∈ S. 



	We will use the decomposition map π Y → (G_v,_v) and then pull back the compact, connected fundamental domain for P's action on (G_v,_v) ∖{ρ} to get a fundamental domain on Ω.
	
	Let C_0 be a connected fundamental domain for the action of P on (G_v, _v) ∖{ρ} as in Proposition <ref>. Without loss of generality, we may increase the size of C_0 such that C_0∩ sC_0 ≠∅ for all s∈ S. We can then repeat the process from Proposition <ref> to construct a connected C.
	
	Since the map π Y → (G_v, _v) defines an upper semicontinuous monotone decomposition, then we know that π(C) is compact and connected. 	The map π is G-equivariant, and therefore since C is a fundamental domain, K is as well. Since C satisfies the desired intersection property, then K does as well.


Using the association between subsets of F and those of Ω, we can establish a way to determine when subsets of Ω are connected.

 
	There exists compact, connected fundamental domains K and C for the action of P on Ω and F, respectively, such that the following property holds: Let  be any subset of P. Then:
	
	

		If ⋃_p∈ pC is connected in F, then ⋃_p∈ pK is connected in Ω.
	




	Choose a compact, connected fundamental domain C for the action of P on F. Let S be the set of p∈ P such that pC∩ C≠∅. Then S is a finite generating set for P. Use Proposition <ref> to pick K such that sK∩ K ≠∅ for all s∈ S. This implies the following, which implies the desired conclusion:	
	

		If C∩ pC ≠∅, then K∩ pK ≠∅
	



We will now fix the compact, connected fundamental domains K and C for P's action on Ω and F, respectively. Using Corollary <ref>, let c' [0,∞)→ Y be a geodesic ray which meets F orthogonally. Using the P-action, we may assume that c'(0)∈ C and c'(∞)∈ K. We will call c'(0)=q_0 and use q_0 as the basepoint for the cone topology.

Let α∈ F. Our method is two-fold. First,  associate connected neighborhoods of α in F^* to neighborhoods of α in Ω, which by Proposition <ref> will be connected. Then, given a neighborhood of α in Ω, find a connected neighborhood of α in F^* to associate inside of the neighborhood in Ω. 

To make this possible, we need to find suitably nice neighborhoods in F^*. 



	Let Z be a (0) space and ξ∈ Z. A neighborhood N of ξ in Z is clean if the following conditions hold:
	
	
		
  * N is connected.
		
  * N=N∩ Z is connected, and
		
  * Every point of Λ:=N∩ Z is a limit point of N.
	


Hruska and Ruane prove  (<cit.>, Proposition 3.3) that for any ξ∈ Z, ξ has a local base of clean connected neighborhoods. We show below that we can pick this neighborhood basis to be clean and connected both in F and F^*. 


	There is a local base of neighborhoods of α in F^* which are clean and connected in F and in F^*. 



	Fix q_0 as the basepoint for the cone topology and let c [0,∞)→ F^* be the ray based at q_0 representing α. It is shown in <cit.> that basic open sets, U(c,r,D), are clean and connected in F^*.  
	
	First note that since c(0)∈ F, c(∞)∈ F and F is convex in F^*, then U(c,r,D)∩F is non-empty. Denote N_F= F∩ U(c,r,D). 
	
	A point p∈ F is in N_F if and only if the geodesic segment from c(0) to p intersects B_D(c(r)). Let γ [0,t]→ F be the geodesic segment [c(0),p], then we can choose s so that d(γ(s),c(r))<D. It follows that the entire geodesic segment [γ(s),γ(t)]⊂ U(c,r,D), so every point of N_F is contained in a connected subset of N_F which intersects the connected ball B_D(c(r)) and thus N_F is connected. All of U(c,r,D) is contained in the closure of N_F⊂F, and therefore is connected as well. Since U(c,r,D) is contained in the closure of N, the limit set requirement is satisfied as well.
	


For a given clean, connected neighborhood N=N∪Λ, let :={p∈ P: pC∩ N≠∅}. Since q_0∈ F,  is non-empty and by Proposition <ref>, ⋃_p∈ pC is connected. 


	Let α∈ F and N be a clean, connected neighborhood with N∩ F = N and N∩ F = Λ'. Let := {p∈ P: pC∩ N≠∅}. Then the neighborhood in Ω of α associated to N is Z= ( ⋃_p∈ pK) ∪Λ'. 
 

