aa r X i v : . [ m a t h . GN ] A p r COARSE SPACES, ULTRAFILTERS AND DYNAMICALSYSTEMS
IGOR PROTASOV
Abstract.
For a coarse space ( X, E ), X ♯ denotes the set of all unboundedultrafilters on X endowed with the parallelity relation: p || q if there exists E ∈ E such that E [ P ] ∈ q for each P ∈ p . If ( X, E ) is finitary then there exists agroup G of permutations of X such that the coarse structure E has the base {{ ( x, gx ) : x ∈ X , g ∈ F } : F ∈ [ G ] <ω , id ∈ F } . We survey and analyze interplaysbetween ( X, E ), X ♯ and the dynamical system ( G, X ♯ ).1991 MSC: 54D80, 20B35, 20F69.Keywords: Coarse spaces, balleans, ultrafilters, dynamical systems.The dynamical ˇ S varc-Milnor Theorem and Gromov Theorem arose at the dawnof Geometric Group Theory.
In both cases, a group or a pair of groups act on somelocally compact spaces, see [22, Chapter 1].The Gromov coupling criterion wastransformed into the powerful tool in coarse equivalences (see references in [23]),however some natural questions on the coarse equivalence of groups need moredelicate combinatorial technique, see [4].In this paper, we describe and survey the dynamical approach to coarse spacesoriginated in the algebra of the Stone- ˇ C ech compactification. We identify theStone- ˇ C ech compactification βG of a discrete group G with the set of all ultrafilterson G . The left regular action G on G gives rise to the action of G on βG by( g, p ) gp , gp = { gP : P ∈ p } . In turn on, the dynamical system ( G, βG )induces on βG the structure of a right topological semigroup. The product pq ofultrafilters p, q is defined by A ∈ pq if and only if { g ∈ G : g − A ∈ q } ∈ p. Thesemigroup βG has very rich algebraic structure and the plenty of combinatorialapplications, see nice paper [5],capital book [6] or booklet [9].Let ( X, E ) be a coarse space. We denote by X ♯ the set of all ultrafilters p on X such that each member P ∈ p is unbounded in ( X, E ). Then we define theparallelity equivalence || on X ♯ by p || q if and only if there exists E ∈ E such that E [ P ] ∈ q for each P ∈ p . For p ∈ X ♯ , the orbit p = { q ∈ X ♯ : q || p } lookslike a smile apart of some hidden cat. This cat appears if ( X, E ) is finitary. ByTheorem 3.1, there exists a group G of permutations of X such that E has thebase {{ ( x, gx ) : x ∈ F } : F ∈ [ G ] <ω , id ∈ F } . In this case, X ♯ = X ∗ , X ∗ = βX \ X and p = Gp . But even ( X, E ) is not finitary, X ♯ contains some counterpart of thekernel of a dynamical system, see Theorem 2.3.Our goal is to clarify interplays between ( X, E ), X ♯ and the dynamical system( G, X ∗ ) in order to understand the dynamical nature of some extremal coarsespaces, in particular, tight, discrete and indiscrete.1. Coarse spaces
Given a set X , a family E of subsets of X × X is called a coarse structure on X if • each E ∈ E contains the diagonal △ X , △ X = { ( x, x ) ∈ X : x ∈ X } ; • if E , E ′ ∈ E then E ◦ E ′ ∈ E and E − ∈ E , where E ◦ E ′ = { ( x, y ) : ∃ z (( x, z ) ∈ E, ( z, y ) ∈ E ′ ) } , E − = { ( y, x ) : ( x, y ) ∈ E } ; • if E ∈ E and △ X ⊆ E ′ ⊆ E then E ′ ∈ E ; • S E = X × X .A subfamily E ′ ⊆ E is called a base for E if, for every E ∈ E , there exists E ′ ∈ E ′ such that E ⊆ E ′ . For x ∈ X , A ⊆ X and E ∈ E , we denote E [ x ] = { y ∈ X : ( x, y ) ∈ E } , E [ A ] = [ a ∈ A E [ a ] , E A [ x ] = E [ x ] ∩ A and say that E [ x ] and E [ A ] are balls of radius E around x and A .The pair ( X, E ) is called a coarse space [22] or a ballean [16], [21].For a coarse space ( X, E ), a subset B ⊆ X is called bounded if B ⊆ E [ x ] forsome