A growth dichotomy for group algebras of free abelian by infinite cyclic groups (Developments of Language, Logic, Algebraic system and Computer Science)

A

_{growth dichotomy}

_{for group}

_algebras

of free abelian

by

infinite

_cyclic

_groups

Christopher Briggs

Embry‐Riddle

Aeronautical

_University

email:briggscl@erau.edu

Abstract

We

_{study growth}

in

_{algebras, especially}

uniform

_{exponential growth, extending}

historical results

in the

_topic

of

_growth

_{in groups. We}

_give

a

condition under

which the

_group

_algebra

ofafree

abelian

by

infinite

cyclic

_group

has uniform

exponential growth.

1

Introduction

Givena

finitely

_generated

_{group G and}afinite

generating

set

S=\{g_{1}^{\pm 1}, . . . , g_{T}^{\pm 1}\}

ofG,for each

element _g\in G, write

g=g_{i_{1}}^{n}\ldots g_{i_{k}}^{n_{k}}1

.Define the

length

ofgwithrespecttoStobethe minimal

nonnegative

integer

k for which suchan

expression

of_gis

possible

(the identity

is consideredtoUe

an_{empty word,}hence

oflength

0

_).

The

growthfunction

$\gamma$_{G,S}(n)

:

\mathbb{Z}_{\geq 0}\rightarrow \mathbb{N}

ofG with_respecttoS is the number of elements of G of

_length

atmostnwith_respecttoS.Wewrite

$\gamma$_{G,S}(n)=$\gamma$_{S}(n)

if

G is understood. The

_{growth being exponential}

orbounded

by

a

polynomial

is

independent

of the

chosen

_generating

set, sowe

speak

of groups of

exponential

or

polynomial growth

respectively.

\mathrm{A}

group which is neither is saidtohave intermediate

_growth.

Growth in groupswasintroduced

_{independently}

_Uy

Schwarz in 1955 and Milnor in 1968

_[8,

9].

In 1968 Wolf

_proved

thata

virtualy nilpotent

_{group has}

polynomial

growth,

anda

virtually

polycyclic

group which is not

_{virtualy nilpotent}

has

_{exponential growth}

_[11].

In 1981 Gromov

proved

thata

_{finitely generated}

_{group of}

_polynomial

_growth

is

_{virtually nilpotent [6].}

In 1968 Milnor asked whether there exists groups of intermediate

growth.

The

_question

was

answered

_{affirmatively by}

_Grigorchuk

in 1983 [5]. In 1981 Gromov defineda _group Gtohave

uniform exponential

growth

if

_{\displaystyle \inf_{S}(\lim_{n}$\gamma$_{S}(n)^{1/n})}

> 1. Inthe sameyear Gromov asked: if G

has

_{exponential growth,}

mustit have uniform

_exponential

_{growth [7]?}

The

_question

wasanswered

negatively by

Wilson 2004

_[10].

Inthemeantime,the

_question

wasanswered

positively

for several

classes of groups. In2002

_Alperin

_proved

thata

_virtually

_polycyclic

_{group has either}

_polynomial

oruniform

_{exponential growth}

_[1].

In2005

_Eskin,

_Moses, and Oh

_proved

alinear_groupover a

fieldofcharacteristiczerohas

polynomial

oruniform

exponential growth

[4].

In 2008Breuillard

and Gelander

_proved

thatalinear group of any fieId has

polynomial

oruniform

exponential

growth

[2].

数理解析研究所講究録

2

Growth

in

_algebras

Write

_A=F[S]

todenote that A is

_{generated (as}

an

algebra)

by

thesetS overthefield F. The

growth

function ofA with_respectto S is

_{$\gamma$_{A,S}(n)}

=\displaystyle \dim_{F}(\sum_{i=0}^{n}FS^{i})

. Growthtypeis

indepen‐

dent of the

_generating

set, soaswith_groups,wedefine

polynomial, exponential,

and intermediate

growth

for

_algebras.

Also as with groups, acommutative

algebra

has

polynomial growth,

and

the free

_algebra

on atleasttwoletters has

_{exponential growth.}

_{We say}an

_algebra

A has

uniform

exponential growth

if

\displaystyle \inf_{S}$\gamma$_{A,S}(n)^{\frac{1}{n}}>1.

