Coarse structures on groups defined by conjugations
aa r X i v : . [ m a t h . G M ] O c t COARSE STRUCTURES ON GROUPS DEFINED BYCONJUGATIONS
IGOR PROTASOV, KSENIA PROTASOVA
Abstract.
For a group G , we denote by ↔ G the coarse space on G endowedwith the coarse structure with the base {{ ( x, y ) ∈ G × G : y ∈ x F } : F ∈ [ G ] <ω } , x F = { z − xz : z ∈ F } . Our goal is to explore interplays between algebraicproperties of G and asymptotic properties of ↔ G . In particular, we show that asdim ↔ G = 0 if and only if G/Z G is locally finite, Z G is the center of G . For aninfinite group G , the coarse space of subgroups of G is discrete if and only if G isa Dedekind group.20E45, 54D80Keywords: coarse structure defined by conjugations, cellularity, FC-group, ul-trafilter. 1. Introduction
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 ;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 [13] or a ballean [10], [12].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 . Let G be a group of permutations of a set X . We denote by X G the set X endowed with the coarse structure with the base {{ ( x, gx ) : g ∈ F } : F ∈ [ G ] <ω , id ∈ F } . By [7, Theorem 1], for every finitary coarse structure ( X, E ), there exists agroup G of permutations of X such that ( X, E ) = X G . For more general resultsand applications see [8] and the survey [9].Let ( X, E ) be a coarse space. We define an equivalence ∼ on X by x ∼ y ifand only if there exists E ∈ E such that y ∈ E [ x ], so X is a disjoint union ofconnected components. If there is only one connected component then ( X, E ) iscalled connected.Now let G be a group. For x, g ∈ G and F ⊆ G , we denote x g = g − xg , x F = { x y : y ∈ F } , F g = { y g : y ∈ F } .We denote by ↔ G the coarse structure on G endowed with the coarse structurewith the base {{ ( x, y ) ∈ G × G : y ∈ x F } : F ∈ [ G ] <ω } . Evidently, each connectedcomponent A of ↔ G is of the form a G , a ∈ A .We endow G with the discrete topology and indentify the Stone- ˇ C ech compact-ification βG of G with the set of all ultrafilters on G . For A ⊆ G , ¯ A denotes theset { p ∈ βG : A ∈ p } and the family { ¯ A : A ⊆ G } forms a base for open sets of βG . The family of all free ultrafilters on G is denoted by G ∗ . By the universalproperty of βG , every mapping f : G → K , K is a compact Hausdorff space, givesup to the continuous mapping f β : βG → K .The action G on G by conjugations extends to the action G on βG : if g ∈ G , p ∈ βG then p g = { g − P g : P ∈ g } . We use this dynamical approach to theconjugasy in groups initiated in [11].In section 2 and 3, we characterize groups G such that the coarse space ↔ G isdiscrete, n -discrete and cellular. In section 4, we show that every finitary coarsespace admits an asymorphic embedding to ↔ G for an appropriate choice of a group G . In section 5, we characterize groups with discrete space of subgroups.2. Discreteness
Let ( X, E ) be a coarse space. We say that a subset B of X is bounded if thereexist a finite subset F of X and E ∈ E such that B ⊆ E [ F ] and note that thefamily of all bounded subset of X is a bornology, i.e. an ideal in the Booleanalgebra of subsets of X containing all finite subsets.We say that a subset A of X is • discrete if, for every E ∈ E , there exists a bounded subset B of X suchthat E A [ a ] = { a } for each a ∈ A \ B ; OARSE STRUCTURES ON GROUPS DEFINED BY CONJUGATIONS 3 • n- discrete , n ∈ N if, for every E ∈ E , there exists a bounded subset B of X such that | E A [ a ] | ≤ n for each a ∈ A \ B . Theorem 1.
For an infinite group G , the following conditions are equivalent(i) G is Abelian;(ii) p G = { p } for each p ∈ G ∗ ;(iii) ↔ G is discrete.Proof. The equivalence ( i ) ⇐⇒ ( ii ) is proved in [11, Proposition 1.1], (i) = ⇒ (iii) is evident.( iii ) = ⇒ ( ii ). We assume that p x = p for some p ∈ G ∗ , x ∈ G and pick P ∈ p such that P x ∩ P = ∅ . Let B be a finite subset of X . We take a ∈ P \ B and notethat a x = a so ↔ G is not discrete. ✷ Theorem 2.
