aa r X i v : . [ m a t h . G R ] F e b COARSE SELECTORS OF GROUPS
IGOR PROTASOV
Abstract.
For a group G , exp G denotes the set of all non-empty subsets of G . Weextend the finitary coarse structure of G from G × G to exp G × exp G and say thata macro-uniform mapping f : exp G → exp G (resp. f : [ G ] → G ) is a global selector(resp. 2-selector) of G if f ( A ) ∈ A for each A ∈ exp G (resp. A ∈ [ G ] ). We provethat a group G admits a global selector iff G admits a 2-selector and iff G is a finiteextension of an infinite cyclic subgroup or G is countable and locally finite. Introduction and results
The notions of selectors went from
Topology . Let X be a topological space, exp X denotes the set of all non-empty closed subsets of X endowed with some (initially, theVictories) topology, F be a non-empty subset of exp X . A continuous mapping f : F → X is called an F -selector of X if f ( A ) ∈ A for each A ∈ F . The question on selectors oftopological spaces was studied in a plenty of papers, we mention only [1], [4], [9], [10].Formally, coarse spaces, introduced independently and simultaniously in [17] and [13],can be considered as asymptotic counterparts of uniform topological spaces. But actu-ally, this notion is rooted in Geometry, Geometric Group Theory and
Combinatorics ,see [17, Chapter 1], [6, Chapter 4] and [13]. Every group G admits the natural finitarycoarse structure which, in the case of finitely generated G , can be viewed as the metricstructure of a Cayley graph of G . At this point, we need some basic definitions.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 } of 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 .Elements E ∈ E of the coarse structure are called entourages on X .For x ∈ X and E ∈ E the set E [ x ] := { y ∈ X : ( x, y ) ∈ E} is called the ball of radius E centered at x . Since E = S x ∈ X ( { x } × E [ x ]), the entourage E is uniquely determinedby the family of balls { E [ x ] : x ∈ X } . A subfamily E ′ ⊆ E is called a base of the coarsestructure E if each set E ∈ E is contained in some E ′ ∈ E ′ .The pair ( X, E ) is called a coarse space [17] or a ballean [13], [16]. We note that coarsespaces can be considered as asymptotic counterparts of uniform spaces, see [13, Section1.1]. A coarse space ( X, E ) is called connected if, for any x, y ∈ X , there exists E ∈ E suchthat y ∈ E [ x ].A subset Y ⊆ X is called bounded if Y ⊆ E [ x ] for some E ∈ E and x ∈ X . If ( X, E ) isconnected then the family B X of all bounded subsets of X is a bornology on X . We recallthat a family B of subsets of a set X is a bornology if B contains the family [ X ] <ω of allfinite subsets of X and B is closed under finite unions and taking subsets. A bornology B on a set X is called unbounded if X / ∈ B . A subfamily B ′ of B is called a base for B if,for each B ∈ B , there exists B ′ ∈ B ′ such that B ⊆ B ′ .Each subset Y ⊆ X defines a subspace ( Y, E| Y ) of ( X, E ), where E| Y = { E ∩ ( Y × Y ) : E ∈ E} . A subspace ( Y, E| Y ) is called large if there exists E ∈ E such that X = E [ Y ],where E [ Y ] = S y ∈ Y E [ y ].Let ( X, E ), ( X ′ , E ′ ) be coarse spaces. A mapping f : X → X ′ is called macro-uniform if for every E ∈ E there exists E ′ ∈ E ′ such that f ( E ( x )) ⊆ E ′ ( f ( x )) for each x ∈ X . If f is a bijection such that f and f − are macro-uniform, then f is called an asymorphism .If ( X, E ) and ( X ′ , E ′ ) contain large asymorphic subspaces, then they are called coarselyequivalent. Given a coarse spaces ( X, E ), we denote by exp X the set of all non-empty subsets of X and endow exp X with the coarse structure exp E with the base { exp E : E ∈ E} ,where ( A, B ) ∈ exp E ⇔ A ⊆ E [ B ] , B ⊆ E [ A ] . The coarse space ( exp X, exp E ) is called the hyperballean of ( X, E ), for hyperballeanssee [2], [3], [14], [15].Now we are ready to the key definition. Let ( X, E ) be coarse space, F be a non-emptysubspace of exp X . A macro-uniform mapping f : F −→ X is called an F - selector of( X, E ) if f ( A ) ∈ A for each A ∈ F . In the case F = exp X , F = B \{ } , F = [ X ] we geta global selector , a bornologous selector and a respectively. The investigationof selectors of coarse was initiated in [11], [12].Every group G with the identity e can be considered as the coarse spaces ( G, E ), where E is the (right) finitary coarse structure with the base {{ ( x, y ) : x ∈ F y } : F ∈ [ G ] <ω , e ∈ F } . Every metric d on a set X defines the coarse structure E d on X with the base {{ ( x, y ) : d ( x, y ) ≤ r } : r > } . Given a connected graph Γ, Γ = Γ[ V ], we denote by d the pathmetric on the set V of vertices of Γ and consider Γ as the coarse space ( V, E d ). We recallthat Γ is locally finite if the set { y : d ( x, y ) ≤ } if finite for each x ∈ V .Our goal is to prove the following theorem. Theorem 1.
