aa r X i v : . [ m a t h . M G ] S e p Coarse compactifications of proper metric spaces
Elisa Hartmann ∗ September 18, 2020
Abstract
This paper studies coarse compactifications and their boundary.We introduce two alternative descriptions to Roe’s original definition of coarse compacti-fication. One approach uses bounded functions on X that can be extended to the boundary.They satisfy the Higson property exactly when the compactification is coarse. The otherapproach defines a relation on subsets of X which tells when two subsets closure meet on theboundary. A set of axioms characterizes when this relation defines a coarse compactification.Such a relation is called large-scale proximity.Based on this foundational work we study examples for coarse compactifications Hig-son compactification, Freudenthal compactification and Gromov compactification. For eachexample we characterize the bounded functions which can be extended to the coarse com-pactification and the corresponding large-scale proximity relation.We provide an alternative proof for the property that the Higson compactification isuniversal among coarse compactifications. Furthermore the Freudenthal compactificationis universal among coarse compactifications with totally disconnected boundary. If X ishyperbolic geodesic proper then there is a closed embedding ν ( R + ) × ∂X → ν ( X ). Its imageis a retract of ν ( X ) if X is a tree. Contents ∗ Department of Mathematics, Karlsruhe Institute of Technology INTRODUCTION Elisa Hartmann
This paper studies a class of compactifications of proper metric spaces which contains the Higsoncompactification, Gromov compactification and a coarse version of the Freudenthal compactifica-tion. With every such space X one can associate a coarse structure . In [Roe03] Roe introduceda class of compactifications which are compatible with this structure. We provide an equivalentdefinition: Definition 1.
Let X be a proper metric space and ¯ X a compactification of X . Then ¯ X is a coarse compactification if for every two nets ( x i ) i , ( y i ) i ⊆ X , such that ( x i , y i ) i is an entouragein X , both nets have the same limit points on the boundary.Instances of coarse compactifications have been studied by many authors, see e.g. [MY15,Kee94, Pro19, FS03, Pro11, PS15, Pro05, KB02, KH16, KH15, Cor19]. In this paper we give twonew descriptions of coarse compactifications which are equivalent to the original one. One ofthe two equivalent definition starts with a relation on subsets of X , the other definition usesbounded functions on X which can be extended to the compactification.Every coarse compactification ¯ X of a proper metric space X gives rise to a relation r ¯ X onsubsets of X which tells when the closure of two sets meet on the boundary. Specifically denoteby ∂X the boundary ¯ X \ X . If A, B ⊆ X are subsets then Ar ¯ X B ⇔ ( ¯ A ∩ ∂X ) ∩ ( ¯ B ∩ ∂X ) = ∅ . A set of axioms tells if a relation on subsets of X comes from a coarse compactification. Such arelation is then called large-scale proximity in the sense of the following definition. Definition 2. If X is a proper metric space a relation r on subsets of X is called large-scaleproximity if1. if B ⊆ X then B ¯ rB if and only if B is bounded;2. ArB implies
BrA for every
A, B ⊆ X ;3. if A, A ′ ⊆ X are subsets and E ⊆ X is an entourage with E [ A ] ⊇ A ′ , E [ A ′ ] ⊇ A then ArB implies A ′ rB for every B ⊆ X ;4. if A, B, C ⊆ X then ( A ∪ B ) rC if and only if ( ArC or BrC );5. if
A, B ⊆ X with A ¯ rB then there exist C, D ⊆ X with C ∪ D = X and C ¯ rA, D ¯ rB .Conversely given a large-scale proximity relation r we can construct a coarse compactification¯ X r which induces this relation on subsets of X . This is done in Definitions 23,29.An entirely different approach characterizes coarse compactifications via the C ∗ -algebra C r ( X ) of bounded continuous functions on X that can be extended to the boundary of thecompactification. Every set of bounded continuous functions A on X generates the smallestcompactification ¯ X A such that functions in A can be extended to the boundary. More specifi-cally we introduce a property on bounded continuous functions: See Definition 9 which defines a coarse structure given a metric space INTRODUCTION Elisa Hartmann
Definition 3.
A bounded continuous function ϕ : X → R is called Higson if for every entourage E ⊆ X the map dϕ | E : E → R ( x, y ) ϕ ( x ) − ϕ ( y )vanishes at infinity.Given a compactification ¯ X denote by C ¯ X ( X ) the algebra of bounded continuous functionson X that can be extended to ¯ X . They must be Higson if ¯ X is coarse. Conversely if the algebraof bounded functions on X that can be extended to the boundary of the compactification consistsof Higson functions then ¯ X is coarse.We summarize those results in the following theorem: Theorem A. If X is a proper metric space and ¯ X a compactification of X the following state-ments are equivalent: • The compactification ¯ X is coarse. • The relation r ¯ X on subsets of X is a large-scale proximity relation. • Every function in C ¯ X ( X ) is Higson.Moreover given a large-scale proximity relation r the compactification ¯ X r is coarse. If all func-tions in an algebra of bounded functions A on X are Higson they generate a coarse compactifi-cation ¯ X A . In Theorem 17 we translate Roe’s original definition of coarse compactification to a definitionwhich is more suitable for us. The equivalence of statements 1,2 in Theorem A is shown inTheorem 36. The equivalence of statements 1,3 in Theorem A is shown in Theorem 37.We investigate three specific examples: The Higson compactification, the Freudenthal com-pactification and the Gromov compactification.
Example 4.
The Higson compactification hX = ν ( X ) ∪ X of a proper metric space X ischaracterized by the following large-scale proximity relation: Two subsets A, B ⊆ X are called close , written A f B , if there exists an unbounded sequence ( a i , b i ) i ⊆ A × B and some R ≥ d ( a i , b i ) ≤ R for every i .Every Higson function on X can be extended to the Higson corona ν ( X ). Example 5.
The Freudenthal compactification εX = Ω X ∪ X of a proper metric space X ischaracterized by the following large-scale proximity relation f f : Two subsets A, B ⊆ X don’thave a same end, written A f f B , if there exist A ′ ⊇ A, B ′ ⊇ B with A ′ ∪ B ′ = X and A ′ f B ′ .Let x ∈ X be a basepoint. A bounded continuous map ϕ : X → R is called Freudenthal iffor every R ≥ K ≥ d ( x, y ) ≤ R, d ( x , x ) ≥ K, d ( x , y ) ≥ K implies ϕ ( x ) = ϕ ( y ). We write C f ( X ) for the ring of Freudenthal functions on X . Every boundedfunction that can be extended to the Freudenthal compactification is Freudenthal and everyFreudenthal function can be extended to the boundary of the Freudenthal compactification. Example 6.
If a metric space X is hyperbolic, proper the Gromov compactification ¯ X = ∂X ∪ X is defined. The associated large-scale proximity relation f g is defined by A f g B if there aresequences ( a i ) i ⊆ A, ( b i ) i ⊆ B such thatlim inf i,j →∞ ( a i | b j ) = ∞ . INTRODUCTION Elisa Hartmann If X is hyperbolic a continuous function ϕ : X → R is called Gromov if for every ε >
K > x | y ) > K → | ϕ ( x ) − ϕ ( y ) | < ε. Every Gromov function can be extended to the Gromov boundary and every function that canbe extended to the Gromov boundary is Gromov.
Remark . We establish functoriality in the following way. In Proposition 38 we show theassociation of a coarse compactification is in a way contravariant on coarse maps. We can alwayspull back a coarse compactification along a coarse map. The reverse direction push out is notalways possible. We can glue coarse compactifications along a coarse cover though which is donein Proposition 40. Then Lemma 41 shows the poset of coarse compactifications is a sheaf on theGrothendieck topology of coarse covers on X .We now describe the results on the specific examples in detail. In particular the boundaryof the Higson compactification retains information about the coarse structure since the Higsoncorona is a faithful functor [Har19b]. This way not much information is lost if we restrict ourattention to the boundary of a coarse compactification when studying coarse metric spaces. TheHigson corona ν ( X ) of X is connected if X is one-ended. Aside from that and from beingcompact and Hausdorff the Higson corona does not have many nice property. The topology ofthe Higson corona does not have a countable base and is in fact is never metrizable [Roe03]. Weprovide a new proof that the Higson corona is universal among coarse compactifications. Theoriginal result can be found in [Roe03]. Theorem B. ( Roe ) If X is a proper metric space the Higson compactification of X is universalamong coarse compactifications of X . This means a compactification of X is a coarse compact-ification if and only if it is a quotient of the Higson compactification, where the quotient maprestricts to the identity on X . This in particular implies that the boundary of every coarse compactification of X is connectedif X is one-ended.The space of ends of a topological space is well known and dates back to Freudenthal’sworks [Fre31, Hop44, Fre45]. We construct a version of Freudenthal compactification on coarseproper metric spaces given both descriptions via a large-scale proximity relation and via boundedfunctions. The Proposition 49 shows both the topological and the coarse version of Freudenthalcompactifiaction agree on proper geodesic metric spaces. The space of ends gives informationabout the number of ends of a coarse metric space [Har17]. It is both metrizable and totallydisconnected.For the space of ends we can obtain a similar result as in the case of the Higson coronaregarding universality. Theorem C. If X is a proper metric space the boundary of the Freudenthal compactification Ω X = εX \ X of X is totally disconnected. If ( ¯ X, X ) is another coarse compactification whoseboundary is totally disconnected then it factors through εX . This means there is a surjective map εX → ¯ X which is continuous on the boundary and the identity on X . The topology of the Gromov compactification is metrizable [KB02]. The usual description ofits boundary is via geodesic rays or sequences that converge to infinity. We investigate in whichway the Higson corona can be recovered from the Gromov boundary.
