Coarse Freundenthal compactification and ends of groups
aa r X i v : . [ m a t h . M G ] F e b COARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OFGROUPS
YUANKUI MA AND JERZY DYDAK
February 10, 2021
Contents
1. Introduction 12. Coarse compactifications 23. Glacial oscillations 34. Coarse Freundenthal compactification 75. Ends of groups 96. Ends of coarse spaces 12References 15
Abstract.
A coarse compactification of a proper metric space X is any com-pactification of X that is dominated by its Higson compactification. In thispaper we describe the maximal coarse compactification of X whose corona is ofdimension 0. In case of geodesic spaces X , it coincides with the Freundenthalcompactification of X . As an application we provide an alternative way ofextending the concept of the number of ends from finitely generated groups toarbitrary countable groups. We present a geometric proof of a generalizationof Stallings’ theorem by showing that any countable group of two ends containsan infinite cyclic subgroup of finite index. Finally, we define ends of arbitrarycoarse spaces. Introduction
Historically, as noted in [5] on p.287, ends are the oldest coarse topological notionand were used by Freundenthal in 1930 in his famous compactification (see [15] forinformation about theorems in this section):
Theorem 1.1.
Suppose X is a σ -compact locally compact and locally connected Haus-dorff space. It has a compactification ¯ X such that ¯ X \ X is of dimension and ¯ X dominates any compactification ˆ X of X whose corona is of dimension . Definition 1.2. A Freundenthal end is a decreasing sequence { U i } i ≥ i of componentsof sets X \ K i , where K i are compact, K i ⊂ int ( K i +1 ) for each i ≥
1, and ∞ [ i =1 K i = X. Date : February 10, 2021.2000
Mathematics Subject Classification.
Primary 54D35; Secondary 20F69.
Key words and phrases. dimension, coarse geometry, ends of groups, Freundenthal compacti-fication, Higson corona.
The space of ends of X is denoted by Ends ( X )The topology on X ∪ Ends ( X ) is induced by the following basis:1. It includes all open subsets of X with compact closure,2. It includes any component U of X \ K i union all ends containing U . Theorem 1.3. (Freudenthal) A path connected topological group has at most twoends.
Theorem 1.4. (H. Hopf ) Let G be a finitely generated discrete group acting on aspace X by covering transformations. Suppose the orbit space B := X/G is com-pact. Then (i) and (ii), below, hold.(i) The end space of X has 0, 1 or 2 (discrete) elements or is a Cantor space.(ii) If G also acts on Y satisfying the hypotheses above, then X and Y have home-omorphic end spaces. Conclusion (ii) suggests to regard the end space of X as an invariant of the group G itself: Definition 1.5.
Let p : X → B be a covering map with compact base B and thegroup of covering transformations G . The end space of G is Ends ( G ) := Ends ( X ) . When applied to a Cayley graph of G , it gives the standard definition of endsof finitely generated groups (see [5], p.295). See [12] for basic results in this theoryand see [13] for more general facts in coarse geometry related to groups.In this paper we will define ends of arbitrary countable groups by generalizingthe construction of the Higson corona. In the case of coarse spaces we generalizeFreudenthal’s method to define their space of ends.E. Specker [16] defined ends of arbitrary groups using Stone’s duality theorem.See a very nice paper [4] of Yves Cornulier describing properties of the space ofends of infinitely generated groups. We consider Specker’s approach highly non-geometric. Additionally, our way of defining ends of spaces leads directly to viewthem as coronas of certain compactifications (large scale compactifications in caseof coarse spaces). A future paper will demonstrate the equivalence of Specker’sdefinition of ends of groups and our definition of them.The authors are grateful to Ross Geoghegan and Mike Mihalik for their help inunderstanding classical theory of ends of finitely generated groups.2. Coarse compactifications
In this section we define the concept of a coarse compactification of a propermetric space X and we give necessary and sufficient condition for the Freundenthalcompactification to be a coarse one. Definition 2.1. A coarse compactification of a proper metric space X is any com-pactification ¯ X of X that is dominated by its Higson compactification. Equiva-lently, any continuous function f : ¯ X → R restricts to a slowly oscillating function f | X : X → R .Recall g : X → R is slowly oscillating if for any r, ǫ > K of X such that x, y ∈ X \ K and d ( x, y ) < r implies | g ( x ) − g ( y ) | < ǫ . OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 3
The
Higson compactification h ( X ) of X is the one induced by all continuous andslowly oscillating functions f : X → [0 , g : h ( X ) → R are slowly oscillating when restricted to X and every continuousand slowly oscillating function f : X → [0 ,
1] extends over h ( X ) to a continuousfunction.Let us show a necessary and sufficient condition for the Freundenthal compacti-fication to be a coarse compactification. Theorem 2.2.
Suppose ¯ X is the Freundenthal compactification of a proper, con-nected, and locally connected metric space X . The following conditions are equiva-lent:1. ¯ X is a coarse compactification of X ,2. For each m > and each bounded subset K of X there is a bounded subset L ⊃ K of X such that for every x ∈ X \ L the m -ball B ( x, m ) is contained in acomponent of X \ K .Proof.
2) = ⇒ g : ¯ X → R is continuous and ǫ >
0. Since dim( ¯ X \ X ) = 0there are mutually disjoint open sets U i , i = 1 , . . . , n , such that ¯ X \ X ⊂ n S i =1 U i and the diameter of each g ( U i ) is less than ǫ . Notice K := X \ n S i =1 U i is a compactsubset of X , so given m > L ⊃ K such that B ( x, m ) is contained in acomponent of X \ K if x ∈ X \ L and that component is contained in some U i .Therefore | g ( x ) − g ( y ) | < ǫ if y ∈ B ( x, m ) and g is slowly oscillating.1) = ⇒ X is a coarse compactification of X , K is a bounded subset of X , and m >
0, then assume existence of two sequences x n , y n such that d ( x n , y n ) < m foreach n ≥ x n ∈ X \ K i is contained in a component C i of X \ cl ( K ), y n is containedin a component D i of X \ cl ( K ), where C i = D i . Since there are only finitely manyunbounded components of X \ cl ( K ), we may assume C i = C and D i = D forinfinitely many i . Also, we may assume ¯ x is the limit of the sequence { x i } i ≥ in¯ X , ¯ y is the limit of the sequence { y i } i ≥ in ¯ X . Those two points are differentcontradicting d ( x n , y n ) < m for each n ≥
1. Indeed, give ¯ X a metric ρ and define f : X → R as the distance to ¯ x . It is extendible over ¯ X , so it is slowly oscillating.Therefore | f ( x n ) − f ( y n ) | → f ( x n ) → f ( y n ) → ρ (¯ x, ¯ y ) = 0. (cid:3) Corollary 2.3. If X is a proper geodesic space, then its Freundenthal compactifica-tion is a coarse compactification of X .Proof. Given a bounded subset K of X and given m >
0, put L := B ( K, m ) andnotice B ( x, m ) is a subset of X \ K if x / ∈ L . Therefore B ( x, m ) is a subset of acomponent of X \ K if x / ∈ L . (cid:3) Glacial oscillations
In this section we define a concept in the spirit of slowly oscillating functionsand we use it to introduce coarse Freundenthal compactifications later on.
