Recurrent points and discrete points for elementary amenable groups

VOL. 15 NO. 4 (1992) 733-740

RECURRENT

POINTS

AND

DISCRETE

POINTS

FOR

ELEMENTARY

AMENABLE

GROUPS

MOSTAFANASSAR

Jodrey School ofComputerScience AcadiaUniversity

Wolfville,NovaScotia

BOP 1X0 Canada

(Received January 18, 1989 and in revised form March 31, 1992)

ABSTRACT. Letfig be the Stone-Cech compactification ofagroup

G, A

athe set of all

almostperiodicpoints in

G, K

a

c[U {

supp

eLIM(G)}]

and

R

athe set of all recurrent pointsin

fiG.

In

this paperwewillstudythe relationships between

K

a and

R

a,

andbetween

A

aand

R

a.

Wewillshow that forany infiniteelementaryamenable group

G, A

a

R

aand

R

a-

K

a

=/=

.

Key words and Phrases: Topological group, recurrent point, elementary amenable

groups.

1980 AMSsubject classification code. 43A60.

1.

INTRODUCTION

Let G be a discrete group,

M(G)

the Banach space of all bounded real valued

functionson Gwiththe supremum norm, and

M(G)*

the conjugate Banach spaceon

M(G).

A

group Giscalled left amenableifthereexistsa mean on

M(G)

which isleft

invariant

[4].

Denote by

LIM(G)

the set of left invariant means on

M(G).

By

using

Rieszrepresentationtheorem,then thereexistsaboundedregularBorelmeasure_#on

flS

such that

qo(f)

fd

i,a,

feM(G).

s

Forasubset

H

ofaleft amenable group

G

definetheupperdensityasfollows:

sup

{o(X,

)"

,LIM(G)}

sup

(H)

{/()

oeLIM(G)

}

where

X.

isthecharacteristic functionofH.

DEFINITIONS:

(1)

Foreach

A

C

G,

let

.

c.e’A-G.

The set

i.

isclosed andopen in

,SG-G.

Thenapoint

wel

issaid to bealmost periodicifforeveryneighbourhood U ofwthere

exist

A,K

C G with

K

finite, Aw C U, and

G

KA. Denote by

Ae"

the set of all

almost periodicpoints in

fiG.

(2)

we,SG

isrecurrentifwheneverU isaneighbourhoodofwthen the set

{xe.G

xwet.

is infinite. Let

R

G _{betheset ofall recurrent pointsin/G.}

(3) we.fiG

isdiscrete if its orbit

0(w)

isdiscrete withrespecttothesubspacc topology

of

fiG.

Let

D

G_{be the set of all such points in}

fiG.

(4) wefiG

is stronglydiscrete if there exists aneighbourhood U ofw such that xU ’lyU where x,

yeG,

x .y. Thesetof allstronglydiscretepoints will be denoted by

DSe’.

Let

WR

G _fig-SD

G,

_{the set of all weak recurrent points}

of/G.

If

G

isafinite group, then

Ae"

D

e"

K

G _{G fig and}

R

G

WR

G

.

Therefore,weareonlyinterested in infinitegroups. Fortheremainderofthis paper,G

stands foraninfinite group. LEMMAI.1.

(1)

A

G_C

R

G

C

WR

G,

SD

G

C

D

G_and

D

G

fiG-

R

G.

(2)

IfGisamenable, then

A

G

C

K

G_CWR

G. [4]

Asmentioned inDay

[3],

the familyofamenable groupsisclosed under the following four standardprocessesof constructing newgroups from givenones:

(a)

subgroup,

(b)

factor group,

(c)

groupextension,

(d)

expandingunion

(or

direct

limit). Denote

by

EG

the family ofelementaryamenable groups.

It

isthe smallestfamil.yof groups containing allfinitegroups and allabeliangroups, andisclosedunder

(a)-

(d).

As pointedout by Chou

[2],

the groups in EG can be constructed from abelian

groups andfinitegroups by applyingprocesses

(c)

and

(d)

only.

Moreover,

everyperiodic groupinEGislocallyfinite, andsince it isknown thatevery infinitelocallyfinitegroup

containsaninfinite abeliansubgroup

(Robinson

[5;

p.

95]),

wehave thefollowing:

LEMMA

1.2

(Chou).

IfGis aninfinitegroup in

EG,

then

G

containsaninfinite

2.

R0

SETS AND RECURRENT POINTS:

In this section wewill show that .4a

c

R

a and

R

a-

K

a

://=

,

for any infinite

elementary amenable groupG.

DEFINITION.

A

subset

A

ofanamenable groupGiscalled

R0-set

if:

(R0-1)

d(A)

0, and

(R0"2)

C

R

c"7

.

PROPOSITION 2.1: Ifanamenable group GcontainsanR0-set, then

A

a

c

R

a and

R

c"

K

a

7

.

PROOF: If

A

is an

R0-set

in

G,

then by

(R0" 2)

there exists

we.

gl

R

a since

-(A)

O,

wfK

a.

Theproofiscompleted.

