Spectra of non-commutative dynamical systems and graphs related to fractal groups

Texte intégral

(1)

Article

Reference

Spectra of non-commutative dynamical systems and graphs related to fractal groups

BARTHOLDI, Laurent, GRIGORCHUK, Rostislav

Abstract

We study spectra of noncommutative dynamical systems, representations of fractal groups, and regular graphs. We explicitly compute these spectra for five examples of groups acting on rooted trees, and in three cases obtain totally disconnected sets.

BARTHOLDI, Laurent, GRIGORCHUK, Rostislav. Spectra of non-commutative dynamical systems and graphs related to fractal groups. Comptes rendus mathématique , 2000, vol.

331, no. 6, p. 429-434

DOI : 10.1016/S0764-4442(00)01658-X arxiv : math/0012174v1

Available at:

http://archive-ouverte.unige.ch/unige:12527

Disclaimer: layout of this document may differ from the published version.

1 / 1

(2)

arXiv:math/0012174v1 [math.GR] 18 Dec 2000

Spetra of non-ommutative dynamial systems and

graphs related to fratal groups

Laurent BARTHOLDI, Rostislav I. GRIGORCHUK

SetiondeMathématiques,UniversitédeGenève,CP240,1211 Genève24,Suisse

E-mailaddress: Laurent.Bartholdimath.unige.h

SteklovInstituteofMathematis,Gubkina8,Mosow117966,Russia

E-mailaddress: grigorhmi.ras.ru

(Transmisle2juin2000)

Abstrat. We study spetra of nonommutative dynamial systems, representations of fratal groups, and

regulargraphs. Weexpliitlyomputethese spetra forveexamplesofgroupsatingon rooted

trees,andinthreeasesobtaintotallydisonnetedsets.

Spetres de systèmes dynamiques non-ommutatifs et de graphes

assoiés à des groupesfratals

Résumé. Nousétudionslesspetresdesystèmesdynamiquesnonommutatifs,dereprésentationsdegroupes

fratals,etdegraphesréguliers. Nousalulonsesspetrespourinqexemplesdegroupesagissant

surdesarbresenrainés,etdanstroisasobtenonsdesensemblesomplètementdéonnetés.

Version française abrégée

Le but deettenote est d'étudierle spetrede systèmesdynamiques non ommutatifs, i.e. desystèmes

engendrés par plusieurs transformations qui ne ommutent pas néessairement. Elle résume les résultats

de[1℄.

Soit

G

ungroupeengendréparunensemblesymétriqueni

S

.Unsystèmedynamique,noté

(S, X, µ)

,est

uneationde

G

surunespaemesuré

X

préservantlalasse

[µ]

delamesure

µ

de

X

.Ilinduitnaturellement une représentationunitaire

π

de

G

dans

L 2 (X, µ)

donnéepar

(π(g)f )(x) = p

g(x)f (g −1 x)

,

g(x)

est la

dérivéedeRadon-Nikodým

dgµ(x)/dµ(x)

.Lespetrede

(S, X, µ)

estlespetredel'opérateurdetypeHeke

H π = 1

| S | X

s∈S

π(s) ∈ B (L 2 (X, µ)).

Plusgénéralement,lespetred'unereprésentationunitaire

π : G → U ( H )

d'ungroupeavesystèmegénéra-

teurxéestlespetredel'opérateur

H π

ommei-dessus.

Ononsidère,pourinqexemplesdegroupesdonnésparleurationsurunarbrerégulierenrainé

T

,leur

représentation

π

dans

L 2 (∂ T , µ)

µ

estlamesuredeBernoulliuniforme.Cettereprésentations'approxime parlesreprésentations

π n

surlessommetsàdistane

n

delarainede

T

.

Nousmontronsque

π

sedéomposeenunesommedereprésentations

π n ⊖ π n−1

dedimensionsnies, et

spec(π) = S

n≥0 spec(π n )

.