Definition 3.1. A glacial scale on a metric space X is a sequence of pairs { ( K i , n i ) } ∞ i =1 of bounded subsets of X and natural numbers such that for each pair ( K, r ) con-sisting of a bounded subset of X and r > i such that K ⊂ K i and n i > r . YUANKUI MA AND JERZY DYDAK
Given a glacial scale S = { ( K i , n i ) } ∞ i =1 , a chain of points x , . . . , x n in X iscalled an S - chain if for each i ≤ n − m ≥ x i , x i +1 / ∈ K m and d ( x i , x i +1 ) ≤ n m . Definition 3.2.
A function f : X → R is glacially oscillating if for each ǫ > S with the property that | f ( x ) − f ( x n ) | < ǫ for each S -chain x , . . . , x n . Observation 3.3.
One can introduce the concept of a subset A of X to be S -connectedand reword the above definition as requiring that the diameter of f ( A ) is less than ǫ for each S -connected subset A of X . Proposition 3.4. If ( X, d ) is an ultrametric space, then every slowly oscillating func-tion f : X → R is glacially oscillating.Proof. Recall that (
X, d ) is an ultrametric space if every triangle in X is isoscelesand the lengths of two equal sides are at least the size of the third side. Equivalently, d ( x, y ) ≤ max( d ( x, z ) , d ( y, z )) for all points x, y, z ∈ X .If f : X → R is slowly oscillating and ǫ >
0, then we can choose an increasingsequence { K n } n ≥ of non-empty bounded subsets of X such that B ( K n , n ) ⊂ K n +1 for each n ≥ | f ( x ) − f ( y ) | < ǫ if x, y / ∈ K n and d ( x, y ) ≤ n . Let S := { ( B ( K n , n ) , n ) } n ≥ and suppose x , . . . , x n is an S -chain. Pick j ≤ n − d ( x j , x j +1 ) is the maximum of all d ( x i , x i +1 ), i ≤ n −
1. Let M be the smallestinteger satisfying d ( x j , x j +1 ) ≤ M . Notice x j , x j +1 / ∈ B ( K M , M ), the distancefrom x to either x j or x j +1 is at most M , the distance from x n to either x j or x j +1 is at most M , hence d ( x , x n ) ≤ M . That implies x , x n / ∈ K M , in particular | f ( x ) − f ( x n ) | < ǫ . That proves f is glacially oscillating. (cid:3) Proposition 3.5.
Suppose f : X → X is close to id X and h : X → R . If h ◦ f isglacially oscillating, then so is h .Proof. f : X → X being close to id X means there is r > d X ( f ( x ) , x ) < r for all x ∈ X . Given ǫ > S = { ( K i , n i ) } i ≥ so that | h ◦ f ( x ) − h ◦ f ( x ) | < ǫ/ x , x ∈ X that can be connected by an S -chain in X . We may assume n > r by truncating S . Let S ′ = { B ( K i , r ) , n i } i ≥ .Now, given any S ′ -chain x , . . . , x n notice x , f ( x ) and x n , f ( x n ) are S -chains.Therefore | h ( x ) − h ( x ) | < ǫ . (cid:3) Proposition 3.6.
Suppose f : ( X, d X ) → ( Y, d Y ) is a coarse, large scale continuousfunction and g : Y → R .a. If g is glacially oscillating, then so is g ◦ f .b. If g ◦ f is glacially oscillating and f is a coarse equivalence, then g is glaciallyoscillating.Proof. f being coarse means f − ( K ) is bounded for each bounded subset K of Y . f being large scale continuous means that for each m ≥ M > d X ( x, y ) < m implies d Y ( f ( x ) , f ( y )) < M .a. Given ǫ > S = { ( K i , n i ) } i ≥ so that | g ( y ) − g ( y ) | < ǫ for any y , y ∈ Y that can be connected by an S -chain in Y . Put C i = f − ( K i )and let m i be the maximum of natural numbers such that d X ( x , x ) ≤ m i implies d Y ( f ( x ) , f ( x )) ≤ n i . Notice C = { ( C i , m i ) } i ≥ is a glacial scale in X and the im-age under f of any C -chain is an S -chain. Therefore, if x and y can be connected bya C -chain, f ( x ) and f ( y ) can be connected by an S -chain and | g ( f ( x )) − g ( f ( y )) | < ǫ . OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 5 b. Choose f ′ : Y → X that is coarse, large scale continuous such that f ◦ f ′ isclose to id Y . By a), g ◦ f ◦ f ′ is glacially oscillating and by 3.5, so is g . (cid:3) Corollary 3.7. If ( X, d ) is a metric space of asymptotic dimension , then everyslowly oscillating function f : X → R is glacially oscillating.Proof. As shown in [2] there is an ultrametric space coarsely equivalent to (
X, d ).Apply 3.6. (cid:3)
Definition 3.8.