E ∈ E and x ∈ X . The family B ( X, E ) of all bounded subsets of ( X, E ) iscalled the bornology of ( X, E ).A coarse space ( X, E ) is called finitary, if for each E ∈ E there exists a naturalnumber n such that | E [ x ] | < n for each x ∈ X .We classify subsets of a coarse space ( X, E ) by their size. A subset A of X iscalled • large if E [ A ] = X for some E ∈ E ; • small if L \ A is large for each large subset L ; • thick if, for each E ∈ E , there exists a ∈ A such that E [ a ] ⊆ A ; • prethick if E [ A ] is thick for some E ∈ E ; • thin (or discrete) if, for each E ∈ E , there exists a bounded subset B of X such that E A [ a ] = { a } for each a ∈ A \ B . OARSE SPACES, ULTRAFILTERS AND DYNAMICAL SYSTEMS 3
For finitary coarse spaces, the dynamical unification of above definitions will begiven in Section 3.Following [17], we say that two subets
A, B of X are • close (write AδB ) if there exists E ∈ E such that, A ⊆ E [ B ], B ⊆ E [ B ]; • linked (write AλB ) if either
A, B are bounded or there exist unboundedsubsets A ′ ⊆ A , B ′ ⊆ B such that A ′ δB ′ .We say that a coarse space E ∈ E is • δ -tight if any two unbounded subsets of X are close; • λ - tight if any two unbounded subsets of X are linked; • indiscrete if E ∈ E has no unbounded discrete subsets; • ultradiscrete if { X \ B : B ∈ B ( X, E ) } is an unltrafilter.We note that λ -tight spaces appeared in [2] under the name utranormal , δ -tightsubsets are called extremely normal in [14]and hypernormal in [1].An unbounded coarse space is called maximal if it is bounded in every strongercoarse structure. By [18, Theorem 3.1], every maximal coarse space is δ -tight. Aballean ( X, E ) is δ -tight if and only if every subset of X is large. If a λ -tight spaceis not indiscrete then it contains an ultradiscrete subspace [14, Theorem 2.2], soevery finitary λ -tight space is indiscrete.2. Ultrafilters
Let X be a discrete space and let βX denotes the Stone − ˇ Cech compactif ication of X . We take the points of βX to be the ultrafilter on X , with the points of X identified with the principal ultrafilters, so X ∗ = βX \ X is the set of all freeultrafilters. The topology of βX can be defined by stating that the sets of theform ¯ A = { p ∈ βX : A ∈ p } , where A is a subset of X , are base for the opensets. The universal property of βX states that every mapping f : X −→ Y , where Y is a compact Hausdorff space, can be extended to the continuous mapping f β : βX −→ X .Given a coarse space ( X, E ), we endow X with the discrete topology and denoteby X ♯ the set of all ultrafilters p on X such that each member P ∈ p is unbounded.Clearly, X ♯ is the closed subset of X ∗ and X ♯ = X ∗ if ( X, E ) is finitary.Following [10], we say that two ultrafilters p, q ∈ X ♯ are parallel (and write p || q )if there exists E ∈ E such that E [ P ] ∈ q for each P ∈ p . By [10, Lemma 4.1, 1], || is an equivalence on X ♯ . We denote by ∼ the minimal (by inclusion) closed (in X ♯ × X ♯ ) equivalence on X ♯ such that || ⊆∼ . The quotient ν ( X, E ) of X ♯ by ∼ iscalled the Higson corona of ( X, E ). For p ∈ X ♯ , we denote p = { q ∈ X ♯ : q || p } , ˘ p = { q ∈ p : q ∼ p } . IGOR PROTASOV
A function f : ( X, E ) −→ R is called slowly oscillating if, for every E ∈ E and ǫ >