We now need to show three things: Z is a neighborhood of α, it is connected, and when N can be chosen to be `small,' then Z is `small' as well.

 
	The set Z associated to N is a neighborhood of α and is connected.



	First, we show that Z is a neighborhood of α. Let c [0,∞)→ F^* be the ray based at q_0 representing α and fix some D>0. Since N is a neighborhood of α, we can choose R large enough so that U(c,R,D+κ)∩F^* is contained in N, and therefore U(c,R,D)∩F^* is contained in N as well. The P-translates of K form a null family, so there is a neighborhood V of α with V⊂ U(c,R,D) such that if pK∩ V ≠∅, then pK⊂ U(c,R,D).
	
	Now it suffices to show that V⊂Z. For η∈ V, either η∈Ω or η∈ F. In the former case, η∈ pK for some p∈ P and so pK⊂ U(c,R,D). In particular, p· q_∞∈ U(c,R,D) and by Lemma <ref>, p· q_0∈ U(c,R,D+κ). Since q_0∈ C and U(c,R,D+κ)∩F^*⊂N, then pC∩ N ≠∅. Therefore, η∈Z by the construction.
	
	In the latter case, η∈ F∩ V, so
	
	
    η∈ U(c,R,D)∩ F ⊂ U(c,R,D+κ)∩ F ⊂N∩ F = Λ'⊂Z

	Therefore, V⊂Z and so it is a neighborhood of α in Ω.
	
	Now we must show that Z is connected. By Proposition <ref>, Z∩Ω is connected. Therefore it suffices to show that every point of Λ' is a limit point of Z∩Ω. Let ξ∈Λ'. Since N is clean, ξ is the limit of a sequence x_i∈ N. Each x_i∈ p_iC for some p_i∈ P. Fix ϵ>0 and note that since x_i→ξ, x_i∈ U(ξ,R,ϵ) for all but finitely many i. Let A be the diameter of C, then p_i· q_0∈ U(ξ,R,ϵ+A) for all but finitely many i, and so by Lemma <ref>, p_i· q(R)∈ U(ξ,R,ϵ+A+δ) for all but finitely many i. Therefore, p_i· q(∞)∈ U(ξ,R,ϵ+A+δ) and thus p_i· q(∞) converge to ξ. Since p_i· C ∩ N ≠∅, then p_i· K∩Z≠∅ and so ξ is the limit of points in Z∩Ω as desired.


 
	The neighborhood Z associated to N can be made arbitrarily small by choosing N to be a sufficiently small neighborhood of α in F^*



	Let U be a neighborhood of α in Ω. Since the P-translates of K form a null family. There is a neighborhood V⊂ U such that if pK∩ V is non-empty, then pK⊂ U. Since V is a neighborhood of α, there are constants R>0 and D>0 such that U(c,R,D+κ)⊂ V and U(c,R,D)∩ F⊂ V. Then, by Lemma <ref>, U(c,R,D) is a neighborhood of α in F^* such that if p· q_0∈ U(c,R,D), then p· q_∞∈ U(c,R,D+κ) and so pK⊂ U. Furthermore, since the P-translates of C form a null family, there is a W⊂ U(c,R,D) such that if pC∩ W is non-empty, then pC⊂ U(c,R,D). Finally, there is a clean, connected neighborhood N within W.
	
	It remains to show that the associated neighborhood Z is contained in U. Let η∈Z. By our choice of U(c,R,D), if η∈ F, then η∈ V and therefore in U. If η∈Ω, then η∈ pK for some p, so pK∩Z is non-empty and thus pC∩ N is non-empty. This means that pC∩ W≠∅ so pC⊂ U(c,R,D) and thus pK intersects V non-trivially. By the choice of V, pK⊂ U and so η∈ U as desired. Thus Z is contained in U.
	
	
	


Recall the definitions of weakly locally connected and locally connected:


	Let M be a topological space. M is weakly locally connected at m∈ M if for every open set V, there  is a connected (and not necessarily open) set N⊂ V with m in the interior of N.
	
	We say a space M is weakly locally connected if it is weakly locally connected at every point.




	Let M be a topological space. M is locally connected at m∈ M if for every open set V containing m, there is a connected open set U with m∈ U ⊂ V. 
	