Theorem D.
Let X be a hyperbolic geodesic proper metric space. Then there is a closed embed-ding Φ : ν ( Z + ) × ∂X → ν ( X ) . The image of Φ is a retract if X is a tree. NOTIONS IN COARSE GEOMETRY Elisa Hartmann
All three instances of coarse compactification are functorial in that the boundary is a coarseinvariant. The Higson corona and space of ends are even functors on coarse maps. This way weassociate compact Hausdorff spaces to coarse proper metric spaces which serve in the classificationof proper metric spaces according to their coarse geometry. This gives access to topologicalmethods that can be used in the coarse setting.
We consider metric spaces as coarse objects. The book [Roe03] introduces a more general notionof coarse spaces, which describes coarse structure in an abstract way. Since we only considerproper metric spaces as examples we do not need to do this here.
Definition 8.
A metric space X is proper if the closure ¯ B in X of every bounded subset B ⊆ X is compact. Definition 9.
Let (
X, d ) be a metric space. Then the coarse structure associated to d on X consists of those subsets E ⊆ X × X for whichsup ( x,y ) ∈ E d ( x, y ) < ∞ . We call an element of the coarse structure entourage . In what follows we assume the metric d tobe finite for every ( x, y ) ∈ X × X . Definition 10.
A map f : X → Y between metric spaces is called • coarsely uniform if E ⊆ X being an entourage implies that f × ( E ) is an entourage ; • coarsely proper if and if A ⊆ Y is bounded then f − ( A ) is bounded. • coarse if it is both coarsely uniform and coarsely properTwo maps f, g : X → Y between metric spaces are called close if f × g (∆ X )is an entourage in Y . Here ∆ X denotes the diagonal in X × X .If S ⊆ X × X, T ⊆ X are subsets of a set we write S [ T ] := { x : ∃ y ∈ T, ( x, y ) ∈ S } and T c = { x ∈ X : x T } . Notation 11.
A map f : X → Y between metric spaces is called • coarsely surjective if there is an entourage E ⊆ Y × Y such that E [im f ] = Y ; • coarsely injective if for every entourage F ⊆ Y the set ( f × ) − ( F ) is an entourage in X .Two subsets A, B ⊆ X are called not coarsely disjoint if there is an entourage E ⊆ X such thatthe set E [ A ] ∩ E [ B ]is not bounded. We write A f B in this case. 5 THE ORIGINAL DEFINITION Elisa HartmannRemark . We study metric spaces up to coarse equivalence. For a coarse map f : X → Y between metric spaces the following statements are equivalent: • There is a coarse map g : Y → X such that f ◦ g is close to id Y and g ◦ f is close to id X . • The map f is both coarsely injective and coarsely surjective.We call f a coarse equivalence if one of the equivalent statements hold. Notation 13. If X is a metric space and U , . . . , U n ⊆ X are subsets then ( U i ) i are said to coarsely cover X if for every entourage E ⊆ X × X the set E [ U c ] ∩ · · · ∩ E [ U cn ]is bounded. In this chapter we introduce the class of compactifications which are coarse.
Definition 14. A compactification of a proper metric space (or more generally a locally compactHausdorff topological space) is an open embedding i : X → ¯ X such that i ( X ) is dense in ¯ X . Weidentify X with the dense open set i ( X ) ⊂ ¯ X .Now we define when a compactification is coarse. The original definition was given in [Roe03,Theorem 2.27, Definition 2.28, Definition 2.38]. We reproduce a slight modification of it. Definition 15.
Let X be a proper metric space. If ¯ X is a compactification of X with boundary ∂X then the sets E ⊆ X × X with¯ E ∩ ( ∂X × ¯ X ∪ ¯ X × ∂X ) ⊆ ∆ ∂X define the topological coarse structure associated to ¯ X .A coarse compactification of X is a compactification whose topological coarse structure isfiner than the originally given coarse structure on X .Note in [FOY18, Definition 1.1] a coarse compactification of a proper metric space has beendefined as a metrizable compactification ¯ X of X equipped with a continuous map f : hX → ¯ X which is the identity on X . This definition is different from our definition. Definition 16.
Let X be a metric space. Two subsets A, B ⊆ X are called close if there existsan unbounded sequence ( a i , b i ) i ⊆ A × B and some R ≥ d ( a i , b i ) ≤ R for every i . Wewrite A f B in this case. By [Har19a, Lemma 9, Proposition 10] the relation f is a large-scaleproximity relation. Theorem 17.
Let X be a proper metric space and ¯ X be a compactification of X . Then ¯ X iscoarse if and only if for every two subsets A, B ⊆ X the relation A f B implies ¯ A ∩ ¯ B = ∅ .Proof. Suppose ¯ X is coarse. Let A, B ⊆ X be two subsets with A f B . Then there existunbounded subsequences ( a i ) i ⊆ A, ( b i ) i ⊆ B such that ( a i , b i ) i is an entourage. Then ( a i , b i ) i ∩ ( ∂X × ¯ X ∪ ¯ X × ∂X ) ⊆ ∆ ∂X . Thus if p ∈ ∂X is a limit point of ( a i ) i then it is also a limit point of( b i ) i . Note ( a i ) i ∩ ( ∂X ) = ∅ since ( a i ) i is unbounded. This way we have shown ( a i ) i ∩ ( b i ) i = ∅ .This implies ¯ A ∩ ¯ B = ∅ . 6 LARGE-SCALE PROXIMITY RELATIONS Elisa Hartmann
Now suppose for every two subsets
A, B ⊆ X the relation A f B implies ¯ A ∩ ¯ B = ∅ . Let E ⊆ X × X be an entourage and ( x i , y i ) i ⊆ E be a net such that ( x i ) i → p ∈ ∂X and( y i ) i → q ∈ ¯ X . If q ∈ X then there is an infinite subnet of ( y i ) i contained in a ball around q .Then an infinite subnet of ( x i ) i is contained in a (larger) ball around q , thus would have a limitpoint in this ball. This way we can conclude q ∈ ∂X .If ( x i ) i ∩ ( y i ) i is bounded then remove those finitely many elements in the intersection andobtain ( x i k ) k , ( y i k ) k subnets with the same limit points. Now ( x i k ) k f ( y i k ) k which implies( x i k ) k ∩ ( y i k ) k = ∅ . This implies p = q .If ( x i ) i ∩ ( y i ) i is not bounded then the subnet in the intersection converges to both p and q .Thus p = q .This way we have shown ¯ E ∩ ( ∂X × ¯ X ∪ ¯ X × ∂X ) ⊆ ∆ ∂X . Thus ¯ X is coarse. In this chapter we study a relation on subsets of a proper metric space X which induce thetopology of a compactification ¯ X . The relation is large-scale proximity as defined below exactlywhen the compactification ¯ X is coarse. Given a large-scale proximity relation r on X we aregoing to present two constructions of spaces ∂ r X, ∂ ′ r X which happen to be boundaries of a coarsecompactification ¯ X r which induces the relation r on subsets of X . Definition 18.
A relation r on subsets of a metric space is called large-scale proximity if1. if B ⊆ X then B ¯ rB if and only if B is bounded;2. ArB implies
BrA for every
A, B ⊆ X ;3. if A, A ′ ⊆ X are subsets and E ⊆ X is an entourage with E [ A ] ⊇ A ′ , E [ A ′ ] ⊇ A then ArB implies A ′ rB for every B ⊆ X ;4. if A, B, C ⊆ X then ( A ∪ B ) rC if and only if ( ArC or BrC );5. if
A, B ⊆ X with A ¯ rB then there exist C, D ⊆ X with C ∪ D = X and C ¯ rA, D ¯ rB . Remark . Note a large-scale proximity relation on a metric space is an instance of a coarseproximity relation as defined in [GS19, Definition 2.2]. Compare this notion with the notion ofproximity relation [NW70], [Wil70]. The axiom 3 of Definition 18 is the characteristic for ourapplication on coarse metric spaces.
Lemma 20.
Let X be a metric space. Every large-scale proximity relation on X is finer thanthe relation close f .Proof. Let r be a large-scale proximity relation on a metric space X . If A f B then there existunbounded ( a i , b i ) i ⊆ X and an entourage E ⊆ X such that E [( a i ) i ] ⊇ ( b i ) i and E [( b i ) i ] ⊇ ( a i ) i . This implies ( a i ) i r ( b i ) i by axiom 3 of Definition 18. By axiom 4 of Definition 18 therelation ArB holds.Now we construct a topological space ∂ ′ r X given a large-scale proximity relation r . This willturn out to be the boundary of a coarse compactification. The topology on this construction iseasier to describe than in the other equivalent definition which will follow below. Definition 21.