For a group G , the following conditions are equivalent(i) p G is finite for each p ∈ G ∗ ;(ii) there exists a natural number n such that | p G | ≤ n for each p ∈ G ∗ ;(iii) there exists a natural number m such that | a G | ≤ m for each a ∈ G ∗ ;(iv) the commutant [ G, G ] of G is finite.Proof. See Theorem 3.1 in [11]. ✷ Theorem 3.
Given a group G , the coarse space ↔ G is n -discrete for some n ∈ N if and only if [ G, G ] is finite.Proof. We assume that ↔ G is n -discrete and show that [ G, G ] is finite. To applyTheorem 2, it suffices to prove that | p G | ≤ n for each p ∈ G ∗ .We assume the contrary: there exists p ∈ G ∗ and g , . . . , g n +1 ∈ G such thatthe ultrafilters p g , . . . , p g n +1 are distinct. We choose P ∈ p such that the subsets p g , . . . , p g n +1 are pair-wise disjoint. Given an arbitrary bounded subset B of G ,we pick a ∈ P \ B . Then a g , . . . , a g n +1 are distinct so ↔ G is not n -discrete.On the other hand, if [ G, G ] is finite then there exists m ∈ N such that | a G | ≤ m for each a ∈ G , see Theorem 2( iii ). We partition G into subset P , . . . , P m suchthat | P i ∩ a G | ≤ a ∈ G and i ∈ { , . . . , m } . For any E ∈ E and a ∈ G ,we have | E [ a ] | ≤ m so G is not m -discrete. ✷ IGOR PROTASOV, KSENIA PROTASOVA
We recall that G is an F C -group if the set a G is finite for each a ∈ G . Clearly, G is an F C -group if and only if each connected component of ↔ G is bounded. Question 4.
Let G be a group such that each connected component of ↔ G isdiscrete. Is G an F C -group?
We show that the answer to Question 4 is affirmative provided that G is finitelygenerated.Let F be a finite subset of G such that F = F − , e ∈ F , e is the identity of G and F generates G . We assume that each connected component of ↔ G is discrete,take an arbitrary element g ∈ G and show that g G is finite. We act on g byconjugations from x ∈ F , write each g x as a word in F of minimal length, deleteduplicates (i.e. words which define the same elements) and get a subset A . Thenwe repeat this procedure for each element g ∈ A and get a subset A , A ⊆ A .Since F is finite, by the assumption there exists n ∈ N such that A n +1 = A n . Thismeans that g G = A n . 3. Cellularity
A coarse space ( X, E ) is called cellular if E has a base consisting of equivalencerelations. By [12, Theorem 3.1.3], ( X, E ) is cellular if and only if asdim ( X, E ) = 0.Applying Theorem 3.1.2 from [12] we get(1) ↔ G is cellular if and only if, for every finitely generated subgroup H of G ,there exists a finite subset F of G such that g H ⊆ g F for each g ∈ G .We recall that a group G is locally normal if each finite subset of G is containedin some finite normal subgroup and use the following characterization [2](2) G is an FC-group if and only if G/Z G is locally normal and each elementof G is contained in finitely generated normal subgroup, Z G is the center of G .A group G is called locally finite if each finite subset of G generates a finitesubgroup. Theorem 5.
For a group G , ↔ G is cellular if and only if G/Z G is locally finite.Proof. We suppose that ↔ G is cellular and show(3) for every element a ∈ G of infinite order there exists n ∈ N such that a n ∈ Z G .We denote by A the subgroup of G generated by a and use (1) to choose a finitesubset F of G such that g A ⊆ g F for each g ∈ G . Let | F | = n . Since | g A | ≤ n , a k g = ga k for some k ≤ m . We put n = m !. OARSE STRUCTURES ON GROUPS DEFINED BY CONJUGATIONS 5
By (1), every finitely generated subgroup H of G is an FC-group. By (3), H/ ( H ∩ Z G ) is a torsion group. Applying (2), we conclude that H/ ( H ∩ Z G ) isfinite. Hence, G/Z G is locally finite.Now let G/Z G is locally finite. We take an arbitrary finitely generated subgroup H of G , choose a set h , . . . , h n of representatives of right cosets of H by H ∩ Z G ,put F = { h , . . . , h n } and note that g H = g F for each g ∈ G . Applying (1), weconclude that ↔ G is cellular. ✷ Remark 6.
