On asymorphisms of groups

Igor Protasov 1  and Serhii Slobodianiuk 2
  • 1 Department of Cybernetics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
  • 2 Department of Mechanics and Mathematics, Kyiv University, Volodymyrska 64, 01033, Kyiv, Ukraine
Igor Protasov and Serhii Slobodianiuk

Abstract

Let G, H be groups and let κ be a cardinal. A bijection f:GH is called an asymorphism if, for any X[G]<κ, Y[H]<κ, there exist X[G]<κ, Y[H]<κ such that for all xG and yH, we have f(Xx)Yf(x), f-1(Yy)Xf-1(y). For a set S, [S]<κ denotes the set {SS:|S|<κ}. Let κ and γ be cardinals such that 0<κγ. We prove that any two Abelian groups of cardinality γ are κ-asymorphic, but the free group of rank γ is not κ-asymorphic to an Abelian group provided that either κ<γ or κ=γ and κ is a singular cardinal. It is known [] that if γ=κ and κ is regular, then any two groups of cardinality κ are κ-asymorphic.

1 Introduction

Following [7, 10], we say that a ball structure is a triple =(X,P,B), where X, P are non-empty sets, and for all xX and αP, B(x,α) is a subset of X which is called a ball of radius α around x. It is supposed that xB(x,α) for all xX, αP. The set X is called the support of , P is called the set of radii.

Given any xX, AX, αP, we set

B*(x,α)={yX:xB(y,α)},B(A,α)=aAB(a,α),B*(A,α)=aAB*(a,α).

A ball structure =(X,P,B) is called a ballean if

  1. for any α,βP, there exist α,βP such that, for every xX,
    B(x,α)B*(x,α),B*(x,β)B(x,β),
  2. for any α,βP, there exists γP such that, for every xX,
    B(B(x,α),β)B(x,γ),
  3. for any x,yX, there exists αP such that yB(x,α).

We note that a ballean can be considered as an asymptotic counterpart of a uniform space, and could be defined [11] in terms of the entourages of the diagonal ΔX in X×X. In this case a ballean is called a coarse structure. For categorical look at the balleans and coarse structures as “two faces of the same coin” see [3].

Let =(X,P,B), =(X,P,B) be balleans. A mapping f:XX is called a -mapping if, for every αP, there exists αP such that, for every xX, f(B(x,α))B(f(x),α).

A bijection f:XX is called an asymorphism between and if f and f-1 are -mappings. In this case and are called asymorphic.

Let =(X,P,B) be a ballean. Each subset Y of X defines a subballeanY=(Y,P,BY), where BY(y,α)=YB(y,α). A subset Y of X is called large if X=B(Y,α), for αP. Two balleans and with supports X and X are called coarsely equivalent if there exist large subsets YX and YX such that the subballeans Y and Y are asymorphic. In the proof of Theorem 2, we use the following equivalent definition: and are coarsely equivalent if there is a -mapping f:X1X2 such that f(X1) is large and, for every αP there exists αP such that f-1(B(f(x),α))(x,α) for each xX.

We recall [8] that an ideal in the Boolean algebra of all subsets of a group G is a group ideal if AB-1 for any A,B. We suppose that contains all finite subsets of G and denote by (G,) the ballean =(G,,B), where P(g,A)=(A{e})g, e is the identity of G.

For an infinite group G and an infinite cardinal κ, κ|G|, we denote by [G]<κ the group ideal {AG:|A|<κ}.

We say that two groups G and H are κ-asymorphic (κ-coarsely equivalent) if the balleans (G,[G]<κ) and (H,[H]<κ) are asymorphic (coarsely equivalent).

In the case κ=0, we say that G and H are finitarily asymorphic and finitarily coarsely equivalent respectively. We note that finitely generated groups are finitarily coarsely equivalent if and only if G and H are quasi-isometric [5, Chapter 4].