Let r be a large-scale proximity relation on a metric space X . A system F ofsubsets of X is called an r -ultrafilter if 7 LARGE-SCALE PROXIMITY RELATIONS Elisa Hartmann A, B ∈ F implies
ArB ;2. if
A, B ⊆ X are subsets with A ∪ B ∈ F then A ∈ F or B ∈ F ;3. X ∈ F .Denote by δ r X the set of r -ultrafilters. If A ⊆ X is a subset then define cl ( A ) := {F ∈ δ r X : A ∈ F} . Lemma 22. If X is a metric space the ( cl ( A ) c ) A ⊆ X constitute a base for a topology on δ r X .Proof. First we show the base elements cover δ r X : Since ∅ is bounded ∅ ¯ rX . This implies ∅ 6∈ F for every r -ultrafilter F . Thus δ r X = cl ( ∅ ) c .Now we show for every element in the intersection of two base elements there is a base elementwhich contains the element and is contained in the intersection: Let A, B ⊆ X be two subsets.Let F ∈ cl ( A ) c ∩ cl ( B ) c be an element. Then A
6∈ F , B
6∈ F thus ( A ∪ B )
6∈ F . This implies
F ∈ cl ( A ∪ B ) c ⊆ cl ( A ) c ∩ cl ( B ) c . Definition 23.
Define the topology on δ r X to be the topology generated by ( cl ( A ) c ) A ⊆ X .Now define a relation λ r on δ r X : F λ r G if A ∈ F , B ∈ G implies ArB . The quotient by thisequivalence relation ∂ ′ r X = δ r X/λ r is called r -boundary 1 . Lemma 24.
The space ∂ ′ r ( X ) is a compact Hausdorff topological space.Proof. The proof of [Har19c, Theorem 26] with f replaced by r implies that ∂ ′ r ( X ) is compact.Now we show ∂ ′ r ( X ) is Hausdorff. Let F , G be two r -ultrafilters with F ¯ λ r G . Thus there exist A ∈ F , B ∈ G with A ¯ rB . Then there exist C, D ⊆ X with C ∪ D = X and C ¯ rA, D ¯ rB . Then G ∈ cl ( D ) c , F ∈ cl ( C ) c . Also: cl ( C ) c ∩ cl ( D ) c = ( cl ( C ) ∪ cl ( D )) c = cl ( X ) c = ∅ . This completes our discussion of ∂ ′ r ( X ). We now define another topological space ∂ r X given alarge-scale proximity relation. This space is homeomorphic to ∂ ′ r X . Compared with the previousmodel the points on ∂ r X are easier to describe.Let R ≥ X is called R -discrete if d ( x, y ) ≥ R for every x = y . If X is a metric space an R -discrete for some R > S ⊆ X is called a Deloneset if the inclusion S → X is coarsely surjective. Every metric space contains a Delone set. Definition 25.
Let r be a large-scale proximity relation on a proper metric space X and S ⊆ X a Delone subset. Denote by ˆ S the set of nonprincipal ultrafilters on S . If A ⊆ S is a subsetdefine cl ( A ) := {F ∈ ˆ S : A ∈ F} . Then define a relation r on subsets of ˆ S : π rπ if for every A, B ⊆ S the relations π ⊆ cl ( A ) , π ⊆ cl ( B ) imply ArB . Lemma 26.
The relation r on ˆ S is a proximity relation.Proof. The proof of [Har19c, Theorem 23] with r in place of f applies.8 LARGE-SCALE PROXIMITY RELATIONS Elisa Hartmann
As before we define a topology on ˆ S : Definition 27.
The relation r on subsets of ˆ S determines a Kuratowski closure operator¯ π = {F ∈ ˆ S : {F} rπ } . Now define a relation λ r on ˆ S : F λ r G if A ∈ F , B ∈ G implies ArB . Lemma 28.
The relation λ r is an equivalence relation on ˆ S .Proof. The relation is obviously symmetric and reflexive. We show transitivity. Let F , F , F be nonprincipal ultrafilters on X such that F λ r F and F λ r F . We show F λ r F . Assume theopposite. There are A ∈ F , B ∈ F with A ¯ rB . Then there exist C, D ⊆ X with C ∪ D = X and C ¯ rA, D ¯ rB . Now C ∈ F or D ∈ F . If C ∈ F this contradicts F λ r F and if D ∈ F thiscontradicts F λ r F . Definition 29.
Now the r -boundary 2 is defined ∂ r ( X ) = ˆ S/λ r as the quotient by λ r .We check this definition does not depend upon the choice of Delone set S : Lemma 30. If T ⊆ X is another Delone subset then ˆ S/λ r = ˆ T /λ r are homeomorphic.Proof. Suppose
S, T ⊆ X are two Delone sets. Without loss of generality assume S ⊆ T is asubset. Then there exists a map ϕ : T → S with H := { ( t, ϕ ( t )) : t ∈ T } an entourage and ϕ ◦ i = id S where i : S → T is the inclusion. There is an induced map ϕ ∗ : ˆ T → ˆ S F 7→ { A ⊆ S : ϕ − ( A ) ∈ F} . We show ϕ ∗ is continuous and respects λ r . If π , π ⊆ ˆ T are subsets with π rπ and A, B ⊆ S are subsets with ϕ ∗ π ⊆ cl ( A ) , ϕ ∗ π ⊆ cl ( B ) then π ⊆ cl ( ϕ − ( A )) , π ⊆ cl ( ϕ − ( B )). Thus ϕ − ( A ) rϕ − ( B ). Since H [ ϕ − ( A )] = A, H − [ A ] = ϕ − ( A ) and H [ ϕ − ( B )] = B, H − [ B ] = ϕ − ( B ) this implies ArB . Thus ϕ ∗ π rϕ ∗ π .Let F , G ∈ ˆ T be elements with F λ r G . Let A ∈ ϕ ∗ F , B ∈ ϕ ∗ G be elements. Then ϕ − ( A ) ∈F , ϕ − ( B ) ∈ G . Thus ϕ − ( A ) rϕ − ( B ). This implies ArB by the above.Now the inclusion i induces a map i ∗ ˆ S → ˆ T in a similar way. We show just as for ϕ ∗ that i ∗ is continuous and respects λ r . We have ϕ ∗ ◦ i ∗ ( F ) = F for every F ∈ ˆ S and i ∗ ◦ ϕ ∗ ( G ) λ r G forevery G ∈ ˆ T since A, B ∈ G implies
ArB which implies ϕ − ( A ∩ S ) rB .Comparing the first and the second model, we can prove: Proposition 31. If X is a proper metric space the map Φ : ∂ r X → ∂ ′ r X [ σ ] [ { A ⊆ X : ArB ∀ B ∈ σ } ] is a homeomorphism.Proof. Let S ⊆ X be a Delone subset. If σ is an ultrafilter on S then the collection { A ⊆ X : ArB ∀ B ∈ σ } is an r -ultrafilter on X by a proof similar to that of [Har19c, Theorem 17].If σ, τ are two ultrafilters on S with σλ r τ then σ, τ are in particular r -ultrafilters on S and σλ r (Φ( σ )) | S , τ λ r (Φ( τ )) | S . By transitivity of λ r we obtain (Φ( σ )) | S λ r (Φ( τ )) | S . This impliesΦ( σ ) λ r Φ( τ ) Thus Φ is well defined. 9 LARGE-SCALE PROXIMITY RELATIONS Elisa Hartmann
Now we show Φ is injective: Let σ, τ be nonprincipal ultrafilters on S with Φ( τ ) λ r Φ( σ ). Then σλ r τ since τ ⊆ Φ( τ ) , σ ⊆ Φ( σ ).Now we show Φ is surjective: Let F be an r -ultrafilter on X . Without loss of generalityassume S ∈ F . Then by [NW70, Lemma 5.3] there exists an ultrafilter σ on X such that S ∈ σ and σ ⊆ F . Then σ | S is mapped by Φ to the class of F .If A ⊆ S is a subset we show cl ( A ) := { [ σ ] : A ∈ σ } is a closed subset of ∂ r X . Let σ ∈ ˆ S be an element with σr cl ( A ). Then for every B ∈ σ we obtain BrA . Thus A ∈ Φ( σ ).By [NW70, Lemma 5.7] there exists an ultrafilter τ on S with τ ⊆ Φ( σ ) and A ∈ τ . This implies τ λ r σ . Thus [ σ ] ∈ cl ( A ). In fact the ( cl ( A ) c ) A ⊆ S constitute a base for the topology on ∂ r ( X ).If A ⊆ S is a subset then Φ( cl ( A )) = cl ( A ). Thus Φ is a closed map.If A ⊆ X then there exists an entourage E ⊆ X such that E [ A ] ⊆ S and A, E [ A ] are finiteHausdorff distance apart. Then Φ − ( cl ( A )) = cl ( E [ A ]). Thus Φ is continuous. Proposition 32. If r, s are two large-scale proximity relations on a proper metric space X and s is finer than r then there is a quotient map ∂ r ( X ) → ∂ s ( X ) . Proof.
Let S ⊆ X be a Delone subset.If F , G are nonprincipal ultrafilters on S then F λ r G implies ( A ∈ F , B ∈ G implies ArB ).Thus
AsB for every A ∈ F , B ∈ G which implies F λ s G .Now we show id ˆ S : ( ˆ S, r ) → ( ˆ S, s ) is continuous. If π , π ⊆ ˆ S are subsets then π rπ implies( A, B ⊆ S with π ⊆ cl ( A ) , π ⊆ cl ( B ) implies ArB ). Then
AsB if π ⊆ cl ( A ) , π ⊆ cl ( B ). Thus π sπ .Since id ˆ S is surjective the induced map on quotients is surjective.Since for every subset A ⊆ S the map id ˆ S maps cl ( A ) to cl ( A ) the induced map on quotientsis closed.Now we produce the compactification of a proper metric space X with the boundary ∂ r X given a proximity relation r . Define ¯ X r = X ⊔ ∂ r X as a set. Closed sets on ¯ X r are generatedby ( ¯ A ∪ cl ( A )) A ⊆ X , where the closure ¯ A of A is taken in X . Proposition 33. If X is a proper metric space and r a large-scale proximity relation on X then ¯ X r is a compactification of X with boundary ∂ r X .Proof. . This topology is compact by the first part of the proof of [Har19b, Theorem 20] with f replaced by r . The spaces ∂ r X, X appear as subspaces of ¯ X r . The inclusion X → ¯ X r is densesince X ∪ cl ( X ) = ¯ X r . Remark . The statement in Proposition 32 can be strengthened: If X is a proper metric spaceand r, s are close relations on X with s finer than r then there is a quotient map ¯ X r → ¯ X s whichis the unique continuous map extending the identity on X . Proof.