0, there exists a bounded subset B of X such that diamf ( E [ x ]) < ǫ for each x ∈ X \ B .We recall [10] that a coarse space ( X, E ) is normal if any two asymptoticallydisjoint subsets A, B of X have disjoint asymptotic neighbourhoods. Two subsets A, B of X are called asymptotically disjoint if E [ A ] ∩ E [ B ] is bounded for each E ∈ E . A subset U of X is called an asymptotic neighbourhood of a subset A if E [ A ] \ U is bounded for each E ∈ E . By [10, Theorem 2.2], ( X, E ) is normal if andonly if, for any two disjoint and asymptotically disjoint subsets A, B of X , thereexists a slowly oscillating function f : X −→ [0 ,
1] such that f | A = 0, f | B = 1.By [11, Proposition 1], p ∼ q if and only if h β ( p ) = h β ( q ) for every slowlyoscillating function h : ( X, E ) −→ [0 , . By [4, Theorem 7], ( X, E ) is normal if and only if ∼ = cl || .By [17, Theorem 9 and Corollary 10], if λ ( X, E ) = λ ( X, E ′ ) then Higson coronas of( X, E ) and ( X, E ′ ) coincide and if ( X, E ) is normal then ( X, E ′ ) is normal.By [21, Theorem 2.1.1] a coarse space ( X, E ) is metrizable if E has a countablebase. If λ ( X, E ) = λ ( X, E ′ ) and ( X, E ) is metrizable then ( X, E ′ ) needs not to bemetrizable [17, Theorem 3]. Question 2.1 [17].
Let δ ( X, E ) = δ ( X, E ′ ) and ( X, E ) is metrizable. Is ( X, E ′ ) metrizable? If the answer to Question 2.1 would be positive then E = E ′ ,Let ( X, E ) be a coarse space. We say that a subset S of X ♯ is invariant if p ⊆ S for each p ∈ S . Every non-empty closed invariant subset of X ♯ contains a minimalby inclusion closed invariant subset. We denote K ( X ♯ ) = S { M : M is minimal closed invariant subset of X ♯ } . Theorem 2.2.
For p ∈ X ♯ , clp is a minimal closed invariant subset if and onlyif, for every P ∈ p , there exists E ∈ E such that p ∈ ( E [ P ]) ♯ .Proof. Apply arguments proving this statement for metric spaces [13, Theorem3.1]. ✷ Theorem 2.3.
For q ∈ X ♯ , q ∈ clK ( X ♯ ) if and only if each subset Q ∈ q isprethick.Proof. Apply arguments proving Theorem 3.2 in [13]. ✷ Theorem 2.4.
Let p, q be ultrafilters from X ♯ such that p , q are countable and clp ∩ clq = ∅ . Then either clp ⊆ clq or clq ⊆ clp. Proof.
Apply arguments proving Theorem 3.4 in [13]. ✷ OARSE SPACES, ULTRAFILTERS AND DYNAMICAL SYSTEMS 5 Dynamical systems
By a dynamical system we mean a pair (
G, T ), where T is a compact space, G is a group of homeomorphisms of G .The following two theorems make a bridge between coarse spaces and dynamicalsystems. For usage of Theorem 3.1 in corona constructions see [3].Let G be a transitive group of permutations of a set X . We denote by X G theset X endowed with the coarse structure with the base. {{ ( x, gx ) : g ∈ F } : F ∈ [ G ] <ω , id ∈ F } . Theorem 3.1.
For every finitary coarse space ( X, E ) , there exists a group G ofpermutations of X such that ( X, E ) = X G . Proof.
Theorem 1 in [12],for more general results see [8], [15]. ✷ Theorem 3.2. If ( X, E ) , ( X, E ′ ) are finitary coarse spaces and || ( X, E ) = || ( X, E ′ ) then E = E ′ .Proof. Theorem 15 in [17]. ✷ If ( X, E ) = X G , we say that X G is the G - realization of ( X, E ). Each G -realization of ( X, E ) defines the dynamical system ( G, X ∗ ) with the action ( g, p ) gp , gp = { gP : P ∈ p } . We note that p = Gp for each p ∈ X ∗ . Since the parallelity || is defined by means of entourages, the partition of X ∗ into G -orbits does notdepend on G -realizations of ( X, E ). It follows that if some property formulatedin terms of G -orbits of ( G, X ∗ ) is proved for some G -realization of ( X, E ) thenit holds for any G -realization. Moreover, by Theorem 3.2, every finitary coarsestructure can be uniquely recognized by the set of orbits in X ∗ .Given a finitary coarse space ( X, E ), its G -realization of X G , a subset A ⊆ X and p ∈ X ∗ , we define the p -companion of A by △ p ( A ) = A ∗ ∩ Gp.