A subset A of a metric space X is coarsely clopen if A and X \ A are coarsely disjoint, i.e. the characteristic function χ A of A is slowly oscillating on X .A basic property of coarsely clopen subsets of a metric space X is the following: Lemma 3.9. If A and C are coarsely clopen subsets of X , then so are A ∩ C , A \ C ,and A ∪ C .Proof. Notice χ A ∩ C = χ A · χ C , χ A \ C = χ A − χ A · χ C , and χ A ∪ C = χ A \ C + χ C \ A + χ A · χ C are all slowly oscillating if both χ A and χ C are slowly oscillating. (cid:3) Here is a description of coarsely clopen subsets of proper metric spaces:
Proposition 3.10. If ( X, d ) is a proper metric space and A is a subset of its Higsoncompactification h ( X ) , then A ∩ X is coarsely clopen in X if and only if cl ( A ) ∩ ( h ( X ) \ X ) and cl ( X \ A ) ∩ ( h ( X ) \ X ) are disjoint, where the closures are takenin h ( X ) .Proof. If cl ( A ) ∩ ( h ( X ) \ X ) and cl ( X \ A ) ∩ ( h ( X ) \ X ) are disjoint, then one cannotproduce two disjoint sequences, S := { x n } in A and S { y n } in X \ A such that x n → ∞ (that means each bounded subset K of X contains only finitely manymembers of the sequence) and d ( x n , y n ) < M for some M > n ≥ h ( X ) \ X as otherwise the characteristic function of cl ( S ) in cl ( S ∪ S ) extendsover h ( X ) to a continuous function f : h ( X ) → [0 ,
1] and f | X is slowly oscillatingcontradicting f ( x n ) = 1, f ( y n ) = 0 for all n ≥ C := A ∩ X is coarsely clopen, then the closure cl X ( C ) of C intersects cl X ( X \ C ) along a bounded subset K , so we may find a bounded opensubset U of X such that the characteristic function χ ( A \ U ) is continuous on X \ U and is slowly oscillating. It extends to a continuous function on h ( X ) \ U which isthe characteristic function of h ( X ) ∩ cl ( C \ U ) (the closure taken in h ( X )) whenrestricted to h ( X ) \ U . That proves cl ( A ) ∩ ( h ( X ) \ X ) and cl ( X \ A ) ∩ ( h ( X ) \ X )being disjoint. (cid:3) Proposition 3.11.
Given a subset A of a metric space X the following conditionsare equivalent:1. The characteristic function χ A of A is glacially oscillating,2. A is coarsely clopen,3. There is a glacial scale S with the property that any S -chain starting at a pointof A is completely contained in A .Proof.
1) = ⇒
2) is clear as χ A of A is glacially oscillating implies χ A of A is slowlyoscillating which is equivalent to A being coarsely clopen. YUANKUI MA AND JERZY DYDAK
2) = ⇒ n ≥ K n containing B ( x , n ) suchthat given two points x, y ∈ X \ K n at distance less than n , x ∈ A implies y ∈ A .Put S = { ( K n , n ) } n ≥ .3) = ⇒ ǫ > S -chain x , . . . , x n in X one has χ A ( x ) = χ A ( x n ). (cid:3) Corollary 3.12.
Given finitely many coarsely clopen subsets A i of a metric space X ,there is a glacial scale S with the property that any S -chain starting at a point ofsome A j is completely contained in A j .Proof. For each j ≤ n pick a glacial scale S j = { ( K ji , k ji } i ≥ with the propertythat any S j -chain starting at a point of A j is completely contained in A j . Define K i = n S j =1 K ji and k i = min( k i , . . . , k ni ). Notice that for S = { ( K i , k i ) } i ≥ any S -chain starting at a point of some A j is completely contained in A j . (cid:3) Corollary 3.13. If ( X, d ) is a metric space, then any slowly oscillating function f : X → R whose image is finite is glacially oscillating.Proof. Notice point inverses of f are coarsely clopen. Therefore f is a linear com-bination of glacially oscillating functions and is itself glacially oscillating. (cid:3) Proposition 3.14. If X is a geodesic space and f : X → R , then the followingconditions are equivalent:1. f is glacially oscillating,2. For each ǫ > there is a bounded subset K of X such that for every component C of X \ K the diameter of f ( C ) is at most ǫ .Proof.
1) = ⇒ ǫ > S = { ( K i , n i ) } i ≥ with the prop-erty that | f ( x ) − f ( x n ) | < ǫ/ S -chain x , . . . , x n . Put K = B ( K , n )and notice that every two points x, y in a component C of X \ K can be connectedby an n -chain in X \ K . That chain is also an S -chain, so | f ( x ) − f ( y ) | < ǫ/ C ) ≤ ǫ .2) = ⇒ n ≥ K n containing B ( x , n ) ∪ B ( K n − , n ) such that the diameter of f ( C ) is at most 1 /n for each component C of X \ K n . Given ǫ > k such that 1 /k < ǫ and consider S = { ( K i + k , i + k ) } i ≥ . Notice that any S -component C is contained in a componentof X \ K k , so diam( f ( C )) < ǫ . (cid:3) Lemma 3.15.
Suppose { K i } i ≥ is an increasing sequence of bounded subsets of ametric space X , n i is a strictly increasing sequence of natural numbers, and A i isan n i -connected subset of X \ K i for each i ≥ . If A i ∩ ( X \ K i +1 ) ⊂ A i +1 for each i ≥ , then A := ∞ S i =1 A i is a coarsely open subset of X .Proof. Suppose x n ∈ A , x n → ∞ , y n / ∈ A and d ( x n , y n ) < M for each n ≥
1. Byswitching to subsequences of x n and y n we may assume there is a strictly increasingsequence m ( n ) such that x n , y n / ∈ K m ( n ) for each n ≥ x n would belong to the same bounded subset of X ). Also, we mayassume m (1) > M . There is p ≥ x ∈ A p . If p < m (1), then x ∈ A m (1) ,so we may assume p ≥ m (1). Now, y ∈ A p , a contradiction. (cid:3) OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 7
Proposition 3.16.
Given a bounded function f : X → R on a metric space X , thefollowing conditions are equivalent:1. f is glacially oscillating.2. For every ǫ, M > there exist a bounded subset K of X such that diam( f ( U )) < ǫ for every M -component U of X \ K .3. For every ǫ > there exist a bounded subset K of X and finitely many coarselyclopen subsets U i , i ≤ p , covering X \ K such that diam( f ( U i )) < ǫ for each i ≤ p .Proof.