Assume without loss of generality that X is R -discrete for some R >
0. A net in X converging to a point p ∈ ¯ X can be written as a filter F on X . Then an ultrafilter σ finer than F converges to the same point.Let α : ¯ X r → ¯ X s be a continuous map extending the identity on X . Then α ∗ σ = σ . Thus σ converges to α ( p ). Now p is represented by σ in ∂ r X and α ( p ) is also represented by σ in ∂ s X . Thus α maps a point represented by σ to a point represented by σ and is thus uniquelydetermined. This implies α | ∂ r X is the quotient map of Theorem 32.10 BOUNDED FUNCTIONS Elisa HartmannRemark . Remark 34 and Lemma 20 imply that the Higson compactification is universal amongcoarse compactifications. This recovers [Roe03, Proposition 2.39].If ¯ X is a coarse compactification of a proper metric space X , define a relation r ¯ X on subsetsof X as follows: For a subset A ⊆ X define cl ( A ) = ¯ A ∩ ∂X . If A, B ⊆ X are subsets then Ar ¯ X B if cl ( A ) ∩ cl ( B ) = ∅ . Theorem 36.
Let ¯ X be a coarse compactification of a proper metric space X . Then r ¯ X is alarge-scale proximity relation on X . There is a homeomorphism Φ : ¯ X → ¯ X r extending theidentity on X .Proof. By [GS19, Theorem 6.7] the relation r is a coarse proximity relation. Thus axioms 1,2,4,5of a large-scale proximity relation are satisfied. It remains to show r satisfies axiom 3. Let A, A ′ , B ⊆ X be subsets and let E ⊆ X × X be an entourage with E [ A ] ⊇ A ′ , E [ A ′ ] ⊇ A and ArB . Then cl ( A ) ∩ cl ( B ) = ∅ . Since the compactification is coarse A, A ′ have the same limitpoints on ∂X . Thus cl ( A ) = cl ( A ′ ) which implies cl ( A ′ ) ∩ cl ( B ) = ∅ . Thus A ′ rB .For the last statement we extend the proof of [GS19, Theorem 6.7]. If x ∈ ¯ X \ X is a pointdefine F x := { A ⊆ X : x ∈ cl ( A ) } . Then we defineΦ : ¯ X → ¯ X r x ( x x ∈ X [ F x ] x ∈ ¯ X \ X Now [GS19, Theorem 6.7] showed Φ | ¯ X \ X is a homeomorphism. This implies in particular thatΦ is a bijective map. We show Φ is continuous: Let A ⊆ X be a subset. ThenΦ − ( ¯ A X ∪ cl ( A )) = ¯ A X ∪ { x ∈ ¯ X \ X : F x ∈ cl ( A ) } = ¯ A X ∪ { x ∈ ¯ X \ X : A ∈ F x } = ¯ A X ∪ cl ( A )is a closed set. Here ¯ A X denotes the closure of A in X . Theorem 37.
Let X be a proper metric space. A compactification ¯ X is coarse if and only ifevery bounded continuous function ϕ : X → R that extends to ¯ X is Higson.Proof. Suppose ¯ X is a coarse compactification and assume for contradiction there is a continuousfunction ϕ : ¯ X → R such that ϕ | X is not Higson. Then there is an entourage E ⊆ X and some ε > x k , y k ) k ⊆ E with | ϕ ( x k ) − ϕ ( y k ) | > ε Now ( x k ) k ∩ ( y k ) k = ∅ since the compactification is coarse. This contradicts that ϕ ( x k ) k , ϕ ( y k ) k have disjoint limit points in R .Now we give an alternative proof of this direction using that the Higson corona is universalamong coarse compactifications. Suppose ϕ : X → R is a bounded continuous function thatextends a continuous function ¯ ϕ on ¯ X . Denote by q : hX → ¯ X the quotient map from theHigson corona. Then ¯ ϕ ◦ q : hX → R is an extension of ϕ to hX . This implies ϕ is Higson.Now suppose every bounded continuous function ϕ : X → R that extends to ¯ X is Higson.Let A, B ⊆ X be subsets such that ¯ A ∩ ¯ B = ∅ . Since ¯ X is normal we can use Urysohn’s lemma:11 FUNCTORIALITY Elisa Hartmann there exists a bounded continuous function ϕ : ¯ X → R with ϕ | ¯ A ≡ ϕ | ¯ B ≡
1. Now ϕ | X isHigson. this implies for every entourage E ⊆ X × X there exists a bounded set C ⊆ X with E ∩ ( A × B ) ⊆ C × C. This implies A f B .Let X be a proper metric space. To a coarse compactification ¯ X we can associate a large-scaleproximity relation r on subsets of X such that ¯ X = ¯ X r . We can also associate to ¯ X the set ofbounded functions C r ( X ) that extend to ¯ X r . They must be Higson. Note that C r ( X ) is a ringby pointwise addition and multiplication. Proposition 38.
Let α : X → Y be a coarse map between proper metric spaces and let ¯ Y r bea coarse compactification. If X is R -discrete for some R > the functions C r ( X ) := { ϕ ◦ α : ϕ ∈ C r ( Y ) } determine a coarse compactification on X . It is the same compactification which isinduced by the relation ArB if α ( A ) rα ( B ) .Proof. If ϕ is a Higson function on Y then ϕ ◦ α is continuous and bounded since α is continuousand ϕ ◦ α is continuous and bounded. Since α is a coarse map ϕ ◦ α satisfies the Higson property.Thus C r ( X ) determines a compactification ¯ X which is coarse. The set C r ( X ) is a ring bypointwise addition and multiplication and contains the constant functions. Thus C r ( X ) equalsthe bounded functions which can be extended to ¯ X r .Now we prove the relation r defined on subsets of X is a large-scale proximity relation. Wecheck the axioms of a large-scale proximity relation:1. if B ⊆ X is bounded so is α ( B ) ⊆ Y . Thus α ( B )¯ rα ( B ) which implies B ¯ rB . If A ⊆ X isunbounded then α ( A ) ⊆ Y is unbounded. Thus α ( A ) rα ( A ) which implies ArA .2. symmetry is obvious.3. Suppose
A, A ′ , B ⊆ X are subsets and E ⊆ X × X an entourage with E [ A ] ⊇ A ′ , E [ A ′ ] ⊇ A and ArB . Then α × ( E )[ α ( A )] ⊇ α ( A ′ ) and α × ( E )[ α ( A ′ )] ⊇ α ( A ) and α ( A ) rα ( B ). Then α ( A ′ ) rα ( B ). Thus A ′ rB .4. If ( A ∪ B ) rC then α ( A ∪ B ) rα ( C ). Now α ( A ∪ B ) = α ( A ) ∪ α ( B ) thus α ( A ) rα ( C ) or α ( B ) rα ( C ). This implies ArC or BrC .5. If A ¯ rB then α ( A )¯ rα ( B ). This implies there exist C, D ⊆ Y with C ∪ D = Y and C ¯ rα ( A ) , D ¯ rα ( B ). Then α − ( C ) ∪ α − ( D ) = X and α − ( C )¯ rA, α − ( D )¯ rB .Now we define a map Φ : ¯ X r → R C r ( X ) x ( ( ϕ ◦ α ( x )) ϕ ◦ α x ∈ X ( x - lim ϕ ◦ α ) ϕ ◦ α x ∈ ∂ r X. We show Φ is well-defined: If F λ r G then for every A ∈ F , B ∈ G the relation ArB holds. Then α ( A ) rα ( B ) which implies α ∗ F λ r α ∗ G . Then F - lim ϕ ◦ α = α ∗ F - lim ϕ = α ∗ G - lim ϕ = G - lim ϕ ◦ α FUNCTORIALITY Elisa Hartmann for every ϕ ∈ C r ( Y ).Now we show Φ is injective: if F ¯ λ r G on X then there exist A ∈ F , B ∈ G with A ¯ rB . Thus α ( A )¯ rα ( B ). This implies α ∗ F ¯ λ r α ∗ G . Then there exists some bounded function ϕ ∈ C r ( Y ) with F - lim ϕ ◦ α = α ∗ F - lim ϕ = α ∗ G - lim ϕ = G - lim ϕ ◦ α. Denote for Z = X, Y the evaluation map e : Z → R C r ( Z ) z ( ϕ ( z )) ϕ . Now the following diagram commutes¯ X r α ∗ / / Φ (cid:15) (cid:15) ¯ Y r e ( X ) α ∗ / / e ( Y )where the upper horizontal map maps F ∈ ∂ r X to α ∗ F and x ∈ X to α ( x ) and the lowerhorizontal map maps ( ϕ ◦ α ( x )) ϕ ◦ α to ( ϕ ◦ α ( x )) ϕ . Now both horizontal maps are continuousand open, thus Φ is continuous and open.Since every ultrafilter on X induces an r -ultrafilter on X the map Φ is surjective on e ( X ). Lemma 39.