R0-sets

were first studied by Chou for the case G

Z

where he showed that

Z

containssuchsets

(see

Chou

[1],

P.60, example

2).

In

this sectionwe will show that

every finiteelementary amenable groupcontains

R0-sets.

THEOREM 2.2: Let

{Ak}

bea sequenceofsubsets ofa group

G,

and

{gk}

bea

sequenceofdifferentelementsofGsuch that

1.

A

D

As

D

A3

D..., and

2.

g.A.+a

C

An

for alln. Then Cl

A.

containsarecurrent point.

PROOF.

Let rbe the family ofsequencesof closed subsets of

.

Letr bedefined asfollows: asequenceof closed subsets

{F,

}

of belongstorif for eachheN,

(i) F,,

C

fi-,,,

(ii)

F.+a

C

F.,

(iii)

gufn+l Q

rn,

and

(iv) F,

:

.

Note that ris non-empty since

i..

#er.

r can be orderedin the followingnatural

way:

{F. } _< {F. }

if andonlyif

F.

C

F.

foreachn. Itis easy tocheek that eachchain in r has a lower bound. Then using

Zorn’s

lemma, r has aminimalelement

{If.

}.

Let we 91

K..

Weclaimthatw isarecurrent point, byshowing that if

ft.

isanopen

Let V U

gA.

Consider the sequence

{K,-

V}.

Itsatisfies

(i), (ii),

and using the

9eG

fact thatfor9eG, gV

V,

onesees that

{K,-

V}

satisfies

(iii).

Since

and

{K,}

isminimalin

,

{K,

V}

4,therefore,

{K,

V}

does not satiny

(iv),

i.e., thereexistsn0

su

that

K,

V

wch implies

K,o

V

Gg.

Since

K,

is

compact,thereexistsa,a2,...,

ameG

such that

K.o

c

,,,,

...=

Foreachn

>_

no,g.wegnK.+z

C

K.

C

K.

o, thusby

(*)gnweaiA

forsome 1

_<

_<

m.

Then clearly, thereexists i0,i0

_<

ra such that g,weaioAfor infinitely many n. Thus

in ni o. horefore

REMARK:

WhenG

Z,

theabovetheoremis contained inChou

[1].

Theideaof

theproofof theabovetheorem follows from

[1,

Proposition

3.1].

LEMMA

2.3

(Chou

[1]):

The additivegroupof integers

Z

contains an

R0-set.

Forprime numberp,let

z(f)

{./f

0

<

.

< f;

.z,.

0,

z,

2,...}

be thesubgroupof

Q/Z

generated by

--’n=l 2,

pn

LEMMA

2.4: Foreachprime numberp,

A(p

)

containsan

R0-set.

PROOF:

Let G

A(p).

Then

G

is asubgroup of

Q/Z.

Note that

G

canbe

writtenasG U

H,

where

Hz

C

H2

C...and each 1 2

..p.

I}

H.

0,

p.,p.,.

P"

isacyclic groupoforderp". Forconvenience,wewillwrite

H.

(/.)

(with

theusual addition in

Q

(rood

Z)).

Then following Chou’s construction for

Z

[1],

we define a sequenceof subsets

E.

inG byinductionasfollows:

1

Let

E=,=IE,-/

;1

1 1 1 1 1

+

+

1 1 1

1 1 1 1 1 1 1

+

+

1 1

+

p25’

p

1 1 1 1 1

1 1

p9;p

1 1

p16’p

Considerthe

F-sequence

{F,

}

inGwhere

F.

H,2.

Then

and

n-1

(E)

1i,m

IE"l

glim 2 =0. Thus

E

satisfies

(R0-1).

Wemay chooseasequenceofinfinitesubsets C, inGsuch that

1 1 1 1 1 1 1 1 1

Cl--

p,

-

-}- -{-

--1 1 11 1 1 1 1 1 1 1 1

=-+

+

+

+

+

+

+

+

Ca=

,+--,+p--,...,

andsoon.

So

EDCI DC

D...

and

C,+

+

p-("+)

CC, for all n.

Then by theorem 2.2,

E

containsarecurrent point, and hence

E

satisfies

(R0 2).

REMARK: If

X

isadiscrete set andYisasubsetof

X,

thenwewillconsider

asasubset of

fiX.

Indeed flyisthe closure of

Y

in

fiX.

LEMMA

2.5: Ifanamenable group

G

contains aninfinitesubgroup

H

ofinfinite

PROOF: Since

[G" H]

oo, the number of disjoint cosets

{tH}

is infinite. For any

eLIM(G),

since

(XH)

(X,H)

iftlH,t2H,...,t.H are distinct cosets, then

(X,,.u,..

u,n.

(X,,

+...

+

(X,n.

n(X,..

<

1. Since n is arbitrary,

(X,,

0so

(H)

0. Thus

H

satisfies

(R0-1).

To

see that

H

satisfies

(R0-2)

in

G,

note that ifweRH and if U is aneighbourhood ofw in /G, then UCl is also a

neighbourhoodofwand

X’

{

hH" hw U

N}

isinfinite. Therefore

{gtG" gu,U}

(D

X)

isinfinite. ThusweR

a.