Theorem 3.3.
For a subset A of ( X, E ) , the following statements hold (1) A is large iff △ p ( A ) = ∅ for each p ∈ X ∗ ; (2) A is thick iff △ p ( A ) = Gp for some p ∈ X ∗ ; (3) A is thin iff |△ p ( A ) | ≤ for each p ∈ X ∗ ;Proof. Theorem 3.1 and 3.2 in [20] . ✷ We recall that a dynamical system (
G, T ) is • minimal if each orbit Gx is dense in T ; IGOR PROTASOV • topologically transitive if some orbit Gx is dense in T .For a dynamical system ( G, T ), ker ( G, T ) denotes the closure of the union ofall minimal closed G -invariant subsets of T . Theorem 2.3 describes explicitely thekernel of the dynamical system ( G, X ∗ ) of X G . Theorem 3.4.
Let ( X, E ) be a finitary coarse space and ( X, E ) = X G . Then ( X, E ) is δ -tight if and only if the dynamical system ( G, X ∗ ) is minimal.Proof. Apply Theorem 3.3(1). ✷ Theorem 3.5.
Let ( X, E ) be a finitary coarse space and ( X, E ) = X G . Thenthe following statements are equivalent (1) ( X, E ) is λ -tight ; (2) for any infinite subset A, B of X , there exist p ∈ X ∗ and g ∈ G such that A ∈ p , B ∈ gp ; (3) for any family { A n : n ∈ ω } of infinite subsets of X , there exists p ∈ X ∗ such that A ∗ n ∩ Gp = ∅ for each n ∈ ω .Proof. (1) = ⇒ (2). Since A, B are linked, there exist A ′ ⊆ A , B ′ ⊆ B and H ∈ [ G ] <ω such that A ′ ⊆ HB ′ . We take p ∈ X ∗ such A ′ ∈ p . Then B ′ ∈ h − p forsome h ∈ H .(2) = ⇒ (3). We choose inductively a sequence ( g n ) n ∈ ω in G and a sequence( C n ) n ∈ ω of subsets of G such that C n ⊆ A n , g n C n ⊆ A n +1 , C n +1 ⊆ g n C n . Let h n = g n g n − . . . g . Then A ∩ h − A ∩ · · · ∩ h − n A n +1 = ∅ for each n ∈ ω. We take an arbitrary ultrafilter p ∈ X ♯ such that A ∩ h − ∩ · · · ∩ h − n A n +1 ∈ p for each n ∈ ω . Then Gp ∩ A ∗ n = ∅ for each n ∈ ω .(3) = ⇒ (1). Evident. ✷ Corollary 3.6. If ( X, E ) = X G and the dynamical system ( G, X ∗ ) is topologi-cally transitive then ( X, E ) is λ -tight. Remark 3.7.
Does there exist a group G of permutations of ω such that ω G is λ -tight and ( G, ω ∗ ) is not topologically transitive? This question can not beanswered in ZFC without additional assumptions. Yes, if t < c and No if t = c ,see [1, Theorems 5.2 and 5.3]. Theorem 3.8.
Let K be a closed nowhere dense subset of ω ∗ . Then thereexists a transitive group G of permutations of ω such that ker ( G, ω ∗ ) = K and theorbit Gp is dense in ω ∗ for each p / ∈ K . OARSE SPACES, ULTRAFILTERS AND DYNAMICAL SYSTEMS 7
Proof.