Let α : X → Y be a coarse map between proper metric spaces and let ¯ X r , ¯ Y r be coarse compactifications of X, Y , respectively. If Ar B implies α ( A ) r α ( B ) then α can beextended to a continuous map α ∗ : ¯ X r → ¯ Y r x ( α ( x ) x ∈ Xα ∗ x x ∈ ∂ r X. If X is R -discrete for some R > and C r ( Y ) ◦ α ⊆ C r ( X ) then α ∗ is the restriction of R C r ( X ) → R C r ( Y ) ( ϕ ( x )) ϕ ( ϕ ◦ α ( x )) ϕ to e ( X ) . Both descriptions of α coincide.Proof. Suppose Ar B implies α ( A ) r α ( B ). Let ϕ ∈ C r ( Y ) be a function and let F be an r -ultrafilter on X . Then α ∗ F is an r -ultrafilter on Y . Thus α ∗ F - lim ϕ exists. This point equals F - lim ϕ ◦ α . Since F was arbitrary the map ϕ ◦ α can be extended to ¯ X r . Thus ϕ ◦ α ∈ C r ( X ).Now suppose C r ( Y ) ◦ α ⊆ C r ( X ). Let A, B ⊆ Y be subsets with A ¯ r B . Then there exists ϕ ∈ C r ( Y ) with ϕ | A ≡ , ϕ | B ≡
0. Then ϕ ◦ α | α − ( A ) ≡ , ϕ ◦ α | α − ( B ) ≡ ϕ ◦ α can beextended to ¯ X r . Thus α − ( A )¯ r α − ( B ).Note the diagram ¯ X r α ∗ / / ¯ Y r e ( X ) α ∗ / / e ( Y )13 FUNCTORIALITY Elisa Hartmann commutes.
Proposition 40.
Let X be a proper metric space. If subsets U , . . . , U n coarsely cover X andeach U i is equipped with a large-scale proximity relation r i such that r i , r j agree on U i ∩ U j thenthe relation r on subsets of X defined by ArB if ( U i ∩ A ) r i ( U i ∩ B ) for some i defines a large-scale proximity relation on X . If X is R -discrete for some R > and for every i there is a ring C s i ( U i ) of Higson functions such that C s i ( U i ) | U j = C s j ( U j ) | U i then the ring C s ( X ) = { ( ϕ i ) i ∈ Y i C s i ( U i ) : ϕ i | U j = ϕ j | U i } consists of Higson functions. If r i = s i for every i then the relation r and the ring of boundedfunctions C s ( X ) describe the same compactification.Proof. We show r is a large-scale proximity relation on X :1. if B ⊆ X is bounded then B ∩ U i is bounded for every i . Thus B ¯ rB . If A ⊆ X is notbounded then there exists some i such that A ∩ U i is not bounded. Then ( A ∩ U i ) r i ( A ∩ U i )thus ArA .2. Symmetry is obvious.3. Without loss of generality assume n = 2. Let A, A ′ , B ⊆ X be subsets and let E ⊆ X × X be an entourage with E [ A ] = A ′ , E − [ A ′ ] = A and ArB . Since U , U coarsely cover X therelation U c f U c holds. Thus there exist C, D ⊆ X with C ∪ D = X and C f U c , D f U c .Now A = ( A ∩ C ) ∪ ( A ∩ D ). Thus by axiom 4 ( A ∩ C ) rB or ( A ∩ D ) rB . Suppose theformer holds. Since A ∩ C ⊆ U ∪ B ′ where B ′ is bounded we have ( A ∩ C ∩ U ) r ( B ∩ U ).Now E [ A ∩ C ] ⊆ U ∪ B ′′ where B ′′ is bounded. Then ( E [ A ∩ C ] ∩ U ) r ( B ∩ U ). Thus A ′ rB by axiom 4.4. If ( A ∪ B ) rC then ( A ∪ B ) r i C for some i . Thus Ar i C or Br i C . This implies ArC or BrC .If
ArC or BrC then ( A ∩ U i ) r i ( C ∩ U i ) for some i or ( B ∩ U j ) r j ( C ∩ U j ) for some j . Thisimplies (( A ∪ B ) ∩ U i ) r i ( C ∩ U i ) or (( A ∪ B ) ∩ U j ) r j ( C ∩ U j ). Thus ( A ∪ B ) rC .5. Without loss of generality assume n = 2 and U , U cover X as sets. If A ¯ rB then ( A ∩ U i )¯ r i ( B ∩ U i ) for both i . Thus there exist C i , D i ⊆ U i with C i ¯ r i ( A ∩ U i ) , D i ¯ r i ( B ∩ U i ) and C i ∪ D i = U i for i = 1 ,
2. Then ( A ∩ U ∩ U )¯ r ( C ∪ C ) , ( A ∩ U ∩ U c )¯ r ( C ∪ U ) , ( A ∩ U ∩ U c )¯ r ( C ∪ U ) combine to A ¯ r ( C ∪ C ). Similarly we obtain B ¯ r ( D ∪ D ). Now X = U ∪ U = C ∪ D ∪ C ∪ D . Now we show C s ( X ) consists of Higson functions. Suppose ϕ i ∈ C s i ( U i ) for i = 1 , . . . , n areelements with ϕ i | U j = ϕ j | U i . Then they can be glued to a bounded continuous function ϕ : X → R . Let E ⊆ X × X be an entourage. Then E = ( E ∩ ( U × U )) ∪ · · · ∪ ( E ∩ ( U n × U n )) ∪ A where A ⊆ B × B with B bounded in X . Now ( dϕ ) | E ∩ ( U i × U i ) = ( dϕ | U i ) | E converges to zero atinfinity for every i . This implies ( dϕ ) | E converges to zero at infinity.If r i = s i for every i then ¯ U r i i = e ( U i ) for every i . The ¯ U r i i glue to ¯ X r and the e ( U i ) glue to e ( X ). The global axiom of Lemma 41 implies uniqueness. Thus ¯ X r = e ( X ).14 HIGSON CORONA Elisa Hartmann
Let X be a proper R -discrete for some R > A ⊆ X theposet of coarse compactifications on A is called CC ( A ). If A ⊆ B is an inclusion of subspacesthen there is a poset map CC ( B ) → CC ( A ) induced by the inclusion.The Grothendieck topology determined by coarse covers on a metric space X is called X ct .A contravariant functor F on subsets of X is a sheaf on X ct if for every coarse cover U , U ⊆ U of a subset of X the following diagram is an equalizer F ( U ) → F ( U ) ⊕ F ( U ) ⇒ F ( U ∩ U ) . Lemma 41.
The functor CC on subsets of X is a sheaf on X ct .Proof. Note every subspace of X is proper. If A ⊆ X is a subset we define ¯ A r ≥ ¯ A s if s is finerthan r .If A ⊆ B is an inclusion of subspaces and ¯ B r ∈ CC ( B ) then the restriction map associated tothe inclusion A → B maps ¯ B r ¯ A r | A . Here the relation r | A is defined as Sr | A T if SrT . Then r | A is a large-scale proximity relation on A :1. If S ⊆ A is bounded, then S ¯ rS , thus S ¯ r | A S . If S is unbounded then SrS so Sr | A S .2. Symmetry is obvious.3. If S, S ′ , T ⊆ A are subsets, E ⊆ A is an entourage with E [ S ] ⊇ S ′ , E [ S ′ ] ⊇ S and Sr | A T then E ⊆ B is an entourage in B . Thus S ′ rT which implies S ′ r | A T .4. obvious.5. If S, T ⊆ A are subsets with S ¯ rT then there exist subsets C ′ , D ′ ⊆ B with C ′ ∪ D ′ = B and C ′ ¯ rS, D ′ ¯ rT . Then C := C ′ ∩ A, D := D ′ ∩ A are subsets with C ∪ D = A, C ¯ rS, D ¯ rT .Note if a large scale proximity relation s on B is finer than another large-scale proximity relation r on B then s | A is finer than r | A on A . This makes CC into a functor on the poset of subsets of X to posets.Now we check the global axiom: Let ( U i ) i be a coarse cover of X and let r, s be close relationson X with r | U i = s | U i . Two subsets A, B ⊆ X satisfy ArB if and only if W i (( A ∩ U i ) r ( B ∩ U i ))if and only if W i (( A ∩ U i ) r | U i ( B ∩ U i )) if and only if W i (( A ∩ U i ) s | U i ( B ∩ U i )) if and only if W i (( A ∩ U i ) s ( B ∩ U i )) if and only if AsB .Now we check the gluing axiom: Let U , . . . , U n be a coarse cover of X equipped with coarsecompactifications ¯ U r , . . . , ¯ U r n n such that r i | U j = r j | U i for every ij . Then the proof Proposition 40implies the ¯ U r i i glue to a coarse compactification ¯ X r of X . This section is denoted to the Higson corona. We recall the original description.
Definition 42. ( Higson corona ) Let X be a proper metric space. A bounded continuousfunction ϕ : X → R is called Higson if for every entourage E ⊆ X the map dϕ | E : E → R ( x, y ) ϕ ( x ) − ϕ ( y )vanishes at infinity. Then the compactification hX of X generated by the Higson functions C h ( X )is called the Higson compactification . The boundary of this compactification ν ( X ) = hX \ X iscalled the Higson corona . 15
HIGSON CORONA Elisa Hartmann
The large-scale proximity relation induced on X is the close relation. Remark . If X is a proper metric space then ∂ f ( X ) = ν ( X ). Proof.