H π

a un spetre purement pontuel et son rayon spetral est une valeur propre. On peut dérire ex-

pliitement es spetresomme suit. Pour

λ ∈ R

, soit

J (λ)

l'ensemble de Juliadu polynmequadratique

z 7→ z 2 − λ

:

J(λ) = r

λ ± q λ ± p

λ ± √ . . .

.

NoteprésentéeparJean-PierreSerre

2000Aadémiedes Sienes

(3)

Groupe Spetrede

H π

Desription

G [ − 1 2 , 0] ∪ [ 1 2 , 1]

deuxintervalles

G ˜ [0, 1]

intervallepositif

Γ { 1, 1 4 } ∪ 1 4 (1 ± J (6)) }

union d'un ensemble de Cantor de

mesurenulleet depointsisolés

Γ { 1, − 1 2 , 1 4 } ∪ 1 4

1 ± q

9

2 ± 2J ( 45 16 )

ensembledeCantordemesurenulle

Γ

identiqueà

Γ

Ces aluls impliquent l'existenede graphesà roissanepolynomiale,qui sont lesgraphesde Shreier

degroupesderoissane intermédiaire, et dont lespetredel'opérateurdeMarkovest un quelonquedes

ensemblesi-dessus.Unrésultatanalogueest vraipourlessystèmesdynamiques.

1. Introdution

Thepurposeof thisnoteisthestudyofspetraofnonommutativedynamialsystems(thatis, systems

generated by several transformations that do not neessarily ommute). We produe several examples of

omputationsofsuhspetrawithaninterestingtopologialstruture. Moredetails appearin[1℄.

In the lassial ase dened by asingle aperiodi measure-preserving transformation

T : X → X

, the

spetrumoftheorrespondingoperator

1

2 (U + U −1 )

,with

U ∈ U (L 2 (X ))

,is

[ − 1, 1]

byRohlin'sLemma,but

inthenonommutativeasethespetrummayhavegaps, andevenbeaCantorset.

Wealso studyspetraofinniteregulargraphsand produetherstexampleof aregulargraphwhose

spetrumisaCantorset.

Our dynamial systems (with assoiated Heke operator) arise from ations of fratal groups on the

boundary of the regular rooted tree on whih they at. The graphs (with assoiated Markov operator)

whose spetra we onsider are the Shreiergraphs of these groupsoverparabolisubgroups. Inspeial

ases,thesegraphsaresubstitutionalgraphs andhavepolynomialgrowth.

If theunderlying groupisamenable,then thespetrumofthe thedynamialsystemoinides withthe

spetrumoftheorrespondinggraph.

The omputation of theabove spetrais based onoperator reursions that hold for fratal groupsand

involvesa

1

-dimensional and

2

-dimensionallassialdynamial systemasanintermediatestep. This leads totheappearaneofJuliasetsofquadratimapsinthedesriptionoftheabovespetra.

Both authorswish tothankheartily PierredelaHarpeandAlain Valettefortheirnumerousomments

andontributionstothisnote.

2. Spetraof Dynamial Systems andRepresentations

Let

G

beagroupnitelygeneratedbyasymmetriset

S

. Anon-ommutativedynamialsystem,denoted

(S, X, µ)

, isanationofagroup

G

(generatedby

S

)onaspae

X

andpreserving themeasurelass

[µ]

of

ameasure

µ

on

X

.

Suh adynamialsystemgivesrisetoanaturalunitaryrepresentation

π

of

G

in

L 2 (X, µ)

givenby

(π(g)f )(x) = p

g(x)f (g −1 x),

where

g(x) = dgµ(x)/dµ(x)

is the Radon-Nikodým derivative. The spetrum of the dynamial system

(S, X, µ)

isthespetrumoftheHeketypeoperator

H π = 1

| S | X

s∈S

π(s) ∈ B (L 2 (X, µ))

(thisterminologyomesfrom ananalogywithHekeoperatorsin numbertheory[13℄.) Moregenerally,the

spetrumof aunitary representation

π : G → U ( H )

ofa groupwithagivennite set of generatorsisthe

spetrumoftheoperator

H π

asabovesee[10℄.