1) = ⇒ S = { ( K i , n i ) } ∞ i =1 for f and ǫ/
2. There is j ≥ n j > M . If U is an M -component of X \ K j then any two pointsof U can be connected by an S -chain. Therefore diam( f ( U )) < ǫ .2) = ⇒ x ∈ X . By induction create a sequence n i of natural numberssuch that diam( f ( U )) < ǫ/ i +1 for every 2 i -component U of X \ B ( x , i ) and n i +1 > n i + 2 i for each i ≥ f ( X ) by p intervals I j of size ǫ/
2. Given j ≤ p and i ≥ U ij of all 2 i -components U of X \ B ( x , n i ) such that f ( U ) intersects B ( I j , ǫ/ − ǫ/ j − ),where I j is the j -th interval. Put U j = ∞ S i =1 U ij . By 3.15 each U j is a coarsely clopensubset of X . Notice that diam( f ( U j )) < ǫ .3) = ⇒ ǫ > K of X and finitely many coarselyclopen subsets U i , i ≤ n , covering X \ K such that diam( f ( U i )) < ǫ for each i ≤ n . We may assume U i , i ≤ n , are mutually disjoint by applying 3.9. By 3.12there is a glacial scale S with the property that any S -chain starting at a point ofsome U j is completely contained in U j . We may assume the first bounded set of S contains K . Therefore, for any two points x, y ∈ X joinable by an S -chain one has | f ( x ) − f ( y ) | < ǫ as they are contained in one of sets U i , i ≤ n . Thus f is glaciallyoscillating. (cid:3) Coarse Freundenthal compactification
In this section we introduce the coarse Freundenthal compactification of propermetric spaces in a way similar to the Higson compactification. That approachshould be of use to researchers in geometric group theory. Later on we will presenta different approach that is more suitable for researchers in coarse topology.
Definition 4.1.
The coarse Freundenthal compactification of a proper metric space X is the maximal compactification CF ( X ) of X with the property that any con-tinuous and glacially oscillating function f : X → [0 ,
1] extends over CF ( X ) to acontinuous function. Proposition 4.2. If f : X → Y is a coarse, large scale continuous function betweenproper metric spaces, then it extends uniquely to a continuous function of coarseFreundenthal compactifications.Proof. Given a continuous and glacially oscillating function g : Y → [0 , g ◦ f is also continuous and glacially oscillating by 3.6. Thus f extends to a continuousfunction of coarse Freundenthal compactifications and that extension is unique as X is dense in CF ( X ). (cid:3) Proposition 4.3. If ψ ( X ) is a coarse compactification of a proper metric space X such that the corona ψ ( X ) \ X is of dimension , then for every continuous f : ψ ( X ) → R its restriction f | X to X is glacially oscillating. YUANKUI MA AND JERZY DYDAK
Proof.
Given ǫ > C i , i ≤ n , covering ψ ( X ) \ X such that diam( f ( C i )) < ǫ/ i ≤ n . Extend each C i to an opensubset U i of ψ ( X ) such that diam( f ( U i )) < ǫ . Notice each U i is coarsely clopen,so by 3.12 there is a glacial scale S with the property that any S -chain starting atsome U j is completely contained in that particular U j . That proves f | X is glaciallyoscillating as we may assume each bounded subset used by S contains n S i =1 ( X \ U i )which is a compact subset of X . (cid:3) Proposition 4.4.
Suppose ψ ( X ) is a compactification of a proper metric space X . Iffor every continuous f : ψ ( X ) → R its restriction f | X to X is glacially oscillating,then the corona ψ ( X ) \ X is of dimension .Proof. Given two disjoint closed subsets A and A of ψ ( X ) \ X choose a continuousfunction f : ψ ( X ) → [0 ,
1] such that f ( A ) ⊂ { } and f ( A ) ⊂ { } . Put ǫ = 1 / S with the property that any S -chain is mapped by f to a subset of diameter less than 1 /
4. Put U = f − [0 , /
4] and U = f − (3 / , V i , i = 1 ,
2, be the set of all points of X than can be connected to U i by an S -chain. Since we may assume each bounded set in the description of S is compact,each V i is open in X . Notice each V i is coarsely clopen (see 3.11) and V ∩ V = ∅ .Therefore C i := cl ( V i ) ∩ ( ψ ( X ) \ X ) is clopen in ψ ( X ) \ X (see 3.10), it contains A i , and C ∩ C = ∅ . That proves ψ ( X ) \ X is of dimension 0. (cid:3) Corollary 4.5.
The coarse Freundenthal compactification of a proper metric space X is the maximal coarse compactification whose corona is of dimension .Proof. By 4.4 the corona of the Freundenthal compactification is of dimension 0.By 4.3 any coarse compactification whose corona is of dimension 0 is dominated bythe Freundenthal compactification. (cid:3)
Proposition 4.6.
If two proper metric spaces ( X, d X ) and ( Y, d Y ) are coarsely equiv-alent, then their Freundenthal coronas CF ( X ) \ X and CF ( Y ) \ Y are homeomor-phic.Proof. Choose coarse equivalences f : X → Y and g : Y → X such that g ◦ f isclose to id X and f ◦ g is close to id Y . Case 1:
Both X and Y are discrete as topological spaces. Using 4.2 one can seethat f, g induce continuous function CF ( f ) : CF ( X ) → CF ( Y ), CF ( g ) : CF ( Y ) → CF ( X ). It is known that the induced functions h ( f ) : h ( X ) → h ( Y ), h ( g ) : h ( Y ) → h ( X ) have the property that h ( g ) ◦ h ( f ) is the identity on h ( X ) \ X and h ( f ) ◦ h ( g ) isthe identity on h ( Y ) \ Y . Since coarse Freundenthal compactifications are dominatedby Higson compactifications, CF ( g ) ◦ CF ( f ) is the identity on CF ( X ) \ X and CF ( f ) ◦ CF ( g ) is the identity on CF ( Y ) \ Y . Case 2: X is a discrete subset of Y and i : X → Y is a coarse equivalence. Givena glacially oscillating function k : X → [0 , k ′ : Y → [0 ,
1] by [10] (see [9] for an earlier version of that resultnot involving continuity). k ′ being close to a glacially oscillating function (namely k ◦ r , where r : Y → X is a coarse inverse of i ) is itself glacially oscillating by 3.5.That means the closure cl ( X ) of X in CF ( Y ) equals CF ( X ) and CF ( X ) \ X = CF ( Y ) \ Y . General Case:
Choose discrete subsets X ′ of X and Y ′ of Y such that inclusions X ′ → X and Y ′ → Y are coarse equivalences. Apply Case 2 and then Case 1. (cid:3) OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 9
Corollary 4.7. If ( X, d ) is a proper geodesic space, then its Freundenthal compacti-fication is the coarse Freundenthal compactification.Proof. Apply 3.14. (cid:3) Ends of groups
In this section we show that the number of elements of the coarse Freundenthalcorona of countable groups generalizes the number of ends of finitely generatedgroups.Given a countable group G one considers all proper metrics d on G that areleft-invariant (that means d ( g · h , g · h ) = d ( h , h ) for all g, h , h ∈ G ). It turnsout id : ( G, d ) → ( G, d ) is always a coarse equivalence for any such metrics d and d . Be aware that considering right-invariant metrics d (while keeping d left-invariant) may lead to id : G → G not being a coarse equivalence (see [3]).If G is finitely generated, then any word metric will do. Example 5.1.