This follows from by [Har19b, Theorem 20].We provide an alternative proof: Let
A, B ⊆ X be subsets. If A f B then by [NW70,Theorem 5.14] there exists a f -ultrafilter F on X with A, B ∈ F . Thus
F ∈ cl ( A ) ∩ cl ( B ) is notempty. If on the other hand A f B then F ∈ cl ( A ) implies F 6∈ cl ( B ). Thus cl ( A ) ∩ cl ( B ) = ∅ isempty. This way we have shown that f is the unique relation on subsets of X that tells whenthe closure of two subsets meet on the boundary of the Higson compactification. Proposition 44. If X is a one-ended proper metric space then ν ( X ) is connected. This impliesthat every coarse compactification of X is connected.Proof. Recall that a metric space has at most one end if for every A ⊆ X we have A f A c orone of A, A c is bounded. Suppose π ⊆ ν ( X ) is a clopen subset. Then π f π c . Then there exist A, B ⊆ X with π ⊆ cl ( A ) , π c ⊆ cl ( B ) and A f B . By the proof of Theorem 52 the inclusion A ∪ B → X is coarsely surjective. Thus one of A, B is bounded which implies one of π, π c is theempty set.Now we select Higson functions which separate coarsely disjoint subsets of X . A close exam-ination shows they together with the constant functions already generate the Higson functions.Let X be a proper metric space. For every two subsets A, B ⊆ X with A f B we define ϕ A,B : X → R x d ( x, A ) d ( x, A ) + d ( x, B )where we assume without loss of generality d ( A, B ) >
0. If F ⊆ C ∗ ( X ) is a subset A ( F ) denotesthe intersection of all algebras in C ∗ ( X ) which contain F . Proposition 45.
There is an isomorphism of C ∗ -algebras A (( ϕ A,B ) A f B ∪ { } ) = C h ( X ) . Here the closure is in C ∗ ( X ) with the sup -metric.Proof. Suppose
A, B ⊆ X are subsets with A f B . By [DKU98, Lemma 2.2] the function ϕ A,B is Higson. Thus we have shown ( ϕ A,B ) A f B ⊆ C h ( X ).Now we show ( ¯ ϕ A,B ) A f B separates points of ν ( X ): Let F , G ∈ ν ( X ) be points with F ¯ λ f G .Then there exist A ∈ F , B ∈ G with A f B . Then¯ ϕ A,B ([ F ]) = F - lim ϕ A,B = 0 = 1= G - lim ϕ A,B = ¯ ϕ A,B ([ G ]) . Thus ¯ ϕ A,B separates F , G .By [BY82, Theorem 2.1] the ( ϕ A,B ) A f B generate the compactification hX . Now we use [BY82,Theorem 3.4] and obtain the result. 16 HIGSON CORONA Elisa Hartmann
Call a metric space W with { x, y ∈ W : d ( x, y ) ≤ R, x = y } finite for every R > discretecoarse [Roe03, Example 2.7]. Remark . If X is an unbounded proper metric space then it contains a sequence ( x i ) i ⊆ X with d ( x i , x j ) > i for j < i . Thus ( x i ) i is discrete coarse. It is easy to check every boundedfunction on ( x i ) i is Higson. Then h (( x i ) i ) = β ( N ) and ν (( x i ) i ) = β ( N ) \ N . Since h, ν preservesmonomorphisms h (( x i ) i ) , ν (( x i ) i ) arise as subspaces of h ( X ) , ν ( X ). Thus ν ( X ) contains a copy of β ( N ) \ N and hX contains a copy of β ( N ). This fact has already been proved in [Kee94, Theorem3]. Proposition 47. If X is a proper metric space then the union of cl ( W ) over every discretecoarse subspace W of X is dense in ν ( X ) .Proof. Define Φ : G W ⊆ X discrete ν ( W ) → ν ( X ) F 7→ i ∗ F where i : W → X is the inclusion. We showΦ ∗ : C ( ν ( X )) → C ( G W ⊆ X discrete ν ( W ))is injective. Note C ( ν ( X )) = C h ( X ) /C ( X ) and C ( G W ν ( W )) = Y W C ( W )= Y W C h ( W ) /C ( W ) . Let ϕ ∈ C h ( X ) be a Higson function. We need to show if ( ϕ ◦ Φ) W ∈ C ( W ) for every discretesubset W ⊆ X then ϕ ∈ C ( X ). Assume for contradiction that ϕ does not converge to zero atinfinity. Then there exists ε > i ∈ N there is some x i B ( x , i ) (Here x ∈ X is a fixed point and B ( x , i ) denotes the ball of radius i around x ) with the property | ϕ ( x i ) | ≥ ε .Now choose a subsequence ( x i k ) k with { x i k : k } discrete. Then ϕ | { x ik : k } C ( { x i k : k } ). Sincebounded functions on ν ( X ) separate points from closed sets we have shown that the closure ofim Φ is ν ( X ).The closure of im Φ is ν ( X ) since S W cl ( W ) ⊆ cl ( A ) implies the inclusion i : A → X iscoarsely surjective (Every unbounded subset of X contains a discrete subset).Suppose X is R -discrete for some R >
0. If X is not discrete there always exists an ultrafilteron X which does not contain a discrete subspace. Define a filter F = { X \ W : W ⊆ X discrete or finite } Then F is a filter:1. If X \ W, X \ V ∈ F then W, V are discrete or finite. This implies W ∪ V is discrete orfinite, thus ( X \ W ) ∩ ( X \ V ) = X \ ( V ∪ W ) ∈ F .2. If X \ W ∈ F and X \ W ⊆ X \ V then W is discrete or finite and V ⊆ W . This implies V is discrete or finite. Thus X \ V ∈ F .If X is not discrete or finite then F is a proper filter. Then there exists an ultrafilter finer than F , it does not contain a discrete subspace. 17 SPACE OF ENDS Elisa Hartmann
The space of ends Ω( X ) of a topological space is the boundary of the Freudenthal compactification ε ( X ). In this chapter we will study a coarse version of the Freudenthal compactification whichcoincides with the topological version of the Freudenthal compactification for a large class ofproper metric spaces.Recall [Wil70, Problem 41B]: Definition 48. ( Freudenthal compactification, topological version ) Let X be a rim-compact Tychonoff space. Define a relation δ on subsets of X by A ¯ δB for A, B ⊆ X if there isa compact subset K ⊆ X such that X \ K = G ∪ H is a disjoint union of two open subsets with¯ A ⊆ G, ¯ B ⊆ H .The Smirnov compactification of the proximity space ( X, δ ) is called the
Freudenthal com-pactification .Its boundary is zero dimensional.Let
A, B ⊆ X be subsets of a metric space. Define A f f B if there exist A ′ ⊇ A, B ′ ⊇ B with A ′ ∪ B ′ = X and A ′ f B ′ . Proposition 49.
Let X be a proper geodesic metric space. Then it is rim-compact Tychonoff.If A, B ⊆ X are two subsets then AδB if and only if ¯ A ∩ ¯ B = ∅ or A f f B .Proof. Since X is a metric space every point x ∈ X has a basis of open neighborhoods { ˚ B ( x, ε ) : ε > } , here ˚ B ( x, ε ) denotes the open ball of radius ε around x . Since X is proper the the set B ( x, ε ) \ ˚ B ( x, ε ) ⊆ ˚ B ( x, ε )is compact. Thus X is rim-compact. Note every metric space is Tychonoff.Suppose A, B ⊆ X are two subsets with A ¯ δB . Then there exists a compact set K ⊆ X suchthat X \ K = G ∪ H with appropriate properties. Let R > g ∈ G, h ∈ H arepoints with d ( g, h ) ≤ R then there exists k ∈ K with d ( g, k ) + d ( k, h ) = d ( g, h ) ≤ R. Now K is bounded thus there exists S ≥ , x ∈ X with K ⊆ B ( x , S ). Then g, h ∈ B ( x , S + R ).This proves G f H . Thus A f f B . Since δ is compatible with the topology on X the relation¯ A ∩ ¯ B = ∅ follows.Suppose A, B ⊆ X are two subsets with A f f B and ¯ A ∩ ¯ B = ∅ . The first relation impliesthere are A ′ ⊇ A, B ′ ⊇ B with X = A ′ ∪ B ′ , A ′ f B ′ . Then there exists a bounded set K ′ ⊆ X such that d ( A ′ \ K ′ , B ′ \ K ′ ) >
1. Define A ′′ = S a ∈ A ′ \ K ′ ˚ B ( a, /
4) and B ′′ = S b ∈ B ′ \ K ′ ˚ B ( b, / X is normal there exist open sets U ⊇ ¯ A, V ⊇ ¯ B with U ∩ V = ∅ . The set K := A ′′ c ∩ U c ∩ B ′′ c ∩ V c ⊆ ( A ′ \ K ′ ) c ∩ ( B ′ \ K ′ ) c = K ′ is bounded and closed. Since X is proper this set is compact. We define G = A ′′ ∪ U, H = B ′′ ∪ V .Then G, H are open and disjoint. We have X \ K = A ′′ ∪ U ∪ B ′′ ∪ V = G ∪ H and A ⊆ G, B ⊆ H . Thus we have shown A ¯ δB .18 SPACE OF ENDS Elisa Hartmann
It is easy to see that f f is a large-scale proximity relation. Thus ¯ X f f is a coarse compact-ification of X . By Proposition 49 the space is homeomorphic to ε ( X ) if X is proper geodesicmetric. By slight abuse of notation we write Ω( X ) , ε ( X ) for the coarse versions of the space ofends, Freudenthal compactification as well. Definition 50.