Denition. Agraphisapair

G = (V, E)

ofsets(thevertiesand edges),amap

α : E → V

(thestartofan

edge)andaninvolution

· : E → E

(theinversion). Onedenesthenthe end

ω(e)

ofanedgeby

ω(e) = α(e)

.

The degreeof avertex is

deg(v) = |{ e ∈ E | α(e) = v }|

. The graphis loallynite if

deg(v) < ∞

for all

v ∈ V

, and is regularif

deg(v)

isonstant over

V

. The graph isa tree if in every iruit

(e 1 , e 2 , . . . , e n )

of edges with

ω(e i ) = α(e i+1 )

(indies modulo

n

) there is a redution, i.e. an

i

with

e i = e i+1

. A graph

morphismisapairof maps between thevertexandedgesets thatommute with

α

and

·

.

(4)

Let

Σ = { 1, . . . , d }

beanitesetofardinality

d

. The

d

-regularrootedtree

T d

isthegraphwithvertexset

Σ

,edgeset

Σ × Σ × {±}

,andmaps

α(σ, s, +) = σ

,

α(σ, s, − ) = σs

and

(σ, s, +) = (σ, s, − )

. Its boundary

is

∂ T = Σ N

,the setof innitesequenesover

Σ

.

Suppose nowthat

G

ats by automorphisms ona rooted tree

T

. This ation extends to aontinuous

ationontheboundaryofthetree,whihisaompat,totallydisonnetedspae. Theuniformmeasureon

Σ N

isthemeasure

µ

denedontheylinders

σΣ N

(for

σ ∈ Σ

)by

µ(σΣ ) = d −|σ|

. Itis

G

-invariant,andis theuniqueinvariantmeasureif

G

atstransitivelyoneahlevel

Σ n

ofthetree,orequivalentlyif

(S, Σ N , µ)

isergodi. Notethat allothernondegenerateBernoullimeasuresarequasi-invariant.

Foreah

n ∈ N

,let

π n

betheunitaryrepresentation(ofnitedimension

d n

)of

G

in

2n )

induedby

theationof

G

on the

n

-thlevel,andlet

π

be theunitaryrepresentationof

G

in

L 2N , µ)

. Then

π n

is a

subrepresentationof

π n+1

and of

π

, andthespetraof

π n

onvergetothat of

π

:

spec(π) = [

n∈ N

spec(π n ).

3. Spetra of Graphs

Let

G

bealoallynitegraph. TheMarkov operator of

G

istheoperator

M

on

2 (V )

givenby

(M f)(v) = 1

deg v X

e∈E: α(e)=v

f (ω(e)).

Theoperator

M

is thetransition operatorforthesimplerandomwalkon

G

.

The spetral propertiesof

M

areof great importane;for instane, atheorem ofKesten [11℄ (extended

by Dodziuk and others)laimsthat agraph

G

ofbounded degreeis amenableifand onlyif

1 ∈ spec(M )

.

(Seemoreonamenabilityin Setion5.)

Denition. Let

G

be a group nitely generated by a symmetri set

S

, and let

H

be any subgroup. The

Shreiergraphof

G

withrespetto

H

is the graph

S (G, H, S)

withvertexset

G/H

and edge set

S × G/H

,

andmaps

α(s, gH) = gH

and

(s, gH) = (s −1 , sgH)

. Ithas anatural base-point

H

.

In Subsetion 4.3 the graphs will be labelled. This is simply done by assigning to eah edge

(s, gH)

the

labeling

s

.

If

H = 1

,weobtaintheusualCayleygraphof

(G, S)

. Notethat

S (G, H, S)

isan

| S |

-regulargraph,but

itsautomorphism groupdoesnot neessarilyattransitively onits verties. Indeed, basiallyanyregular

graphisaShreiergraph[12,Theorem5.4℄.