Since

#

A

H

C

RH,

R [3-

#

.

Thus

H

satisfies

(R0" 2)

inG.

COROLLARY

2.6: Ifan amenable groupGisadirect product of infinitelymany

non-trivialsubgroups

{G.

cI},

thenGcontainsan

R0-set.

PROOF:

Toprove that GcontainsanR0-set,weconstuct aninfinitesubgroup

H

ofGof infinite index with theabovelemma. Indeed,writeI

I1

"t9

I2

such that both

I

and

I2

areinfinite, then

H

r

{G,

treI1

}

iswhat wewant.

LEMMA

2.7: Ifaninfinitesuchgroup

H

ofanamenable groupGcontains an/’Q-set

A,

then

A

isalsoan

R0-set

inG.

PROOF:

This isquite obvious since whenever

dH(A)

0 then

d(A)

0 and

R

H _{C_.R}

G.

LEMMA

2.8: If

H

isan amenable subgroupofan amenable group G such that

G/H

hasan

R0-set

A’,

then

8-(A

’)

isan

R0-set

inG

(8

isthe natural homomorphism ofGonto

G/H).

PROOF:

Let

A

8-(A

’)

and

G’

G/H.

Then for each

eLIM(G),

q([A

’.P(XA

.8)

8*p(X,

0since

8*eLIM(G’)

and

dc,(A’)

0. Then

A

satisfies

(R0-1).

The natural homomorphism 8 ofGonto

G’

canbeextendedtoacontinuous map-ping

of/G

onto/G’.

Wewill denotetheextended mapping againby8. Itis not hard to checkthat

8-(R

G’)

C

R

G,

sine

A’-

t3

R

’

#

4, and

A-

gl

R

a

#

either. Thus

A

satisfies

(R0 2),

and

A

isan

R0-set

aswanted.

Before providingourmainresult,wewill needsome definitionsandsomestructure theorems forabeliangroups. Fortheproofssee

[6].

A

subgroup SofGis inGincase nGt3S nSfor everyintegern.

IfGisaperiodic group, thenasubgroup

B

ofG isabasicsubgroup ofGin the followingcases:

(i) B

isthedirectsumof cyclic groups;

(ii) B

is pure in

isdivisible.

THEOREM2.9"

1. Everyperiodic group Gcontains abasicsubgroup B. 2. Everyperiodic group Gisadirectsumof p-groups.

3. Everyperiodicgroupisanextensionofadirectsumof cyclic groupsbyadivisible

group.

4. Everydivisiblesubgroup

D

is adirect sumofcopies of

Q

andofcopies of

Z(p).

We arenow readytostateandprovethemainresults of thispaper.

THEOREM 2.10: If

G

isaninfinite abeliangroup, thenGcontainsan

R0-set.

PROOF: Wewill consider twocases:

1. G: non-periodic. ThenGcontains an infinitecyclicsubgroupwhichcanberegarded

asthe additive group of integers Z. Thusbylemmas 2.3 and2.7, thereexistsan

R0-set.

2. G: periodic. Thenby 2.9,

G

containsabasicsubgroup

B,

sothat

B

is adirect sumof cyclic groups. Herewehave two subcases:

(a)

If

B

isaninfinite subgroup, by corollary 2.6 andlemma 2.7, thereexists an

R0oseto

(b)

If

B

isafinitesubgroup,the question group

G/B

isaninfinitedivisible that canbewrittenas adirect sumof

_<

copies of

Z(P) (see 2.9).

Considerone of these copies andapplylemmas2.4and 2.7 togetan

R0-set

inG. Then the proof of the theoremis completed.

THEOREM 2.11" IfGis aninfiniteelementary amenable group, then Gcontains an

R0-set.

THEOREM2.12: IfGisaninfiniteelementaryamenablegroup, then

R

DA

and

Ra-K

.

Theprooffollows from the above theorem and proposition 2.1.

COROLLARY

2.13: IfGisaninfiniteelementaryamenable group, then

RG

C

w

RG.

Thisfollows immediately from the above theorem and lemma1.1.

CONJECTURE: Everyinfiniteamenable groupGcontainsanR0-set, and therefore

A

CR

and

R

G-K

G:.

ACKNOWLEDGEMENT:

The author would like to thank thereferee for his comments.

REFERENCES

1.

CHOU,

C.,

Minimalsets, Recurrentpoints andDiscrete orbits in

fiN

N,

IllinoisJ.ofMath.,22

#1 (1978)

pp. 54-63.

2.

CHOU, C.,

Elementaryamenable groups, IllinoisJ.ofMath.,24

#

3

(1980)

pp. 396-407.

3.

DAY,

M. M. Amenable semigroups; IllinoisJ. ofMath., 1

(1957)

pp. 509-544. 4.

NASSAR,

M.,

Recurrentandweaklyrecurrent pointin

G.

InternationalJ.of

Math and MathSciences,9

No.

2(1986)

pp. 319-322.

5.

ROBINSON, J.,

Finiteness conditionsand generalized soluble groupsPart

!,

Springer-Verlag,

New

York

(1972).