Let X be a metric space with basepoint x ∈ X . A bounded continuous map ϕ : X → R is called Freudenthal if for every R ≥ K ≥ d ( x, y ) ≤ R, d ( x , x ) ≤ K, d ( x , y ) ≤ K implies ϕ ( x ) = ϕ ( y ). We write C f ( X ) for the ring of Freudenthalfunctions on X . Lemma 51.
Let X be a proper metric space. A bounded continuous function ϕ : X → R isFreudenthal if and only if it can be extended to ¯ X f .Proof. Without loss of generality assume X is R -discrete for some R > ϕ : X → R is Freudenthal. If F is an ultrafilteron X define ¯ ϕ ( F ) = F - lim ϕ . We show ϕ is well defined: Let F , G be ultrafilters on X with F - lim ϕ = G - lim ϕ . Then X = ϕ − (( −∞ , F - lim ϕ + G - lim ϕ )) ∪ ϕ − ([ F - lim ϕ + G - lim ϕ , ∞ )) and ϕ − (( −∞ , F - lim ϕ + G - lim ϕ )) f ϕ − ([ F - lim ϕ + G - lim ϕ , ∞ )). Thus F ¯ λ f G .We show ¯ ϕ is continuous: Choose an Interval I ⊆ R such that im ϕ ⊆ I and consider ϕ as amap X → I . Let S, T ⊆ I be subsets such that ¯ S ∩ ¯ T = ∅ . Then there is some subset C ⊆ I with S ⊆ C, T ⊆ C c and ¯ C ∩ ¯ T = ∅ , C c ∩ S = ∅ . Then we obtain ϕ − ( C ) ⊇ ϕ − ( S ) , ϕ − ( C c ) ⊇ ϕ − ( T )and X = ϕ − ( C ) ∪ ϕ − ( C c ). Now let R ≥ B ⊆ X such that d ( x, y ) ≤ R, ϕ ( x ) = ϕ ( y ) implies x, y ∈ B . Thus if x ∈ ϕ − ( C ) , y ∈ ϕ − ( C c ) and d ( x, y ) ≤ R then x, y ∈ B . This implies ϕ − ( C ) f ϕ − ( C c ). Thus ϕ − ( S ) f f ϕ − ( T ). This shows¯ ϕ − ( S ) ∩ ¯ ϕ − ( T ) = ( ϕ − ( S ) ∪ cl ( ϕ − ( S ))) ∩ ( ϕ − ( T ) ∪ cl ( ϕ − ( T )))= ∅ . Thus ¯ ϕ is continuous.Now we show C f ( X ) separates points of ∂ f ( X ) = ¯ X f \ X . If F , G are ultrafilters on X with F ¯ λ f G then there are A ∈ F , B ∈ G with A f f B . Then there exists C ⊆ X with A ⊆ C, B ⊆ C c and C f C c . Define ϕ : X → R x ( x ∈ C x ∈ C c . Then ϕ is a Freudenthal function. Now the extension ¯ ϕ of ϕ separates F from G . Then by [BY82]the ring C f ( X ) determines the compactification ¯ X f of X . Theorem 52.
Let X be a proper metric space. The boundary of the Freudenthal compactification Ω X = εX \ X of X is totally disconnected. If ( ¯ X, X ) is another coarse compactification whoseboundary is totally disconnected then it factors through εX . The association Ω is a functor thatmaps coarse maps modulo close to continuous maps.Remark . Compare this result with [Pes90, Theorem 1]. The Freudenthal compactificationof a topological space with nice properties is universal among compactifications with totallydisconnected boundary.
Proof.
At first we show Ω X is totally disconnected. It is sufficient to show that there exists a basisconsisting of clopen subsets in ∂ f f ( X ). If A ⊆ X has the property A f A c then cl ( A ) = cl ( A c ) c GROMOV BOUNDARY Elisa Hartmann is both open and closed. Now we show ( cl ( A )) A f A c are a basis for the topology on Ω X . Notealready ( cl ( A ) c ) A ⊆ X are a base for a topology on Ω X . Let A ⊆ X be a subset and F ∈ cl ( A ) c be a f f -ultrafilter. Then there exists B ∈ F with B f f A . Thus there exists A ′ ⊇ A, B ′ ⊇ B with A ′ ∪ B ′ = X and A ′ f B ′ . This implies A ′ f f B thus F ∈ cl ( A ′ ) c ⊆ cl ( A ) c . Now A ′ is of thetype A ′ f A ′ c .Suppose r is a close relation on X such that ∂ r X is totally disconnected. Then there existsa basis of clopen sets on ∂ r X . Let π ⊆ ∂ r X be a clopen subset. Thus π ¯ rπ c . This implies thereexist A, B ⊆ X with A ¯ rB and π ⊆ cl ( A ) , π c ⊆ cl ( B ). In particular A f B and cl ( A ∪ B ) = cl ( A ) ∪ cl ( B ) ⊇ π ∪ π c = ∂ r X. This implies the inclusion A ∪ B → X is coarsely surjective. Thus A f f B which implies π f f π c .Thus the unique map εX → ∂ r X extending the identity on X is well-defined and continuous.Now we show Ω is a functor. Let ϕ : X → Y be a coarse map between metric spaces. It issufficient to show that A f f B implies ϕ ( A ) f f ϕ ( B ). Suppose ϕ ( A ) f f ϕ ( B ). Then there exist A ′ ⊇ ϕ ( A ) , B ′ ⊇ ϕ ( B ) with A ′ ∪ B ′ = Y and A ′ f B ′ . This implies ϕ − ( A ′ ) ⊇ A, ϕ − ( B ′ ) ⊇ B, ϕ − ( A ′ ) ∪ ϕ − ( B ′ ) = X and ϕ − ( A ′ ) f ϕ − ( B ′ ). Thus A f f B . Corollary 54. ( Protasov ) Let X be a proper metric space. If asdim( X ) = 0 then ν ( X ) and Ω( X ) coincide.Remark . Compare this result with [Pro03, Lemma 4.3]. We prove the same result usinguniversal properties.
Proof.
Since asdim( X ) = 0 the space ν ( X ) is zero dimensional by [DKU98], [Dra00]. This implies ν ( X ) is totally disconnected. By Theorem 52 there exists a unique surjective map h ( X ) → ε ( X )which extends the identity on X . Now by Remark 34 there exists a unique surjective map h ( X ) → ε ( X ). Since the composition of both maps h ( X ) → h ( X ) and ε ( X ) → ε ( X ) are uniquesurjective they agree with the identity. This proves the spaces Ω( X ) , ν ( X ) are homeomorphic. The Gromov boundary is the last interesting example in this paper. There is a quotient map fromthe Higson compactification to the Gromov compactification since it is a coarse compactification.We are going to present in this chapter maps in the other direction.If X is a metric space and x ∈ X a fixed point then the Gromov product of two points x, y ∈ X is defined as ( x | y ) := 1 / d ( x, x ) + d ( y, x ) − d ( x, y )) . Definition 56. ( Gromov boundary ) Let X be a proper geodesic hyperbolic metric space. Acontinuous function ϕ : X → R is called Gromov if for every ε >
K > x | y ) > K → | ϕ ( x ) − ϕ ( y ) | < ε. The Gromov functions determine a compactification of X called the Gromov compactification gX . The boundary ∂X = gX \ X is called the Gromov boundary .20
GROMOV BOUNDARY Elisa HartmannRemark . Let X be a proper hyperbolic geodesic metric space. Two sequences ( a i ) i , ( b i ) i ⊆ X converge to the same point on the Gromov boundary ∂X if and only iflim inf i,j →∞ ( a i | b j ) = ∞ . If p ∈ ∂X define U ( p, r ) = { q ∈ ∂ ( X ) : [( x n ) n ] = p, [( y n ) n ] = q, lim inf i,j →∞ ( x i | y j ) ≥ r } and U ( p, r ) = { y ∈ X : [( x n ) n ] = p, lim inf i,j →∞ ( x i | y ) ≥ r } . Then { U ( p, r ) ∪ U ( p, r ) : r ≥ } is a neighborhood basis of p in gX . Proof.
The first part is [FOY18, Proposition 4.3]. The second part is [KB02, Definition 2.13].
Example 58. ( Gromov boundary ) Let
A, B ⊆ X be subsets of a hyperbolic proper metricspace. Define A f g B if there are sequences ( a i ) i ⊆ A, ( b i ) i ⊆ B such thatlim inf i,j →∞ ( a i | b j ) = ∞ . If A f B then there exist unbounded sequences ( a i ) i ⊆ A, ( b i ) i ⊆ B and some R ≥ d ( a i , b i ) ≤ R for every i . This implies lim inf i,j →∞ ( a i | b j ) = ∞ thus A f g B . By [GS19,Proposition 9.8] the relation f g is a coarse proximity relation, Thus axioms 1,2,4,5 of Definition 18hold. Axiom 3 of Definition 18 holds trivially, thus f g is a large-scale proximity relation. Example 59. ( Gromov boundary ) Let X be a hyperbolic geodesic proper metric space. ByRemark 57 we obtain ∂ f g = ∂ ( X ). Here the right side denotes the Gromov boundary of X . Remark . If X is a hyperbolic metric space and γ, δ : Z + → X are quasigeodesic rays in X then γ ( Z + ) f δ ( Z + ) implies there exists some entourage E ⊆ X × X with E [ γ ( Z + )] ⊇ δ ( Z + ) and E [ δ ( Z + )] ⊇ γ ( Z + ). Proof.