G

atsbyleft-multipliationon

G/H

, thevertexsetof

S (G, H, S)

. Theorrespondingunitaryrepresen- tationin

2 (G/H)

isthequasi-regular representation

ρ G/H

.

Suppose nowthat

G

atsonarooted tree

T d

. Fix theray

e = dd · · · ∈ Σ N

. Let

P n

bethestabilizer of

vertexatlevel

n

ofthisray,andlet

P = T

n∈ N P n

bethestabilizeroftheinniteray

e

(itisalledaparaboli

subgroup). We write

M n

the Markov operator of

S (G, P n , S)

and

M

the Markovoperator of

S (G, P, S)

;

theyaretheHeketypeoperatorsassoiatedwiththeationof

G

on

Σ n

and

Σ N

.

4. FratalGroupsandSubstitutional Graphs

Let

G

beagroupatingonatree

T

,andlet

H = T

s∈Σ Stab G (s)

betherstlevel stabilizer. Restriting theationof

H

toeah subtreespannedby

givesanembedding

ψ : H → Aut ( T ) Σ .

Denition. The group

G

is fratalif

ψ

isa subdiretembeddingof

H

in

G Σ

, i.e. if

ψ(H )

lies in

G Σ

and

itsprojetion oneah fatorisonto.

Let now

G

be agroup nitely generated by asymmetri set

S

, and let

(X, x 0 )

be apointed spae on

whih

G

ats. Thegrowth of

X

is thefuntion

γ : N → N

givenby

γ(n) = |{ x ∈ X : x = s 1 . . . s n x 0

forsome

s i ∈ S }| .

X

haspolynomial growth if

γ(n) ≤ n D

forsome largeenough

D

, has exponential growth if

γ(n) ≥ λ n

for

somesmallenough

λ > 1

,andhasintermediategrowthintheremainingases. Thegrowthof

G

isitsgrowth

underitsationonitselfbyleftmultipliation.

Wedesribenowbrieyvearhetypialexamplesof groups.

(5)

4.1. The Groups

G

and

G ˜

. Therstgroupwasintroduedbytheseondauthorin1980[5℄;bothgroups

aton

T 2

. Let

a

be theautomorphismpermuting thetoptwobranhesof

T 2

, and denereursively

b, c, d

by

φ(b) = (a, c)

,

φ(c) = (a, d)

and

φ(d) = (1, b)

. Let

G

bethegroupgeneratedby

{ a, b, c, d }

.

Dene also

˜ b, ˜ c, d ˜

by

φ(˜ b) = (a, ˜ c)

,

φ(˜ c) = (1, d) ˜

and

φ( ˜ d) = (1, ˜ b)

, and let

G ˜

bethe groupgeneratedby

{ a, ˜ b, ˜ c, d ˜ }

. Clearly

G ˜

ontains

G

asthesubgroup

h a, ˜ b˜ c, ˜ c d, ˜ d ˜ ˜ b i

.

4.2.

GGS

Groups. Let

d

beaprime number. Denote by

a

theautomorphismof

T d

permutingylially

thetop

d

branhes. Fixasequene

ǫ = (ǫ 1 , . . . , ǫ d−1 ) ∈ ( Z /d) d−1

. Denereursivelytheautomorphism

t ǫ

of

T d

,written

t ǫ = (a ǫ 1 , . . . , a ǫ d −1 , t ǫ )

,by

t ǫ (σ 1 σ 2 . . . σ n ) =

( σ 1 a ǫ σ 1 (σ 2 . . . σ n )

if

1 ≤ σ 1 ≤ d − 1, σ 1 t ǫ (σ 2 . . . σ n )

if

σ 1 = d.

Then

G ǫ

isthesubgroupof

Aut ( T d )

generatedby

{ a, t ǫ }

. Itisalleda

GGS

group [3℄.