	We say M is locally connected if it is locally connected at every point.


In Lemma <ref> and Lemma <ref>, we do not know if Z is open or not, so we cannot prove local connectivity in Y. Instead, we show the following:

	If α∈ Y ∩ F, then Y is weakly locally connected at α.



	Let U be a neighborhood of α in Y. Since Y = Ω∪ F^*, then U=U_1∪ U_2, with U_1 a neighborhood of α in Ω and U_2 a neighborhood of α in F^*. Combining Lemma <ref> and Lemma <ref>, there is a V_1⊂ U_1 which is connected and contains α. Since F^* is a sphere, it is locally connected, so there is some V_2⊂ U_2 which is connected and contains α. Lastly, since V_1∩ V_2 is non-empty, then their union is connected as well. Thus V_1∪ V_2⊂ U is a connected neighborhood of α.


Combining all of this work leads to the following theorem:


	Let (G,) be a (0) group with isolated flats acting geometrically on X, let G_v be a component vertex in its maximal peripheral splitting, acting geometrically on X_v⊂ X and let Y be as in Definition <ref>. Then Y is path connected. 



	
	It suffices to show that Y is both connected and weakly locally connected at every point. By Theorem 27.16 in <cit.>, a space which is weakly locally connected at every point is locally connected at every point. Also by Theorem 31.2 in <cit.>, a space which is compact, connected, locally connected and metrizable is locally path connected, which implies globally path connected (by <cit.>, Theorem 27.5). Since boundaries of proper metric spaces are compact and metrizable, all we need to show is that Y is connected and weakly locally connected at each point. 
	
	Connectedness follows from Theorem <ref>. Since the map π Y → (G_v,_v) is monotone and (G_v,_v) is connected, then Y is connected as well.
	
	For weakly locally connected, there are three types of points in Y to consider: points in X_v and not a flat, points just in a flat, and points in both X_v and F for some flat F. By Corollary <ref>, if x∈ Y and x∉ F for any flat F, then Y is locally connected at x and therefore weakly locally connected at x. If ξ∈ F^* for some flat F^* and ξ∉ X_v, then Y is locally connected at ξ and so weakly locally connected at ξ. This is because sufficiently small neighborhoods of ξ only contain points in F^*, which is locally connected. Lastly, if α∈ X_v ∩ F for some flat F, then by Proposition <ref>, Y is weakly locally connected at α.


We are almost ready to combine all this work to conclude that (0) groups with isolated flats have path connected visual boundaries. But first we need to show that these groups satisfy the lonely condition.

 
	Suppose G is a 1-ended (0) group with isolated flats acting geometrically on a (0) space X. Let  be the block decomposition coming from the maximal peripheral splitting of , then all irrational rays are lonely.




	For any flat F, let γ_F [0,b]→ X be the geodesic segment from x_0 to the closest point of F. Since F is convex, a closest point exists and is unique. By Theorem <ref>, γ_F∪ F is κ-quaisconvex. In particular, every geodesic starting at x_0 which intersects F must be the in κ-neighborhood of γ_F(b). 
	
	Let c,c' be rays in X based at x_0 with the same, infinite itinerary. Since these rays have the same itinerary, they pass through the same sequence of flats, and must pass through infinitely many flats. But then the rays are 2κ-close infinitely often. By the convexity of the distance metric, the rays must be 2κ-close always, and therefore c and c' are asymptotic.
	



  
	All 1-ended (0) groups with isolated flats have path connected visual boundaries.



	Let (G,) be a (0) group with isolated flats. It suffices to show there is a space X on which G acts which has path connected visual boundary. Let  be the maximal peripheral splitting for (G,), X be the tree-of-spaces on which G acts geometrically, and  the block decomposition coming from this peripheral splitting.
	
	Each peripheral vertex group in the splitting is virtually ^n for some n, so has path connected visual boundary. If each component vertex has path connected visual boundary as well, then by Theorem <ref> X is path connected. By Lemma <ref>, the irrational rays are all lonely, so we can apply the theorem.
	
	If there is a peripheral vertex which does not have path connected visual boundary, then by Theorem <ref>, ∂_R X is path connected. This is true because
	
    ∂_R X = ⋃_v∈ Y_v

	where  is the collection of component vertices in the tree  for the splitting . Therefore, we can apply Proposition <ref> to conclude that X is path connected.


















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