A classification of countable locally finite groups (each finite subset generates finite subgroup) up to finitary asymorphisms is obtained in [6] (cf. [7, p. 103]).

Two countable locally finite groups G1 and G2 are finitarily asymorphic if and only if the following conditions hold:

  1. (i)for every finite subgroup FG1, there exists a finite subgroup H of G2 such that |F| is a divisor of |H|,
  2. (ii)for every finite subgroup H of G2, there exists a finite subgroup F of G1 such that |H| is a divisor of |F|.

It follows that there are continuum many distinct types of countable locally finite groups and each group is finitarily asymorphic to some direct sum of finite cyclic groups.

The following coarse classification of countable Abelian groups is obtained in [2].

Two countable Abelian groups are finitarily coarsely equivalent if and only if the torsion-free ranks of G and H coincide and G and H are either both finitely generated or infinitely generated.

In particular, any two countable torsion Abelian groups are finitely coarsely equivalent.

The referee suggested to mention [1, 4, 12] in connection to the coarse classification of Abelian groups, as well as the asymptotic dimension and its connection to the Hirsh length.

This note is motivated by the following result [9, Theorem 3]. Let G, H be groups of cardinality γ, γ>0. If γ is regular, then G and H are γ-asymorphic.

What happens if γ is singular? Are G and Hκ-asymorphic for uncountable κ<γ?

2 Theorems

Let G be an Abelian group of cardinality γ, γ>0 and let κ be a cardinal such that 0<κγ. Then G is κ-asymorphic to the free Abelian group Aγ of rank γ.

Let G be an Abelian group of cardinality γ, γ>0. Then G is not κ-coarsely equivalent to the free group Fγ of rank γ provided that either 0<κ<γ or κ=γ and γ is a singular cardinal. In particular, G and Fγ are not κ-asymorphic.

The referee asks: “The ‘huge jump’ from Abelian to free (non-Abelian) groups in Theorem 2 gives rise to the natural question of what one can do for groups close to being Abelian (as done in [1] in the case of finitary coarse equivalence).”

In this line, we conjecture that Theorem 1 remains true if G satisfies the following condition: |{x-1gx:xG}|<κ for each gG.

3 Proofs

Proof of Theorem 1.

We choose inductively an increasing chain of subgroups {Gα:α<γ} of G such that

  1. (1)G0={e}, G=α<γGα,
  2. (2)GαGβ for all α<β<γ,
  3. (3)Gβ=α<βGα for every limit ordinal β<γ,
  4. (4)|Gα+1:Gα|=0 for every α<γ.

For each α<γ, we fix some system Xα of representatives of cosets of Gα+1Gα by Gα so Gα+1Gα=XαGα.

We take an arbitrary element gG{e} and choose the smallest subgroup Gα such that gGα. By (3), α=α0+1 for some α0<γ. Then g=g0x0, g0Gα0, x0Xα0. If g0e, we repeat the argument for g0: choose α1 such that g0Gα1+1Gα1 and write g0=g1x1, where g1Gα1, x1Xα1 and so on. Since the set of ordinals less than κ is well ordered, after finite number of steps, we get

g=xs(g)x1x0,xiXαi,i{0,,s(g)},α0>α1>>αs(g).

By the construction, the representation is unique, so we can denote

supt(g)={αs(g),,α1,α0},supt(e)=.

We identify Aγ with the direct sum α<γZα of infinite cyclic groups and, for each α<γ, fix some bijection fα:XαZα{eα}, eα is the identity of Zα. We define a bijection f:GAγ putting f(e)=(eα)α<γ and, for gG{e},

f(g)=f(xs(g)x1x0)=(fαs(g)(xs(g)),,fα1(x1),fα0(x0)).

We show that the bijection f is an asymorphism between the balleans (G,[G]<κ) and (Aγ,[Aγ]<κ).