Thefollowingresultsbelongto folklore(see[6℄):

G ǫ

isaninnitegroupifanonlyif

ǫ 6 = (0, . . . , 0)

. Itis

atorsiongroupifandonlyif

P ǫ i = 0

. Theonlythree innite

GGS

groupsfor

d = 3

areasfollows:

Let

r

be the automorphism of

T 3

dened reursively by

φ(r) = (a, 1, r)

, and let

Γ

be the subgroup of

Aut ( T 3 )

generatedby

{ a, r }

. It wasrstonsideredbyNarainGuptaandJaekFabrykowski[4℄.

Dene

s

by

φ(s) = (a, a, s)

,andlet

Γ

bethesubgroupof

Aut ( T 3 )

generatedby

{ a, s }

.

Dene

t

by

φ(t) = (a, a −1 , t)

,andlet

Γ

bethesubgroupof

Aut ( T 3 )

generatedby

{ a, t }

;itwasrststudied

inthe80'sbyNarainGuptaandSaid Sidki[8,9℄.

Theorem1. The groups

G, G ˜

and all innite

GGS

groups with

d ≥ 3

have intermediate growth. The Shreier graph

S (G, P, S)

orresponding to

G = G , G ˜

orany

GGS

group has polynomialgrowth.

In partiular,

S ( G , P, S)

and

S ( ˜ G , P, S)

have lineargrowth, while

S (Γ, P, S )

,

S (Γ, P, S )

and

S (Γ, P, S)

havepolynomialgrowthofdegree

log 2 (3)

.

4.3. Substitutional Graphs. We givea self-ontaineddesription of the Shreier graphs

S (G, P, S)

for

theexamplesabove,intheformofsubstitutionalrules. Inthissubsetion,allgraphsshallhaveabasepoint,

andshallbeedge-labelled;graphembeddingsmustpreservethelabelings.

Denition. Asubstitutionalruleisatuple

(U, R 1 , . . . , R n )

,where

U

isanite

d

-regularedge-labelledgraph, alledthe axiom,andeah

R i

isaruleoftheform

X i → Y i

,where

X i

and

Y i

areniteedge-labelledgraphs.

Thegraphs

X i

are requiredtohave noommon label. Furthermore, thereisan inlusion, written

ι i

, of the

verties of

X i

inthe vertiesof

Y i

;the degree of

ι i (x)

isthe sameasthe degree of

x

for all

x ∈ V (X i )

,and

allvertiesof

Y i

not inthe image of

ι i

havedegree

d

.

Given a substitutional rule, one sets

G 0 = U

and onstruts iteratively

G n+1

from

G n

by listing all

embeddingsof all

X i

in

G n

(noting that theyare disjoint),and replaingthembytheorresponding

Y i

. If

thebase point

of

G n

isinagraph

X i

,thebase pointof

G n+1

willbe

ι i ( ∗ )

.

Note that this expansionoperation preservesthe degree,so

G n

is a

d

-regular nite graphforall

n

. We

areinterestedin xed pointsof this iterativeproess, orequivalently in aonvergingsequene of balls of

inreasingradiusin the

G n

, andallalimitgraph(whihexistsby[7℄)asubstitutionalgraph.

Theorem2. For the ve examples

G

desribed above, the Shreier graphs

S (G, P, S)

are substitutional graphs.

Asanillustration,hereisthesubstitutionalruleforthegroup

Γ

:

a

a a

ρ σ

τ

-

a a a

2 ρ 1 ρ

0 ρ

a a a

1 σ 0 σ

2 σ

a a a

0 τ 2 τ

1 τ

s s

s

axiom

a a a

2

s

1

s

0

s

(6)

5. Amenability and Spetra

Let

G

be agroupatingonaset

X

. This ationisamenable in thesense of vonNeumann[14℄ ifthere

existsanitelyadditivemeasure

µ

on

X

,invariantundertheationof

G

,with

µ(X) = 1

.

A group

G

isamenable ifitsationonitselfbyleft-multipliationisamenable.

We nowstate the main onnetion betweenthe spetraof our representationsand dynamialsystems.

Reall

π

istherepresentationof

G

on

L 2N )

and

π n

istherepresentationof

G

on

L 2n )

. Sine

π n

ontains

π n−1

wewrite

π n = π n−1 ⊕ π n

.