We take a filter φ on ω such that K = φ and φ has the base { A : A ∈ φ } .We denote by G the group of all permutations g of ω such that there exists A g such that g ( x ) = x for each x ∈ A g . Clearly, G is transitive.If q ∈ K then g ( q ) = q for each g ∈ G so K ⊆ ker ( G, ω ∗ ).We fix p ∈ ω ∗ \ K and take an arbitrary q ∈ ω ∗ , p = q . Let P ∈ p , Q ∈ q and P ∩ Q = ∅ . Since K is nowhere dense, there exists A ∈ φ such that P \ A ∈ p and Q \ A is infinite. By the definition of G , there exists g ∈ G such that g ( P \ A ) = Q \ A . Hence, Gp is dense in ω ∗ and K = ker ( G, ω ∗ ). ✷ Corollary 3.9.
There are c λ -tight finitary coarse spaces on ω which are not δ -tight.Proof. In light of Corollary 3.6, it suffices to notice that there are 2 c freeultrafilters on ω . ✷ Each orbit of a dynamical system from the proof of Theorem 3.8 is either denseor a singleton. We construct a topologically transitive (
G, ω ∗ ) having an infinitediscrete orbit. Example 3.10.
We partition ω into infinite subsets { W n : n ∈ Z } , fix abijection f n : W n −→ W n +1 and denote by f a bijection of ω such that f | W n = f n .For each n ∈ Z , we pick p n ∈ ω ∗ such that W n ∈ p n and denote by S the set ofall permutations g such that, for each g ( x ) = x , x ∈ W n .We take the group G of permutations generated by S ∪ { f } . Then Gp = { p n : n ∈ Z } and Gp is discrete.If p ∈ W ∗ and p = p then Gp is dense in ω ∗ so ( G, ω ∗ ) is topologically transitive. Remark 3.11.
If ( X, E ) is a finitary coarse space, ( X, E ) = X G then, byTheorem 2.3, every infinite subset of ( X, E ) is prethick if and only if ker ( G, X ∗ ) = X ∗ .If every infinite subset of a finitary coarse space ( X, E ) is thick then ( X, E ) isdiscrete. Indeed, if ( X, E ) is not discrete then, by Theorem 3.3(3), there exists q ∈ X ∗ and g ∈ G such that gq = q . We take Q ∈ q such that gQ ∩ Q = ∅ . Itfollows that Q is not thick.We say that a subset A of X G is • sparse if △ p ( A ) is finite for each p ∈ G ∗ ; • scattered if, for each infinite subset Y of A there exists p ∈ Y ∗ such that △ p ( Y ) is finite. Theorem 3.12.
Let ( X, E ) be a finitary coarse space and ( X, E ) = X G . Thenthe following statements are equivalent: IGOR PROTASOV (1) ( X, E ) is indiscrete; (2) for every infinite subset A of X , there exist p ∈ X ∗ and g ∈ G such that A ∈ p , A ∈ gp and p = gp ; (3) every infinite subset A of X is not sparse.Proof. The equivalence of (1) and (2) follows from Theorem 3.3(3), (3) = ⇒ (1)is evident.To show (2) = ⇒ (3), we choose a sequence ( g n ) n ∈ ω in G and sequence ( A n ) n ∈ ω of subsets of A such that A n +1 ⊂ A n , g n A n +1 ∩ A n +1 = ∅ , n ∈ ω and choose p ∈ X ∗ such that A n ∈ p for each n ∈ ω . Then Gp ∩ A ∗ is infinite and A is not sparse. ✷ Theorem 3.13.
A subset A of X G is scattered if and only if Gp is discrete foreach p ∈ A ∗ .Proof. Theorem 5.4 in [20] . ✷ Let G be a group of permutations of a set X . Let ( g n ) n ∈ ω be a sequence in G and let ( x n ) n ∈ ω be a sequence in X such that(1) { g ǫ . . . g ǫ n n x n : ǫ i ∈ { , }} ∩ { g ǫ . . . g ǫ m m x m : ǫ i ∈ { , }} = ∅ for all distinct n, m ∈ ω ;(2) { g ǫ . . . g ǫ n n x n : ǫ i ∈ { , }}| = 2 n +1 for every n ∈ ω .Following [20], we say that a subset Y of X is a piece-wise shifted F P -set ifthere exist ( g n ) n ∈ ω , ( x n ) n ∈ ω satisfying (1), (2) and such that Y = { g ǫ . . . g ǫ n n x n : ǫ i ∈ { , }} , n ∈ ω } . Theorem 3.14.