One way to introduce a proper left-invariant metric d on a countablegroup G that is not finitely generated is as follows:1. Choose generators g i , i ≥ , of G such that g n +1 does not belong to the subgroupof G generated by g , . . . , g n .2. Choose a strictly increasing sequence of positive integers n i ,3. Assign each g ai , where a = ± , the norm n i and put | G | = 0 ,4. Assign to each g ∈ G the norm minimizing the sum of norms of g ai appearing inall possible expressions of g as the product of generators,5. Define d ( g, h ) as the norm of g − · h . Observation 5.2.
A subset A of a countable group G is coarsely clopen if and onlyif it is almost invariant.Proof. Recall that A is almost invariant if for every g ∈ G the symmetric difference A ∆( A · g ) is finite.Put a proper left-invariant metric d on G . Suppose A is coarsely clopen but notalmost invariant. Choose g ∈ G such that A ∆( A · g ) is not finite. Either there isa sequence x n in A diverging to infinity such that x n · g / ∈ A for all n ≥ x n in A diverging to infinity such that x n · g − / ∈ A for all n ≥ y n = x n · g in the first case and y n = x n · g − in the second case. Observe d ( x n , y n ) = d (1 G , g ) for all n ≥ A being coarsely clopen.Suppose A is almost invariant but not coarsely clopen. There is a sequenceof points x n in A diverging to infinity and M > y n ∈ G \ A , d ( x n , y n ) < M for all n ≥
1. Therefore x − n · y n ∈ B (1 G , M ) for all n ≥
1. Since B (1 G , M ) is finite, we may assume, without loss of generality, thatthere is g ∈ B (1 G , M ) such that x − n · y n = g for all n ≥
1. Hence x n · g = y n forall n ≥ A ∆( A · g ) is not finite, a contradiction. (cid:3) Definition 5.3.
The number of ends of a countable group G is the cardinality of theFreundenthal corona CF ( G ) \ G , where G is equipped with a proper left-invariantmetric.Our next result shows that the above definition does generalize the classic defi-nition of the number of ends for finitely generated groups. Proposition 5.4. If G is finitely generated, then its number of ends equals the car-dinality of the Freundenthal corona of any Cayley graph of G .Proof. Equip G with a word metric based on a symmetric set of generators S . Noticethe inclusion G → Σ( G, S ) from G to the Cayley graph is a coarse equivalence, hence CF ( G ) \ G is homeomorphic to the coarse Freundenthal corona of Σ( G, S ) (see 4.6)and that corona is equal to the ends of G by 4.7. (cid:3) Here is another way to introduce the number of ends of countable groups:
Proposition 5.5.
Let G be a countable group.1. G has ends if it is finite.2. G has one end if every almost invariant subset A of G is either finite or itcomplement is finite.3. The number of ends of G is the supremum of n ≥ such that there are n mutually disjoint non-finite almost invariant subsets of G .Proof. If CF ( G ) \ G has at least n points, then it contains at least n non-emptyclopen sets C i that are mutually disjoint. We can extend them to mutually disjointopen subsets U i of CF ( G ) such that C i = cl ( U i ) ∩ ( CF ( G ) \ G ) for each i ≤ n . By3.10 each U i is coarsely clopen and by 5.2 each U i is almost invariant.Cnversely, the existence of n mutually disjoint non-finite almost invariant subsetsof G implies that CF ( G ) \ G contains at least n points. (cid:3) Proposition 5.6. If G is an infinite locally finite group, then its number of ends isinfinite.Proof. Recall that a group H is locally finite if each finite subset of it is contained ina finite subgroup of H . H is locally finite if and only if its asymptotic dimension is 0.By 4.5 the Higson compactification of G is the coarse Freundenthal compactificationof G . It is well-known that the Higson corona of unbounded metric spaces is infinite. (cid:3) Proposition 5.7.
Suppose a group G is the union of an increasing sequence of itsnon-locally finite subgroups { G i } i ≥ . If A is a coarsely clopen infinite subset of G ,then there is n ≥ such that A ∩ G n is infinite.Proof. Suppose A ∩ G i is finite for each i ≥
1. Since G is not of asymptoticdimension 0, there is m ≥ G contains arbitrarily long m -chains. Choose k ≥ B ( A, m ) ∩ B ( G \ A, m ) ⊂ B (1 G , k ), then find m > k such that B (1 G , k ) ⊂ G m and then there is a ∈ A m +1 \ G m . Pick an m -chain C in G that islonger than the number of elements in A m +1 . By translating (i.e. switching from C to g · C for some g ∈ G ) we may assume C starts at 1 G . Notice a · C is completelyoutside of G m , so a · C ⊂ A . Hence a · C ⊂ A m +1 , a contradiction. (cid:3) Definition 5.8. NCC is a shortcut for non-trivial coarsely clopen subsets Y of ametric space X , i.e. those coarsely clopen subsets that are infinite and X \ Y isinfinite. Lemma 5.9.