By [Roe03, Definition 6.16] a map γ : Z + → X is a quasigeodesic ray if there are constants R > , S ≥ R − | i − j | − S ≤ d ( γ ( i ) , γ ( j )) ≤ R | i − j | + S for every i, j ∈ Z + . It follows from [Roe03, Theorem 6.17] that there exists some T ≥ d ( γ ( Z + ) , δ ( Z + )) ≤ T . Remark . Let X be a geodesic metric space and ˜ γ : R + → X a geodesic ray. If γ : Z + → X is close to ˜ γ then it is coarsely injective coarse and the induced map ν ( γ ) : ν ( Z + ) → ν ( X ) is aclosed embedding. Proof.
This is [Har19c, Lemma 39].
Theorem 62.
Let X be a hyperbolic geodesic proper metric space. Then there is a closedembedding Φ : ν ( Z + ) × ∂X → ν ( X ) . GROMOV BOUNDARY Elisa HartmannRemark . Compare this result with [BS07, Theorem 10.1.2] which states that for every propergeodesic hyperbolic metric space the inequalityasdim( X ) ≥ dim ∂ ( X ) + 1holds. Note asdim( Z + ) = 1 and asdim( X ) = dim( ν ( X )) for every proper metric space [DKU98],[Dra00]. By [Mor77, Theorem 3] we obtain dim( ν ( Z + ) × ∂X ) = dim( ∂X ) + 1. Thus we obtaina new proof for the above inequality. Proof.
Let S ⊆ X be an R -discrete for some R > S → X iscoarsely surjective. Let p ∈ ∂ ( X ) be a point. Then p is represented by an isometry ˜ γ : Z + → X .Now choose γ : Z + → S close to ˜ γ . Then γ is a quasigeodesic ray in S . If F is a nonprincipalultrafilter on Z + then γ ∗ F is a nonprincipal ultrafilter on S . DefineΦ : ν ( Z + ) × ∂X → ν ( X )([ F ] , p ) [ γ ∗ F ]Since γ is a coarse map F λ f G implies γ ∗ F λ f γ ∗ G for every F , G ∈ ν ( Z + ). If γ, δ : Z + → S represent the same point in ∂X then there exists K ≥ d ( γ ( n ) , δ ( n )) ≤ K . Thisimplies γ ∗ F λ f δ ∗ F for every F ∈ ν ( Z + ). Thus Φ is a well defined map.Now we show Φ is injective: Let F , G ∈ ν ( Z + ) , γ, δ ∈ ∂X be elements with γ ∗ F λ f δ ∗ G . Then γ ( Z + ) f δ ( Z + ). Then Remark 60 implies that γ, δ represent the same element. Without loss ofgenerality assume that γ = δ . Then γ ∗ F λ f γ ∗ G implies F λ f G by Remark 61.Now we show Φ is open: Denote by p : ν ( Z + ) × ∂ ( X ) → ∂ ( X ) the projection to the secondfactor. Then p ◦ Φ − equals the quotient map q X of Theorem 32 restricted to the image of Φ.Let U ⊆ ∂X be open then Φ( U × ν ( Z + )) = ( p ◦ Φ − ) − ( U )= q − X ( U )is open. If V ⊆ ν ( Z + ) is open then Φ( ∂X × V ) = S γ γ ∗ V is a union of open sets and thus open.Thus we have shown Φ − : Φ( ν ( Z + ) × ∂ ( X )) → ν ( Z + ) × ∂X is bijective and continuous.Since Φ( ν ( Z + ) × ∂ ( X )) is compact and ν ( Z + ) × ∂ ( X ) Hausdorff the map Φ is a homeomorphismonto its image. Proposition 64. If T is a tree then the space Φ( ∂ ( T ) × ν ( R + )) is a retract of ν ( T ) .Proof. We first show that Φ( ∂ ( T ) × ν ( R + )) is a retract of ̟ := S A ⊆ T, | ∂ ( A ) | =1 cl ( A ) ⊆ ν ( T ):If F ∈ cl ( A ) with | ∂ ( A ) | = 1 then there is a geodesic ray γ on T such that F converges to thepoint represented by γ in the Gromov compactification. Since T is CAT(0) and γ ( R + ) is convexand complete [BH99, Proposition II.2.4] provides us with a projection map π : T → γ ( R + ) suchthat d ( x, π ( x )) = d ( x, γ ( R + )) for every x ∈ T .Then { π ( A ) : A ∈ F , ∂ ( A ) = [ γ ] } define a base for a f -ultrafilter F on T :1. If A, B ∈ F then A f B . Thus there exist unbounded sequences ( a i ) i ⊆ A, ( b i ) i ⊆ B with d ( a i , b i ) ≤ R . Then π ( a i ) i ⊆ π ( A ) , π ( b i ) i ⊆ π ( B ). By [BH99, Proposition II.2.4.4)]the map π does not increase distances. Thus d ( π ( a i ) , π ( b i )) ≤ R for every i . Sincelim inf i,j →∞ ( a i | γ ( j )) = ∞ the sequence ( π ( a i )) i is not bounded. This way we have shownthat π ( A ) f π ( B ).2. If A ∪ B ∈ F then A ∪ B = π ( C ) for some C ∈ F . Define A ′ = { a ∈ C : π ( a ) ∈ A } and B ′ = { b ∈ C : π ( b ) ∈ B } . Then π ( A ′ ) = A, π ( B ′ ) = B and A ′ ∪ B ′ = C . Now A ′ ∈ F or B ′ ∈ F which implies A ∈ F or B ∈ F .22 EFERENCES Elisa Hartmann
Now define a map r : ̟ → ν ( T ) F 7→ F . Let F , G ∈ ν ( T ) be two elements with F λ f G . If A ∈ F , B ∈ G then there are A ′ ∈ F , B ′ ∈ G with π ( A ′ ) = A, π ( B ′ ) = B . Now A ′ f B ′ implies π ( A ′ ) f π ( B ′ ) as above. This implies F λ f G .Now we show r is continuous on ̟ : Suppose A, B ⊆ T are subsets with ( cl ( A ) ∩ ̟ ) ∩ ( cl ( B ) ∩ ̟ ) = ∅ . Then r ( cl ( A ) ∩ ̟ ) ∩ r ( cl ( B ) ∩ ̟ ) = ( [ A ′ ⊆ A, | ∂ ( A ) | =1 cl ( π ( A ′ ))) ∩ ( [ A ′ ⊆ A, | ∂ ( A ) | =1 cl ( π ( A ′ )))= [ A ′ ⊆ A, | ∂ ( A ′ ) | =1 ,B ′ ⊆ B, | ∂ ( B ′ ) | =1 ( cl ( A ′ ) ∩ cl ( B ′ )) = ∅ . To see the last inequality choose A ′ := ( a i ) i ⊆ A, B ′ := ( b i ) i ⊆ B unbounded with d ( a i , b i ) ≤ R for every i and some R ≥
0. If necessary we can choose a subsequence of ( a i ) i such that ( a i ) i converges to a point on the Gromov boundary. Then ( b i ) i converges to the same point. Thuswe can assume | ∂ ( A ′ ) | = 1 = | ∂ ( B ′ ) | . Then π ( A ′ ) f π ( B ′ ), in fact both sets are finite Hausdorffdistance apart. Thus cl ( π ( A ′ )) ∩ cl ( π ( B ′ )) = . We just showed r is uniformly continuous withregard to the unique uniformity on the compact space ν ( T ).Note that S A ⊆ T, | ∂ ( A ) | =1 cl ( A ) is dense in ν ( T ): Consider the closure of S A ⊆ T, | ∂ ( A ) | =1 cl ( A ).If B ⊆ T is a subset then there exists a sequence ( b i ) i ⊆ B such that ( b i ) i converges to a point γ ∈ ∂ ( X ) in the Gromov compactification. This means | ∂ (( b i ) i ) | = 1. Thus S A ⊆ T, | ∂ ( A ) | =1 cl ( A ) = ν ( T ).Then [Eng89, Theorem 8.3.10] implies the retract map r can be extended to ν ( T ). Remark . The results in Theorem 62 and Proposition 64 are functorial: If α : T → S is acoarse map between trees such that α ◦ γ is coarsely injective if γ : R + → X is coarsely injectivecoarse then there is a continuous map r ◦ ν ( α ) ◦ Φ : ∂ ( T ) × ν ( R + ) → ∂ ( S ) × ν ( R + )([ γ ] , [ F ]) ([ α ◦ γ ] , [ F ]) . If α is a coarse equivalence then ∂ ( α ) : ∂ ( T ) → ∂ ( S ) is a homeomorphism since the Gromovboundary is a functor on coarse equivalences. This implies r ◦ ν ( α ) ◦ Φ is an isomorphism in thetopological category. [BH99] M. R. Bridson and A. Haefliger.
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Elisa Hartmann,
Department of Mathematics, Karlsruhe Institute of Technology, D-76128 Karl-sruhe Germany
E-mail address , Elisa Hartmann: [email protected]@kit.edu