Every finitely generated subgroup H of a group G is an FC-groupif and only if g H is finite for each g ∈ G . If G/Z G is locally finite then everyfinitely generated subgroup H of G is an FC group. We show that the conversestatement does not hold. Let H = ⊕ i<ω H i be the direct sum of ω copies of Z .We partition ω into consecutive intervals { W i : i < ω } of length | W i | = i + 1.Then we take an automorphism a of H acting on each ⊕{ H m : m ∈ W i } as thecyclic permutations of coordinates, denote by A the cyclic group generated by A and consider the semidirect product G = H ⋋ A . Then every finitely generatedsubgroup of G is an FC-group but a n / ∈ Z G for each n ∈ N so G/Z G is not locallyfinite. 4. Asymorphic embeddings
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 )] foreach x ∈ X . We say that an injective mapping f : X −→ X ′ is an asymorphicembedding if f : X −→ X ′ and f − : f ( X ) −→ X are macro-uniform. Theorem 7.
Every finitary coarse space ( X, E ) admits an asymorphic embed-ding to ↔ G for an appropriate choice of a group G .Proof. We represent ( X, E ) as the coarse space X H for some group H of permu-tations of X , see [7, Theorem 1]. We consider { , } X as a group with point-wiseaddition mod
2. For h ∈ H and χ ∈ { , } X , we put χ h ( y ) = χ ( h − y ). Then wedefine a semidirect product G = { , } X ⋋ H by( χ, h )( χ ′ , h ′ ) = ( χ + χ ′ h , hh ′ )and note that the mapping f : X −→ { , } X , f ( x ) is the characteristic functionof { x } is an asymorphic embedding of ( X, E ) into ↔ G . ✷ If a subset A of a coarse space ( X, E ) is the union of n discrete subsets then A is n -discrete. Theorem 8.
Let G be a countable group. Then every n -discrete subset A of ↔ G can be partitioned into n discrete subsets. IGOR PROTASOV, KSENIA PROTASOVA
Proof.
Use arguments proving this statement in the case of a connected coarsespace with a linearly ordered base [6, Theorem 1.2]. ✷ Theorem 9.
There exists a group G such that ↔ G has 2-discrete subset whichcannot be finitely partitioned into discrete subsets.Proof. By Theorem 6.3 from [3], there exists 2-discrete finitary coarse space on ω which cannot be finitely partitioned into discrete subspaces. Apply Theorem 7. ✷ The space of subgroups
For a group G we denote by S ( ↔ G ) the set S ( G ) of all subgroups of G endowedwith the coarse structure with the base {{ ( X, Y ) ∈ S ( G ) × S ( G ) : Y ∈ X F } : F ∈ [ G ] <ω } ,X F = { g − Xg : g ∈ F } . We recall that G is a Dedekind group if each subgroup of G is normal. A non-abelian Dedekind group is called Hamiltonian. By [1],(4) G is Hamiltonian if and only if G is isomorphic to Q × P , where Q isthe quaternion group, P is an Abelian group without of elements of order 4. Theorem 10.