Suppose G contains three NCC sets that are disjoint. If G is not locallyfinite, then it acts trivially on at most one of the three NCC sets.Proof. G acts trivially on an NCC set E means the symmetric difference E ∆( g · E )is finite for each g ∈ G . OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 11
Suppose G acts trivially on disjoint NCC sets E , E and E is an NCC setsdisjoint from E ∪ E . Using 5.7 we may reduce the proof to G being finitelygenerated. Equip G with a left-invariant word metric d . Find a bounded subset K of G containing 1 G such that if i = j and g ∈ E i \ K , h ∈ E j \ K , then d ( g, h ) > E := G \ ( E ∪ E ∪ E ). Either E is an NCC orit is finite. Find m ≥ x ∈ E i , i ≤
3, of norm at least m , B ( x, · diam ( K ) + 2) is contained in E i . If E is unbounded, require the sameproperty for E , otherwise require that B ( x, diam ( K ) + 1) is disjoint with E .In E find an element g of the norm bigger than m . Hence g · K ⊂ E .Since E ∆( g · E ) is finite, choose g ∈ E of the norm larger than m such that g · g ∈ E . Given a 1-chain c joining g to g ∈ K , it stays in E until it hits K for the first time. Truncate c to include only those elements of G . Now, g · c is a1-chain starting in E and ending in E . Therefore it hits K at certain moment.That means existence of x ∈ E such that g · x ∈ K . Similarly, we can find x ∈ E such that g · x ∈ K . That means g − · K intersects both E and E , acontradiction as that set is contained exclusively in only one of E i , i ≤
4, due tothe norm of g − being larger than m . (cid:3) Theorem 5.10. If G is a countable group, then the number of ends of G is eitherinfinite or at most .Proof. If G is finite, then Ends ( G ) is empty. If G is locally finite and infinite, then Ends ( G ) is infinite by 5.6.Assume G is infinite, not locally finite, its number of ends is finite, and it containsthree NCC sets that are disjoint. Since G acts on its ends via the left multiplication,there is a subgroup H of G of finite index that acts on Ends ( G ) trivially. By 4.6(Case 2) H acts trivially on Ends ( H ) which is equal to Ends ( G ). This contradictsLemma 5.9. (cid:3) Theorem 5.11.
Suppose G is a countable group. If G is the union of an increasingsequence { G n } n ≥ of its subgroups that have finitely many ends (that have at most m ends), then the number of ends of G is at most (is at most m ).Proof. Suppose G has m + 1 mutually disjoint NCC sets E i , i ≤ m + 1. By 5.7 wecan find an index n such that each E i ∩ G n is an NCC set in G n , a contradiction. (cid:3) Theorem 5.12. If G is a countable group with ends, then it is finitely generated.Therefore it contains an infinite cyclic subgroup of finite index.Proof. Suppose G is a countable group with 2 ends that is not finitely generated.We will show that there exists a subgroup H of G of index at most 2 and a strictlyincreasing sequence H n of subgroups of H satisfying the following conditions:1. H is infinite cyclic,2. H n is of finite index in H n +1 for each n ≥ H is the union of all H n , n ≥ G acts on its ends Ends ( G ) and it has a subgroup H acting on Ends ( G ) = Ends ( H ) trivially. Express H as the union of two disjoint NCC sets E , E whichare almost invariant in H . Given a finite subset F of H we can find using 5.7 afinitely generated subgroup H F of H such that both E ∩ H F and E ∩ H F are NCCsets in H F . By a theorem of Mike Mihalik (see [14], Theorem 1.2.12) H F cannothave infinitely many ends as for such groups H F · E is dense in Ends ( H F ) for anyend E . In particular, there is g ∈ H F such that ( g · E ∩ H F ) ∩ E is an NCC set in H F , a contradiction. Thus H F has exactly 2 ends. By a theorem of J.Stallings, H F has an infinite cyclic subgroup of finite index. In particular, if we constructtwo subgroups H F ⊂ H F ′ that way, then H F is of finite index in H F ′ . Using thesefact it is easy to construct the required sequence H n of subgroups of H .Suppose H has an NCC set C . There is m ≥ E ∩ H m and E c ∩ H m are both infinite. Let t be a generator of H . Since both E ∆( E · t ) and E c ∆( E c · t )are finite, there is k > m such that both these sets are contained in H k . Given x ∈ E ∩ H k +1 \ H k one has x · t ∈ E as x · t / ∈ H k . Consequently, x · t n ∈ E for allinteger n . The set A := { x · t n } n ∈ Z is isometric to H as d ( x · t i , x · t j ) = d ( t i , t j ) for all i, j ∈ Z . Let f : H k +1 → H be a coarse inverse of the inclusion H → H k +1 . Since f | A : A → H is a coarse embedding, it must be a coarse equivalence. Thereforethe inclusion A → H k +1 is a coarse equivalence and A ∩ E ought to be an NCC setin A contradicting A ⊂ E . (cid:3) Corollary 5.13.
The group of rational numbers has end. More generally, anycountable subgroup of reals has one end if it is not finitely generated. Ends of coarse spaces
In this section we generalize the concept of Freundenthal ends to arbitrary coarsespaces. See [7] for other ways to introduce ends in coarse spaces.We follow a description of coarse spaces (quite often our terminology is that of large scale spaces ) as in [6]. It is equivalent to Roe’s definition of those spaces in[17].Recall that a star st ( x, U ) of x ∈ X with respect to a family U of subsets of X isdefined as the union of U ∈ U containing x . If A ⊂ X , then st ( A, U ) = S x ∈ A st ( x, U ).Given two families U , V of subsets of X , st ( U , V ) is defined as the family st ( A, V ), A ∈ U . Definition 6.1. A large scale space is a set X equipped with a family LSS of covers(called uniformly bounded covers) satisfying the following two conditions:1. st ( U , V ) ∈ LSS if U , V ∈
LSS .2. If
U ∈
LSS and every element of V is contained in some element of U , then V ∈
LSS .Sets which are contained in an element of
U ∈
LSS are called bounded . Definition 6.2.
A subset A of a large scale space X is coarsely clopen if for everyuniformly bounded cover U of X the set st ( A, U ) ∩ st ( X \ A, U ) is bounded.A non-trivial coarsely clopen subset A of a large scale space X (an NCC-set forshort) is one that is not bounded and X \ A is not bounded. Lemma 6.3. st ( A ∩ A , U ) ∩ st (( A ∩ A ) c , U ) ⊂ st ( A , U ) ∩ st (( A ) c , U ) ∪ st ( A , U ) ∩ st (( A ) c , U ) .Proof. Suppose x ∈ st ( A ∩ A , U ) ∩ st (( A ∩ A ) c , U ). There is y ∈ A ∩ A satisfying x ∈ st ( y, U ) and there is z ∈ A c ∪ A c satisfying x ∈ st ( z, U ). Thus either x ∈ st (( A ) c , U ) or x ∈ st (( A ) c , U ) and we are done. (cid:3) Corollary 6.4.
The intersection of two coarsely clopen subsets of X is coarselyclopen.Proof. Apply 6.3. (cid:3)
OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 13
Definition 6.5.
A topology on X is compatible with the large scale structure on X if there is a uniformly bounded cover of X consisting of open subsets of X . Observation 6.6.