For a group G , the following statements are equivalent ( i ) G admits a global selector; ( ii ) G admits a 2-selector; OARSE SELECTORS OF GROUPS 3 ( iii ) G is a finite extension of an infinite cyclic subgroup or G is countable and locallyfinite (i.e. every finite subset of G generates a finite subgroup). In the prof of Theorem 1 we use the following characterization of locally finite graphsadmitting selectors. By N and Z , we denote the graph on the sets of natural and integernumbers in which two vertices a, b are incident if and only if | a − b | = 1. We note alsothat two graphs are coarsely equivalent if and only if they are quasi-isometric, see [6,Chapter 4] for quasi-isometric spaces. Theorem 2.
For a locally finite graph Γ , the following statements are equivalent: ( i ) Γ admits a global selector; ( ii ) Γ admits a 2-selector; ( iii ) Γ is either finite or coarsely equivalent to N and Z . We prove Theorem 2 in Section 2 and Theorem 1 in Section 3.2.
Proof of Theorem 2
The implication ( i ) ⇒ ( ii ) is evident. To prove ( ii ) ⇒ ( iii ), we choose a 2-selector f of Γ[ V ] and get ( iii ) at the end of some chain of elementary observations.We define a binary relation ≺ on V as follows: a ≺ b iff a = b and f ( { a, b } ) = a .We use also the Hausdorff metric on the set of all finite subsets of V defined by d H ( A, B ) = max { d ( a, B ), d ( b, A ) : a ∈ A, b ∈ B } , d ( a, B ) = min { d ( a, b ) : b ∈ B } . Wenote that the coarse structure on [ V ] is defined by d H . Since f is macro-uniform, thereexists the minimal natural number r such that if A, B ∈ [ V ] and d H ( A, B ) ≤ d ( f ( A ) , f ( B )) ≤ r . We fix and use this r .We recall that a sequence of vertices a , . . . , a m is a geodesic path if d ( a , a m ) = m and d ( a i , a i +1 ) = 1 for each i ∈ { , . . . , m − } . Claim 1.
Let a , . . . , a m be a geodesic path in V and m ≥ r . If a ≺ a r (resp. a r ≺ a ) then a i ≺ a j (resp. a j ≺ a i ) for all i, j such that j − i ≥ r . Let a ≺ a r . By the choice of r , we have a ≺ a r +1 , . . . a ≺ a j and a ≺ a j , . . . a i ≺ a j . Claim 2.
Let v ∈ V , B ( v, r ) = { x ∈ V : d ( x, v ) ≤ r } and U be a subset of V \ B ( v, r ) such that the graph Γ[ U ] is connected. Then either v ≺ u for each u ∈ U or u ≺ v foreach u ∈ U . We take arbitrary u, u ′ ∈ U and choose a , . . . , a k in U such that a = u , a k = u ′ and d ( a i , a i +1 ) = 1 for each i ∈ { , . . . , k − } . Let a ≺ v . By the choice of r , we have a ≺ v , . . . , a k ≺ v . IGOR PROTASOV
Claim 3.
Let u, v, v ′ ∈ V , d ( v, v ′ ) = n and d ( u, v ) > n + r . If u ≺ v (resp. v ≺ u )then u ≺ v ′ (resp. v ′ ≺ u ) .We choose a geodesic path a , . . . , a m from v to v ′ . Let u ≺ v . By the choice of r , u ≺ a , u ≺ a , . . . , u ≺ a n . Claim 4.