Theorem3. Let

G

be agroup atingon aregularrootedtree, with

π

,

π n

and

π n

beasabove.

1.

π

isareduiblerepresentationofinnitedimensionbutsplitsas

π 0 ⊕ L

n≥1 π n

,soallofitsirreduible

omponents arenite-dimensional. Moreover

spec(π) = S

n≥0 spec(π n )

.

However, if

G

isweakbranh (see[2℄), then

ρ G/P

isirreduible.

2. The representations

π n

and

ρ G/P n

are equivalent, so their spetra oinide. The spetrumof

ρ G/P

is

ontainedin

∪ n≥0 spec(π n )

,andtherefore isontainedin the spetrumof

π

.

If moreovereither

P

or

G/P

are amenable, these spetra oinide, andif

P

is amenable, theyare

ontainedin thespetrumof

ρ G

.

3.

H π

hasapure-pointspetrum,anditsspetralradius

r(H π ) = s ∈ R

isaneigenvalue,whilethespetral radius

r(H ρ G/P )

isnotaneigenvalueof

H ρ G/P

. Therefore

H ρ G/P

and

H π

aredierentoperatorshaving

the samespetrum.

6. Resultson Spetra

Sineallthegroupsonsideredhaveintermediategrowth,theyareamenableand

spec ρ G/P = spec π

. We

nowdesribeexpliitlythesespetra. For

λ ∈ R

let

J (λ)

betheJuliaset ofthequadratimap

z 7→ z 2 − λ

:

J (λ) =

 s

λ ± r

λ ± q

λ ± √ . . .

 .

Group Spetrumof

M

Desription

G [ − 1 2 , 0] ∪ [ 1 2 , 1]

twointervals

G ˜ [0, 1]

nonnegativeinterval

Γ { 1, 1 4 } ∪ 1 4 (1 ± J(6)) }

union of Cantor set of null Lebesgue

measureandset ofisolatedpoints

Γ { 1, − 1 2 , 1 4 } ∪ 1 4 1 ± q

9

2 ± 2J ( 45 16 )

Cantorset ofnullLebesguemeasure

Γ

sameasfor

Γ

These omputationsimplythefollowingresults:

Theorem4. 1. There are onneted

4

-regular graphs of polynomial growth, whih is are the Shreier

graphs of groups of intermediate growth, and whose Markov operator's spetrum is any of the above

sets.

2. There arenonommutativedynamial systems generatedby

3

(inthe ase of

G

),

4

(in the aseof

G ˜

)

or

2

transformations, whose spetrumisanyof the above sets.

These spetraare all omputedusing the sametehnique: onsider the representation

π n

of dimension

d n

, givenby

d n × d n

-permutationmatries

π n (s)

, forall

s ∈ S

. Thesematriessatisfyblokidentities,for instaneforthegroup

G

π n (a) =

0 1 d n −1

1 d n −1 0

, π n (b) =

π n−1 (a) 0 0 π n−1 (c)

,

π n (c) =

π n−1 (a) 0 0 π n−1 (d)

, π n (d) =

1 d n −1 0 0 π n−1 (b)

.

Notethatforourveexamples

π n (s)

isexpressed bybloksoftheform

0

,

1

and

π n−1 (s )

.

Dene nowthepolynomial

Q : C S+1 → C

by

Q n ( { X s } , λ) = det X

s∈S

X s π n (s) − λ

!

.

(7)

Thespetrumof

π n

is

{ λ ∈ C | Q n ( |S| 1 , . . . , |S| 1 , λ) = 0 }

. Usingthe aboveblokidentities, it is possible to express

Q n

asrationalexpressionover

Q n−1

,andthereforeto omputeindutively

Q n

andthespetrumof

π n

. Forthegroup

G

,forinstane,wehavefor

n ≥ 2

,setting

α = X a

and

β = X b = X c = X d

,

Q n (α, β, λ) = (3β 2 + 2λβ − λ 2 ) 2 n−2 Q n−1

2

2 λ − λ 2 , λ − β + (λ − β)α 22 + 2βλ − λ 2

.