For an infinite group G , S ( ↔ G ) is discrete if and only if G is aDedekind group.Proof. If each subgroup of G is normal then, evidently, S ( G ) is discrete.We assume that S ( ↔ G ) is discrete and consider two cases. Case 1: G has an element of infinite order. First, we show that every infinitecyclic subgroup of G is invariant. We suppose the contrary and choose an infinitecyclic subgroup A, A = < a > and z ∈ G such that z − az / ∈ A . Since S ( ↔ G ) isdiscrete, there exists m ∈ N such that z − < a n > z = < a n > for each n > m .By the same reason, there exists k ∈ N such that z − < aa n > z = < aa n > foreach n > k . We take an arbitrary n such that n > m , n > k . Then z − a n +1 z =( z − az )( z − a n z ) ∈ < a n +1 > , z − a n z ∈ < a n > , so z − a n z ∈ A , contradicting thechoice of A and z .Second, we take an arbitrary element a ∈ G of infinite order and show that a ∈ Z G . Assuming the contrary, we get z ∈ G such that z − az = a . By aboveparagraph z − az = a − , so z − az = a and ( a n z )( a n z ) = a n z z − a n z = a n z a − n = z for each n ∈ N . Since S ( ↔ G ) is discrete, there exists m ∈ N such that z − ( < a n z >< z > ) z = < a n z >< z > OARSE STRUCTURES ON GROUPS DEFINED BY CONJUGATIONS 7 for each n > m . Hence, z − ( a n z ) z = a − n z ∈ < a n z >< z > and a n ∈ < z > , contradicting z − a n z = a − n .If b is an element of finite order and a is an element of infinite order then ab hasan infinite order because a ∈ Z G , so ab ∈ Z G , b ∈ Z G , and G is Abelian. Case 2:
Every element of G has a finite order. We prove that G is a Dedekindgroup provided that the following condition holds(5) for every finite subset K of G containing the identity e , there exists a ∈ G , a = e such that K ∩ < a > = { e } .We suppose the contrary and choose b ∈ G , z ∈ G such that z − bz / ∈ < b > .Since S ( ↔ G ) is discrete, by (5), there exists a ∈ G , a = e such that z − bz < b > ∩ < a > = { e } , z − < a > z = < a >,b − < a > b = < a >, z − < b >< a > z = < b >< a > . Then z − baz = ( z − bz )( z − az ) ∈ < b >< a > , z − bz ∈ < b >< a > and z − bz ∈ , contradicting the choice of b and z .We denote by π ( G ) the set of all prime divisors of orders of elements of G andput X n = { g ∈ G : g n = e } . If G is not a Dedekind group, by (5), π ( G ) is finite and X p is finite for each p ∈ π ( G ). We prove that G is layer-finite: X n is finite for each n ∈ N . It suffices to verify that X p n is finite for all p ∈ π ( G ), n ∈ N . We supposethat X p m is finite but X p m +1 is infinite. Then there exists a sequence ( a n ) n ∈ ω in G and a ∈ G such that | a n | = p m +1 , | a | = p m and < a n > ∩ < a k > = < a > for all distinct n, k ∈ N . We denote by H the subgroup of G generated by theset { a n : n ∈ ω } and put M = H/ < a > . Since S ( ↔ M ) is discrete, applying (5)and (4) to M , we conclude that M has an infinite Abelian subgroup of exponent p . By the Gr .. u n’s lemma (see [5], p. 398), H has an infinite Abelian subgroup ofexponent p , so X p is infinite and we get a contradiction.Thus, our assumption that G is not a Dedekind group gives G is layer-finite and π ( G ) is finite. Since G is infinite, by the Chernikov’s theorem [4], G has a centralquasi-cyclic p -group A , A = ∪ n ∈ ω < a n >, a pn +1 = a n . We take c, z ∈ G such that z − cz = < c > , | c | = q m , q ∈ π ( G ). Since S ( ↔ G ) is discrete, there exists k ∈ N suchthat, for each n > k , we have z − < a n c > z = < a n c >, a n ( z − cz ) ∈ < a n c > . If q = p then z − cz ∈ < c > , contradicting the choice of c and z . If q = p and n > m , n > k then ( a n c ) p m = a p m n , | a p m n | > p m and z − cz ∈ < a p m n > . Since A iscentral, z − cz = c and z − cz ∈ < c > , contradicting the choice of z, c . The proofis completed. ✷ IGOR PROTASOV, KSENIA PROTASOVA
Remark 11.
Let G be a transitive group of permutations of a set X , St ( x ) = { g ∈ G : gx = x } , x ∈ X . Then the natural mapping x St ( x ) is an asymorphicembedding of the finitary coarse space X G intoIf ( ↔ G ) is cellular then applying (1) we see that S ( ↔ G ) is cellular. Question 12. Is ↔ G cellular provided that S ( ↔ G ) is cellular? References [1] R. Baer,
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