Let a , . . . , a m be a geodesic path in V , v ∈ V , d ( v, { a , . . . , a m } ) = d ( v, a k ) , k > r + 1 , m − k > r + 1 . Then d ( v, a k ) ≤ r . We take the first alternative given by Claim 1, the second is analogical. Then a ≺ a k , a k ≺ a m . Assuming that d ( v, a k ) > r , we can replace v to some point on a geodesicpath from v to a k and get d ( v, a k ) = r + 1. We take the first alternative given by Claim2, the second is analogical. Then v ≺ a , v ≺ a m . But v ≺ a and a ≺ a k contradictClaim 3.We recall that a sequence ( a n ) n<ω in V is a ray if d ( a i , a j ) = j − i for all i < j .Evidently, Γ[ { a n : n < ω } ] is asymorphic to N . Claim 5.
Let ( a n ) n<ω , ( c n ) n<ω be rays in V , A = { a n : n < ω } , C = { c n : n < ω } and A ∩ C = ∅ . Let t , . . . , t k be a geodesic path from a to c , T = { t , . . . , t k } . Assumethat T ∩ { A } = { a } , T ∩ C = { c } . If there exists a finite subset H of V such thatevery geodesic path from a vertex a ∈ A to a vertex c ∈ C meets H then ( A ∪ C ∪ T, d ) is asymorphic to Z . We define a bijection f : A ∪ C ∪ T → Z by f ( c i ) = − i − , f ( t i ) = i, f ( a i ) = i + k + 1and show that f is an asymorphism.If x, y ∈ A ∪ C ∪ T then | f ( x ) − f ( y ) | ≤ d ( x, y ). Hence, f − is macro-uniform.We denote by p = max { d ( a , h ) , d ( b , h ) : h ∈ H } . Then the restriction of f to C ∪ T ∪ { a , . . . , a p } is an asymorphism and the restriction of f to A ∪ T ∪ { c , . . . , c p } is an asymorphism. Let n > p , m > p . Since a geodesic path from c n to a m meets H ,we have d ( a m , c n ) ≤ n − p + m − p = | f ( a m ) − f ( c n ) | − k − p, so f is macro-uniform and the claim is proven.We suppose that V is infinite. Since Γ[ V ] is locally finite, there exists a ray ( a n ) n<ω in V . We put A = { a n : n < ω } . If V \ B ( A, r ) is finite then Γ[ V ] is coarsely equivalentto N .We suppose V \ B ( A, r ) is infinite, take u ∈ V \ B ( A, r ) and show that every path P from u to a point from B ( A, r ) meets B ( { a , . . . , a r +1 } , r + 1). We take a point v ∈ P such that d ( v, A ) = r + 1 and take k such that d ( v, a k ) = r + 1. By Claim 4, OARSE SELECTORS OF GROUPS 5 k ≤ r + 1, so v ∈ B ( { a , . . . , a r +1 } , r + 1). We choose a ray ( c n ) n<ω in V \ B ( A, r )and put C = ( c n ) n<ω . We delete (if necessary) a finite number of points from A so that A, C and T satisfy the assumptions of Claim 5 with F = B ( { a , . . . , a r +1 } , r + 1). Then( B ( A ∪ C ∪ T ) , d ) is coarsely equivalent to Z .We show that V \ B ( A ∪ C, r ) is finite, so Γ[ V ] is coasly equivalent to Z . We supposethe contrary and choose a ray ( x n ) n<ω in V \ B ( A ∪ C, r ). Applying arguments fromabove paragraph, we can construct a subset X of V such that ( X, d ) is coarsely equivalentto a tree T which is a union of three rays with common beginning. Since ( X, d ) has a2-selector, by Proposition 5 from [12], T also admits a 2-selector. On the other hand,Claim 4 states that T does not admit a selector and we get a contradiction.It remains to prove ( iii ) ⇒ ( i ). This is evident if Γ is finite. By [12, Proposition 5],it suffices to show that N and Z admit global selectors.If A ∈ exp N then we put f ( A ) = min A and note that f : exp N → N is a globalselector.Let A ∈ exp Z . If A ∩ { z ∈ Z : z ≤ } 6 = ∅ then we put f ( A ) = max ( A ∩ { z ∈ Z : z ≤ } ). Otherwise, f ( A ) = min A . Then f : exp Z → Z is a global selector.3. Proof of Theorem 1