The simplest non-trivial topology compatible with a large scalestructure is the discrete topology.
Definition 6.7. A topological coarse space is a set equipped with large scale structureand with a compatible topology. Additionally, we assume that the coarse structureis coarsely connected , i.e. the union of two bounded subsets of X is bounded. Lemma 6.8. If A is a coarsely clopen subset of X , then st ( A, U ) is a coarsely clopensubset of X for each uniformly bounded cover U of X .Proof. Notice st ( st ( A, U ) , V ) ⊂ st ( A, st ( U , V )) for any two covers U , V . There-fore st ( st ( A, U ) , V ) ∩ st ( st ( A c , U ) , V ) ⊂ st ( A, st ( U , V )) ∩ st ( A c , st ( U , V )). Since st ( A, U ) c ⊂ A c ⊂ st ( A c , U ) the proof is completed. (cid:3) Lemma 6.9. If A is a coarsely clopen subset of X , then a subset C of A is coarselyclopen provided A ⊂ st ( C, V ) for some uniformly bounded cover V of X .Proof. Observe C ′ := ( st ( A c , V )) c ⊂ C is coarsely clopen by 6.8 and B := C \ C ′ ⊂ A ∩ st ( A c , V ) is bounded as st ( A, V ) ∩ st ( A c , V ) is bounded. Adding a boundedset B to a coarsely clopen subset preserves being coarsely clopen as can be easilyseen. (cid:3) Definition 6.10. An end of a topological large scale space X is a family E of un-bounded, open and coarsely clopen subsets of X that is maximal with respect tothe property of being closed under intersections. Definition 6.11.
Let T be the topology of a topological large scale space X , we extend the topology T over X ∪ Ends ( X ) as follows: Y ⊂ X ∪ Ends ( X ) is declaredopen if Y ∩ X is open in X end for each end E ∈ Y there is an open coarsely clopenset U such that U ∈ E and U ⊂ Y . Proposition 6.12.
The topology on
Ends ( X ) is independent of the topology on X as long as the topology is compatible with the coarse structure.Proof. Suppose U is an open and uniformly bounded cover of X . If E is an endof X in the discrete topology, then there is a unique end E ′ of X containing allsets st ( A, U ), A ∈ E (use 6.8 and 6.9). Therefore Ends ( A ) and Ends ( st ( A, U )) areidentical for any subset A of X and the proof is completed. (cid:3) Recall that a compact space is totally disconnected if its components are sin-gletons. Equivalently, it has a basis of open closed subsets (see [11]) which is ourpreferred point of view.
Proposition 6.13. X ∪ Ends ( X ) is large scale compact.2. Ends ( X ) is compact Hausdorff and totally disconnected.3. X ∪ Ends ( X ) is Hausdorff if X is Hausdorff.Proof. X ∪ Ends ( X ) being large scale compact means that for any open cover { U s } s ∈ S of it there is a finite subset F of S such that Ends ( X ) ⊂ S s ∈ F U s and X \ S s ∈ F U s is a bounded subset of X (see [8]). Claim 1:
Given a family { U s } of open coarsely clopen subsets of X such that Ends ( X ) ⊂ S s ∈ S Ends ( U s ), there is a finite subset F of S such that Ends ( X ) ⊂ S s ∈ F Ends ( U s ) and X \ S s ∈ F U s is a bounded subset of X . Proof of Claim 1:
Consider a uniformly bounded and open cover V of X . Let V s := st ( U s , V ), C s := cl ( U s ) for each s ∈ S . Those are coarsely clopen subsets of X by 6.8 and by 6.9 as C s ⊂ V s for each s ∈ S .Consider the family X \ S s ∈ F C s , F a finite subset of S . It cannot be extendedto an end of X as such an end cannot belong to S s ∈ F Ends ( U s ), so there is F suchthat B := X \ S s ∈ F C s is bounded.To show C := X \ S s ∈ F U s is bounded define B s as st ( U s , V ) ∩ st ( X \ U s , V ). Weplan to show C ⊂ B ∪ S s ∈ F B s . Suppose x ∈ C \ B . There is t ∈ F so that x ∈ C t ,hence x ∈ C t \ U t . Therefore x ∈ st ( U t , V ) ∩ st ( X \ U t , V ) = B t . Thus C is bounded.Now, S s ∈ F Ends ( U s ) must contain Ends ( X ) as otherwise there is an end E con-taining each X \ C s , s ∈ F , hence also containing X \ S s ∈ F C s , a contradiction. Claim 2: X is an open subset of X ∪ Ends ( X ). Proof of Claim 2:
Suppose x ∈ X and E is an end of X . Pick a uniformly boundedcover U of X consisting of open subsets. Let x ∈ V ∈ U . Notice cl ( V ) is bounded(as it is contained in st ( V, U ), so W := X \ cl ( V ) ∈ E and ( W ∪ Ends ( W )) ∩ V = ∅ . Claim 3:
Two different ends of X have disjoint neighborhoods in X ∪ Ends ( X ). Proof of Claim 3:
Suppose E = E are two different ends of X . There is U ∈ E \ E , hence there is V ∈ E such that U ∩ V is bounded in view of 6.4. Choosea uniformly bounded cover U of X consisting of open subsets. Let W := st ( U , U ).By 6.9 both A U := U \ st ( U ∩ V, W ) and A V := V \ st ( U ∩ V, W ) are coarselyclopen. Notice U ′ := st ( A U , U ) ∈ E , V ′ := st ( A V , U ) ∈ E are disjoint, hence U ′ ∪ Ends ( U ′ ) is a neighborhood of E , V ′ ∪ Ends ( V ′ ) is a neighborhood of E andthey are disjoint.1. Follows from Claim 1.2. Ends ( X ) being compact Hausdorff follows from Claims 1-3. Suppose V isa neighborhood of the end E in X ∪ Ends ( X ). Choose an open coarsely clopensubset U so that U ∈ E and U ∪ Ends ( U ) ⊂ V . Choose a bounded and open cover W of X such that st ( U, W ) ∩ st ( X \ U, W ) is bounded. By 6.8 the set st ( X \ U, W )is open and coarsely clopen. Notice Ends ( st ( X \ U, W )) ∩ Ends ( U ) = ∅ and theirunion is Ends ( X ). Thus Ends ( U ) is clopen in Ends ( X ) and Ends ( X ) is totallydisconnected.3. Follows from Claim 3 and the proof of Claim 2. (cid:3) Proposition 6.14.