A subset A of X G is scattered if and only if A does not containpiece-wise shifted F P -sets.Proof.
Theorem 4.4 in [20]. ✷ Theorem 3.15.
Let ( X, E ) be a finitary indiscrete space, ( X, E ) = X G . Thenthere exists p ∈ X ∗ such that the orbit G p is not discrete.Proof. We may suppose that G consists of all permutations g of X such that( x, gx ) ∈ E for some E ∈ E and all x ∈ X .In light of Theorem 3.13 and Theorem 3.14, it suffices to find a piece-wise shifted F P -set in X defined by some sequence ( g n ) n ∈ ω in G and some sequence ( x n ) n ∈ ω in X . OARSE SPACES, ULTRAFILTERS AND DYNAMICAL SYSTEMS 9
Since X is not discrete, there are an infinite subset A ⊂ X and an involution g ∈ G such that A ∩ g A = ∅ and g x = x for each x ∈ X \ ( A ∪ gA ). Pick x ∈ A .Suppose that A , . . . , A n , g , . . . , g n and x , . . . , x n have been chosen. Since A n is not discrete, we can find an infinite subset A n +1 of A n and an involution g n +1 ∈ G such that g n +1 A n +1 ⊂ A n , A n +1 ∩ g n +1 A n +1 = ∅ and g n +1 x = x for each x ∈ X \ ( A n +1 ∪ g n +1 A n +1 ). Pick x n +1 ∈ A n +1 .After ω steps, we get the desired sequences ( g n ) n ∈ ω , ( x n ) n ∈ ω . ✷ Example 3.16.
We show that an indiscrete space needs not to be λ -tight.To this end, we take the set X of all rational number on [0 , G the group of all homeomorphisms of X and consider the finitary space X G . Let A = { a n : n ∈ ω } , B = { b n : n ∈ ω } be subsets of X such that ( a n ) n ∈ ω convergesto 0 and ( b n ) n ∈ ω converges to some irrational number. Then A, B are not linked,so X G is not λ -tight. On the other hand, let A be an infinite subset of X . We taketwo disjoint sequences ( a n ) n ∈ ω , ( b n ) n ∈ ω in A which converge to some point x ∈ X .Then there is a homeomorphism g of X such that g ( a n ) = b n , n ∈ ω . ApplyingTheorem 3.12, we see that X G is indiscrete.As the results, we have got the following line δ -tight = ⇒ topologically transitive = ⇒ λ -tight = ⇒ indiscrete,in which the first and the third arrow can not be reversed, and the second arrowcan be reversed but only under some assumptions additional to ZFC. Question 3.17.
Is a dynamical system ( G, ω ∗ ) minimal provided that ker ( G, ω ∗ ) = ω ∗ and ( G, ω ∗ ) is topologically transitive? The Higson corona of every λ -tight space is a singleton.The following example suggested by Taras Banakh shows that the Higson coronaof finitary indiscrete space needs not to be a singleton. Example 3.18.
Let ( X , E ), ( X , E ) be infinite indiscrete finitary spaces. Weendow the union X of X and X with the smallest coarse structure E such thatthe restrictions of E to X and X coincide with E and E . Then the Higsoncorona of ( X, E ) is not a singleton because the function f defined by f ( x ) = 0, x ∈ X and f ( x ) = 1, x ∈ X is slowly oscillating.A subset A of a coarse space ( X, E ) is called n -thin (or n -discrete) , n ∈ N if foreach E ∈ E there exists a bounded subset B of ( X, E ) such that | E A [ a ] | ≤ n foreach a ∈ A \ B . Every n -thin metrizable coarse space can be partitioned into ≤ n thin subsets [7],but the Bergman’s construction from [19] gives a finitary n -thinspace which can not be partitioned into ≤ n thin subsets. Theorem 3.19.
There exists a group G of permutations of ω such that thecoarse space ω G is 2-discrete but G can not be finitely partitioned into discretesubsets.Proof. Theorem 6.1 in [1]. ✷ Acknowledgements.
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