Let G be a group with the finite system S of generators, S = S − . We recall thatthe Cayley graph Cay ( G, S ) is a graph with the set of vertices G and the set of edges { ( x, y ) : x = y, xy − ∈ S } . We note that the finitary coarse space of G is asymorphicto the coarse space of Cay ( G, S ).Now let G be an arbitrary group. The implication ( i ) ⇒ ( ii ) is evident.We prove ( ii ) ⇒ ( iii ). By [12, Theorem 4], G is countable. Let f be a 2-selector of G . We use the binary relation ≺ on G , defined in Section 2, and consider two cases. Case 1. G has an element a of infinite order. We denote by A the subgroup of G ,generated by A , and show that | G : A | is finite.On the contrary, let | G : A | is infinite. We put S = { e, a, a − } , denote by Γ[ A ]the graph Cay ( A, S ) and choose a natural number r such that if B, C ∈ [ A ] and d H ( B, C ) ≤ d ( f ( A ) , f ( B )) ≤ r . By Claim 1, either a m ≺ a n for all m, n ∈ Z suchthat n − m ≥ r or a n ≺ a m for all m, n ∈ Z such that n − m ≥ r .Since f : [ G ] → G is macro-uniform, there exists a finite subset F of G such that F = F − , e ∈ F and if B, C ∈ [ G ] and A ⊆ SB , B ⊆ SA then f ( A ) ∈ F f ( B ). Since | G : A | is infinite, we can choose h ∈ G \ F A , so
F h ∩ A = ∅ . Then either a n ≺ h foreach n ∈ Z or h ≺ a n for each n ∈ Z . We consider the first alternative, the second isanalogical.Since f is macro-uniform, we can choose m ∈ N , m ≥ r such that e ≺ a m and h ≺ a m ,but h ≺ a m contradicts above paragraph. IGOR PROTASOV
Case 2. G is a torsion group. We suppose that G is not locally finite, choose afinite subset S of G such that the subgroup H , generated by S , is infinite. We denoteΓ[ H ] = Cay ( H, S ). By Theorem 2, Γ[ H ] is coarsely equivalent to N or Z .We take v ∈ Γ[ H ] and denote S ( v, n ) = { u ∈ H : d ( v, u ) = n } , n ∈ N . By [8, Theorem1] or [16, Theorem 5.4.1], there exists a natural number k such that | S ( v, n ) | ≤ k foreach n ∈ N . Hence, H is of linear growth. Applying either [5] or [7], we conclude that H has an element of infinite order, a contradiction with the choice of G .It remains to verify ( iii ) ⇒ ( i ). If G is a finite extension of an infinite cyclic subgroupthen we apply Theorem 2. If G is countable and locally finite, one can refer to Theorem5 in [12], but we prefer to give the following direct proof.We write G as the union of an increasing chain { G n : n < ω } , G = { e } of finitesubgroup. For each n , we choose some system R n , e ∈ R n of representatives of rightcosets of G n +1 by G n , so G n +1 = G n R n . We denote X = { ( x n ) n<ω : x n ∈ R n and x n = e f or all but f initery many n } and define a bijection h : G → X as follows.We put h ( e ) = ( x n ) n<ω , x n = e . Let g ∈ G , g = e . We choose n such that g ∈ G n +1 \ G n and write g = g r n , g ∈ G g , r n ∈ R n . If g = e then we find n , g ∈ G n , r n ∈ R n such that g = g r n . After a finite number k of steps, we get g = r n k . . . r n r n . We put h ( g ) = ( y n ), where y n = r n if n ∈ { n k , . . . , n } , otherwise, y n = e .Now we define a linear order ≤ on X . For each n < ω , we choose some linear order ≤ n on R n with the minimal element e . If ( x n ) n<ω = ( y n ) n<ω then we choose the minimal k such that x n = y n for each n > k . If x k < k y k then we put ( x n ) n<ω < ( y n ) n<ω .We note that ( X, ≤ ) is well-ordered, so every non-empty subset of X has the minimalelement. To define a global selector f : exp G → G , we take an arbitrary A ∈ exp G and put f ( A ) = min h ( A ). References [1] G. Artico, U. Marconi, J. Pelant, L. Rotter, M. Tkachenko,
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I.Protasov: Taras Shevchenko National University of Kyiv, Department of ComputerScience and Cybernetics, Academic Glushkov pr. 4d, 03680 Kyiv, Ukraine
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