To show that f-1 is a -mapping, we take aAγ, K[Aγ]<κ and choose a set I[γ]<κ such that KαIZα. We denote X=αIZα, b=prIa, c=prγIa. Then we have

f-1(Ka)f-1(Xc)=f-1(X)f-1(c)=f-1(X)(f-1(b))-1f-1(b)f-1(c)=f-1(X)(f-1(b))-1f-1(a)f-1(X)(f-1(X))-1f-1(a)

and it is suffices to note that f-1(X)(f-1(X))-1[G]<κ.

The verification that f is a -mapping is more delicate. We take an arbitrary F[G]<κ and denote by Y the smallest subgroup of G containing F and such that if gY and αsupt(g), then XαY. We show that Y[G]<κ. Indeed, Y can be obtained in the following way. For a subset A of G, we denote by A the subgroup generated by A and h(A)=A{Xα:αsupt(g),gA}. We put S0=F, Y0=h(S0) and inductively Sn+1=Yn, Yn+1=h(Sn). Then Y=nωYn.

To conclude the proof, we take an arbitrary gG, put I=tYsupt(t), and write g=g0g1 where supt(g0)I, supt(g1)γI. Then we have

f(Fg)f(Yg0g1)f(Yg1)f(Y)f(g1)f(Y)f(g).

Proof of Theorem 2.

We are going to get a contradiction assuming only that there is a -mapping f:GFγ such that Fγ=Kf(G) for some K[Fγ]<κ. This time the Abelian group G is written additively. In view of Theorem 1, we may suppose that G is a group of exponent 2. Since f is a -mapping, for every yG, there exists Ky[Fκ]<κ such that, for each xG, we have

f(y+x)Kyf(x).

We note that the family {Ky:yG} can be chosen so that f(y)Ky (just join f(y) to Ky) and for some cardinal δ, κδ<γ,

  1. (5)|Ky|<δ for each yG.

If κ<γ, then (5) is evident with δ=κ. We consider the case γ=κ and κ is singular. If (5) could not be satisfied, then, for every δ<γ, there exists aδG such that

|{f(aδ+x)(f(x))-1:xG}|δ.

We denote Yδ={f(aδ+x)(f(x))-1:xG} and use singularity of γ to choose a subset Δ of γ such that |Δ|<γ and the set {|Yδ|:δΔ} is cofinal in γ. We put A={aδ:δΔ}. Since |A||Δ|<γ and f is a -mapping, there is a subset Y of Fγ such that |Y|<κ and {f(A+x)(f(x))-1:xG}Y. Then we have got a contradiction with YδY for each δΔ, |Y|<γ and |Yδ|δ.

We consider Fγ as the group of all reduced group words over the alphabet κ. For gFγ, we denote by alpg the set of all letters α<γ such that α or α-1 occurs in g, and for subset S of Fγ, we put alpS=gSalpg.

Now we show how to find a,bG such that

  1. (6)alpf(a)alpKb,
  2. (7)alpf(b)alpKa.

Since Fγ=Kf(G) for some K[Fγ]<κ, we can choose a subset AG such that |A|=δ and |alpf(A)|=δ. It suffices to take Sf(G) such that |S|=|alpS|=δ and choose AG such that f(A)=S. We take cf(G) such that

alpctAalpKt

and find bG such that f(b)=c. Since |alpf(A)|=δ, there is aA such that alpf(a)alpKb. Clearly, alpf(b)alpKa.

We put f(a+b)=z and note that

  1. (8)f(b)=f(a+(a+b))Kaz,
  2. (9)f(a)=f(b+(a+b))Kbz.

We take ualpf(a)alpKb, valpf(b)alpKa. Then ualpz, valpz. If u occurs in z before v, then, by (9), valpf(a) so valpKa contradicting the choice of v. If v occurs in z before u, then, by (8), ualpf(b) so ualpKb contradicting the choice of u. ∎

Acknowledgements

We thank the referee for constructive comments, corrections and additional references.