Any continuous, coarse and large scale continuous function f : X → Y of topological coarse spaces extends to a continuous map ¯ f : X ∪ Ends ( X ) → Y ∪ Ends ( Y ) . If f, g : X → Y are close, then ¯ f | Ends ( X ) = ¯ g | Ends ( Y ) .Proof. Given an end E of Y the family f − ( E ) consists of unbounded coarselyclopen subsets of X , so any end of X containing that family is mapped by ¯ f to E .It is clear ¯ f is continuous. OARSE FREUNDENTHAL COMPACTIFICATION AND ENDS OF GROUPS 15
Suppose f, g : X → Y are close. There is an open uniformly bounded cover U of Y with the property that f ( x ) ∈ st ( g ( x ) , U ) for all x ∈ X . Suppose ¯ f ( E ) = ¯ g ( E ) forsome end E of X . As in 6.8 there are V ∈ ¯ f ( E ) and W ∈ ¯ g ( E ) such that st ( V, U ) ∩ st ( W, U ) = ∅ . Therefore f − ( V ) ∩ g − ( W ) = ∅ contradicting f − ( V ) , g − ( W ) ∈ E . (cid:3) Corollary 6.15.
If two topological coarse spaces X and Y are coarsely equivalent,then Ends ( X ) is homeomorphic to Ends ( Y ) .Proof. By 6.12 we may assume both X and Y are equipped with the discretetopology. Apply 6.14. (cid:3) Lemma 6.16. If ( X, d ) is a metric space and f : X → R is glacially oscillating, thenfor each compact subset C of an open set U ⊂ R there is a coarsely clopen subset A of X such that f − ( C ) ⊂ A ⊂ f − ( U ) .Proof. Choose ǫ > B ( C, ǫ ) ⊂ U . Then choose a glacial scale S such that | f ( x ) − f ( y ) | < ǫ if x and y can be connected by an S -chain. Define A as all points in X that can be connected to f − ( C ) by an S -chain. Clearly, f − ( C ) ⊂ A ⊂ f − ( U )and A is coarsely clopen by 3.11. (cid:3) Theorem 6.17. If ( X, d ) is a proper metric space, then id X : X → X extends toa homeomorphism from the coarse Freundenthal compactification CF ( X ) to X ∪ Ends ( X ) .Proof. It suffices to show that for any glacially oscillating function f : X → [0 , E of X the set C := T A ∈ E cl ( f ( A )) consists of one point. Suppose thereare two different points a, b ∈ C and put ǫ = | a − b | /
4. By 6.16 there is a coarselyclopen subset A ′ of X such that f − [ a − ǫ, a + ǫ ] ⊂ A ′ ⊂ f − ( a − ǫ, a + 3 ǫ ). The set K := f − [ a − ǫ, a + ǫ ] cannot be bounded as in such case removing it from elementsof E would contradict a ∈ C . Similarly, the complement of A ′ cannot be bounded.Since A ′ / ∈ E , there is A ∈ E such that A ∩ A ′ is bounded. Hence A \ A ′ ∈ E and b / ∈ cl ( f ( A \ A ′ )), a contradiction. (cid:3) Remark . Now we can extend the definition of the space of ends of arbitrarygroup G by giving it the following large scale structure: uniformly bounded coversare those that refine covers of the form { g · F } g ∈ G for some finite subset F of G .The same can be done for locally compact topological groups. Instead of F beingfinite we consider neighborhoods of 1 G with compact closure. References [1] M. Bridson and A. Haefliger,
Metric spaces of non-positive curvature , Springer- Verlag,Berlin, 1999.[2] N.Brodskiy, J.Dydak, J.Higes, and A.Mitra,
Dimension zero at all scales , Topology and itsApplications, 154 (2007), 2729–2740.[3] N.Brodskiy, J.Dydak, and A.Mitra,
Coarse structures and group actions , Colloquium Math-ematicum 111 (2008), 149–158.[4] Yves Cornulier,
On the space of ends of infinitely generated groups , Topology Appl. 263(2019) 279-298[5] C. Drutu, M. Kapovich,
Geometric group theory , Colloquium publications, Vol. 63, AmericanMathematics Society (2018).[6] J.Dydak and C.Hoffland,
An alternative definition of coarse structures , Topology and itsApplications 155 (2008) 1013–1021 [7] Jerzy Dydak,
Ends and simple coarse structures , Mediterranean Journal of Mathematics(2020) 17: 4, arXiv:1801.09580[8] Jerzy Dydak,
Unifying large scale and small scale geometry , arXiv:1803.09154 [math.MG][9] J. Dydak, A. Mitra,
Large scale absolute extensors , Topology and Its Applications 214 (2016),51–65. arXiv:1304.5987[10] Jerzy Dydak, Thomas Weighill,
Extension Theorems for Large Scale Spaces viaCoarse Neighbourhoods , Mediterranean Journal of Mathematics, (2018) 15: 59. https://doi.org/10.1007/s00009-018-1106-z [11] R. Engelking,
Theory of dimensions finite and infinite , Sigma Series in Pure Mathematics,vol. 10, Heldermann Verlag, 1995.[12] R. Geoghegan,
Topological methods in group theory , Graduate Texts in Mathematics, vol.243, Springer, New York, 2008.[13] M. Gromov,
Asymptotic invariants for infinite groups , in Geometric Group Theory, vol. 2,1–295, G. Niblo and M. Roller, eds., Cambridge University Press, 1993.[14] Mike Mihalik,
A Manual for Ends, Semistability and Simple Connectivity at Infinity forGroups and Spaces , to appear[15] Georg Peschke,
The Theory of Ends , Nieuw Archief voor Wiskunde, 8 (1990), 1–12[16] E. Specker,
Endenverb¨ande von Raumen und Gruppen , Math. Ann. 122, (1950). 167–174.[17] J. Roe,
Lectures on coarse geometry , University Lecture Series, 31. American MathematicalSociety, Providence, RI, 2003.
Xi’an Technological University, No.2 Xuefu zhong lu, Weiyang district, Xi’an, China710021
Email address : [email protected] University of Tennessee, Knoxville, TN 37996, USA
Email address : [email protected] Xi’an Technological University, No.2 Xuefu zhong lu, Weiyang district, Xi’an, China710021
Email address ::