Inallourasesthepolynomials

Q n

anbeexpliitlyomputed;againforthegroup

G

,wehavethe

Lemma1. Dene

Φ 0 = α + 3β − λ, Φ 1 = − α + 3β − λ,

Φ 2 = − α 2 − 3β 2 − 2βλ + λ 2 , Φ n = Φ 2 n−1 − 2(2α) 2 n −2

for

n ≥ 3.

Thenfor all

n ∈ N

wehave

Q n (α, β, λ) = Φ 0 Φ 1 · · · Φ n

.

We show below theset ofvanishing points of

Q 6 (X a = X a −1 = α, X r = X r −1 = 1, λ)

for thegroup

Γ

,

andtheShreiergraph

S (Γ, P 6 , { a ±1 , r ±1 )

.

λ

α

-2 -1 0 1 2 3 4

-2 -1 1 2

On montrei-dessus l'ensemble despointsd'annulation de

Q 6 (X a = X a −1 = α, X r = X r −1 = 1, λ)

pour

legroupe

Γ

,etlegraphede Shreier

S (Γ, P 6 , { a ±1 , r ±1 )

.

Referenes

1. LaurentBartholdiandRostislavI.Grigorhuk,OnthespetrumofHeketypeoperatorsrelatedtosomefratalgroups,to

appearinPro.SteklovInst.Math.,1999.

2. ,Sous-groupesparaboliquesetreprésentationsdegroupesbranhés,submitted,2000.

3. GilbertBaumslag,Topisinombinatorialgrouptheory,LeturesinMathematis,ETHZürih,BirkhäuserVerlag,Basel,

1993.

4. JaekFabrykowski andNarainD.Gupta,Ongroupswith sub-exponentialgrowthfuntions,J.IndianMath. So. (N.S.)

49(1985),no.3-4,249256.

5. RostislavI.Grigorhuk,OnBurnside'sproblemonperiodigroups,Funktsional.Anal.iPrilozhen.14(1980),no.1,5354,

Englishtranslation: FuntionalAnal.Appl.14(1980),4143.

6. ,Justinnitebranhedgroups,HorizonsinProniteGroups(DanSegal,MarkusP.F.duSautoy,andAnerShalev,

eds.),Birkhaüser,Basel,2000,pp.121179.

7. RostislavI. Grigorhuk and Andrzej›uk, On the asymptoti spetrum of random walks on innite families of graphs,

Proeedingsofthe ConfereneRandomWalksandDisretePotentialTheory (Cortona)(M.Piardelloand W.Woess,

eds.),SymposiaMathematia(CambridgeUniversityPress),2228June1997,(toappear),pp.134150.

8. NarainD.GuptaandSaidN.Sidki,OntheBurnsideproblemforperiodigroups,Math.Z.182(1983),385388.

9. ,Someinnite

p

-groups,AlgebraiLogika22(1983),no.5,584589.

10. Pierre de laHarpe, A. Guyan Robertson, and AlainValette, On the spetrum of thesum of generators for a nitely

generatedgroup,IsraelJ.Math.81(1993),no.1-2,6596.

11. HarryKesten,Symmetrirandomwalksongroups,Trans.Amer.Math.So.92(1959),336354.

12. AlexanderLubotzky,Cayleygraphs:Eigenvalues,expandersandrandomwalks,Surveysinombinatoris,1995(Stirling),

CambridgeUniv.Press,Cambridge,1995,pp.155189.

13. Jean-PierreSerre,Répartitionasymptotiquedesvaleurspropresdel'opérateurdeHeke

T p

,J.Amer.Math.So.10(1997),

no.1,75102.

(8)

14. JohnvonNeumann,ZurallgemeinenTheoriedesMasses,Fund.Math.13(1929),73116and333,=Colletedworks,vol.

I,pages599643.

Figure

Updating...

Sujets connexes :