References

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    Banakh T., Cencelj M., Repovš R. and Zarichnyi I., Coarse classification of Abelian groups and amenable shift-homogeneous mentric spaces, Q. J. Math. 65 (2014), 1127–1144.

    • Crossref
    • Export Citation
  • [2]

    Banakh T., Higes J. and Zarichnyi M., The coarse classification of countable abelian groups, Trans. Amer. Math. Soc. 362 (2010), 4755–4780.

    • Crossref
    • Export Citation
  • [3]

    Dikranjan D. and Zava N., Some categorical aspects of coarse spacaes and balleans, Topology Appl., to appear.

  • [4]

    Dranishnikov A. and Smith J., Asymptotic dimensions of discrete groups, Fundam. Math. 189 (2006), 27–34.

    • Crossref
    • Export Citation
  • [5]

    de la Harpe P., Topics in Geometric Group Theory, University Chicago Press, Chicago, 2000.

  • [6]

    Protasov I. V., Morphisms of ball structures of groups and graphs, Ukr. Mat. Zh. 53 (2002), 847–855.

  • [7]

    Protasov I. and Banakh T., Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser. 11, VNTL, Lviv, 2003.

  • [8]

    Protasov I. V. and Protasova O. I., Sketch of group balleans, Mat. Stud. 22 (2004), 10–20.

  • [9]

    Protasov I. V. and Tsvietkova A., Decomposition of cellular balleans, Topology Proc. 36 (2010), 77–83.

  • [10]

    Protasov I. and Zarichnyi M., General Asymptology, Math. Stud. Monogr. Ser. 12, VNTL, Lviv, 2007.

  • [11]

    Roe J., Lectures on coarse geometry, Univ. Lecture Series 31, American Mathematical Society, Providence, 2003.

  • [12]

    Smith J., On asymptotic dimension of countable Abelian groups, Q. J. Math. 65 (2014), 1127–1144.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1]

    Banakh T., Cencelj M., Repovš R. and Zarichnyi I., Coarse classification of Abelian groups and amenable shift-homogeneous mentric spaces, Q. J. Math. 65 (2014), 1127–1144.

    • Crossref
    • Export Citation
  • [2]

    Banakh T., Higes J. and Zarichnyi M., The coarse classification of countable abelian groups, Trans. Amer. Math. Soc. 362 (2010), 4755–4780.

    • Crossref
    • Export Citation
  • [3]

    Dikranjan D. and Zava N., Some categorical aspects of coarse spacaes and balleans, Topology Appl., to appear.

  • [4]

    Dranishnikov A. and Smith J., Asymptotic dimensions of discrete groups, Fundam. Math. 189 (2006), 27–34.

    • Crossref
    • Export Citation
  • [5]

    de la Harpe P., Topics in Geometric Group Theory, University Chicago Press, Chicago, 2000.

  • [6]

    Protasov I. V., Morphisms of ball structures of groups and graphs, Ukr. Mat. Zh. 53 (2002), 847–855.

  • [7]

    Protasov I. and Banakh T., Ball structures and colorings of groups and graphs, Math. Stud. Monogr. Ser. 11, VNTL, Lviv, 2003.

  • [8]

    Protasov I. V. and Protasova O. I., Sketch of group balleans, Mat. Stud. 22 (2004), 10–20.

  • [9]

    Protasov I. V. and Tsvietkova A., Decomposition of cellular balleans, Topology Proc. 36 (2010), 77–83.

  • [10]

    Protasov I. and Zarichnyi M., General Asymptology, Math. Stud. Monogr. Ser. 12, VNTL, Lviv, 2007.

  • [11]

    Roe J., Lectures on coarse geometry, Univ. Lecture Series 31, American Mathematical Society, Providence, 2003.

  • [12]

    Smith J., On asymptotic dimension of countable Abelian groups, Q. J. Math. 65 (2014), 1127–1144.

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