Thereare

examples

of

algebras

of uniform

exponential growth:

Golod‐Shafarevich

algebras,

_group

algebras

of Golod‐Shafarevich groups, and any

algebra graded

by

\mathbb{N} with

_{exponential growth}

have

uniforn

_exponential

_growth

_[3].

An

_example

ofan

_algebra

of nonuniform

_exponential

_growth

is

the group

algebra

of Wilson’s groupof nonuniform

_{exponential growth}

over afieldof characteristic

0.

We are

particularly

interested in the

growth

of group

algebras,

as FG has

exponential,

poly‐

nomial,

orintermediate

growth

if and

_only

ifG does

_{respectively.}

IfFG has uniform

_exponential

growth,

sodoes G. Itisunclear, and it isa

motivating question

of the author’s

research,

whether

theconverseistrue.

3

Main result

The main resultconcerns acondition under whichthe group

algebra

ofafreeabelian

by

infinite

cyclic

group has uniform

exponential growth.

The desired end result is an extensionto _group

algebras

of

_Alperin’s

result thata

polycyclic

_{group has either}

polynomial

oruniform

exponential

growth.

The free abelian

_by

infinite

_cyclic

_groupisa

_building

block of the po

_ycyclic

_group, soit

makessense to

_{begin Uy}

_attempting

toestablish thata_group

algebra

ofafree abelian

by

infinite

cyclic

group is of

polynomial

oruniform

exponential

growth.

Henceforth,

let

_{$\Gamma$=G\rangle\triangleleft_{ $\sigma$}\mathbb{Z}}

beafree abelian

_by

infinite

_cyclic

_group. LetF beafield. Since

the action of $\sigma$on GcanUedescribed

_by

a_matrix,we canrefertothe

_eigenvalues

of $\sigma$.

By

aresultof

_{Alperin [1],}

combined with the fact that F $\Gamma$ has

_{polynomial growth}

if and

_only

if $\Gamma$

_does,

wehavethe result

Theorem 1

_{Ifall eigenvalues of}

$\sigma$havenorm1,then the_group

algebra

F $\Gamma$ has

polynomial

growth.

The

_{following simple}

result is often useful:

Lemma 2

_{Uniform exponential growth}

_lifts

_from

_homomorphic

_images

_for

_groups and

_algebras

[31

\cdot

The main result is:

Theorem 3

_If

$\sigma$hasareal

eigenvalue

$\lambda$ with

| $\lambda$|>1

,then F $\Gamma$ has

uniform exponential growth.

The_yetunsolvedcasetoestablish the

_{growth dichotomy (polynomial}

oruniform

exponential)

for free abelian

_by

infinite

_cyclic

_groupsis thecasewhen all

_eigenvalues

are

complex

butnotall

eigenvalues

havenorm1.

References

[1]

R.

_Alperin,

_{Uniform exponential growth of polycyclic}

_groups, Geom. Dedicata 92

₍₂₀₀₂₎

105−113

[2]

E. Breuillard andT. Gelander,

Uniform

independence for

linear_{groups, Invent.}Math. 173

(2008)

no.2,225−263

[3]

C.

_{Briggs, Examples of Uniform Exponential}

Growth in

_Atgebras,

J.

_Algebra

_Appl.,

DOI:

http:

//\mathrm{d}\mathrm{x}

_.doi.org

/10.1142/\mathrm{S}0219498817502413

[4]

A. Eskin, S Mozes and H. _Oh, On

_{uniform exponential growth for}

linear_groups, Invent.

Math. ló0

₍₂₀₀₅₎

no. 1,1−30

[5]

R.I.

_Grigorchuk,

On Milnor’s

_{problem of}

_group

_growth,

Soviet. Math. Dokl. 28

₍₁₉₈₃

\rangle23−26

[6]

M. Gromov,

Groups

of polynomial growth

and

_expanding

_maps, Inst. Hautes

Études

Sci.

Publ. Math. 53

(1981)

53−73

[7] M.

_Gromov,

Structures

_metriques

_pourlesvarietesriemanniennes,

CEDIC,

Paris,

1981,

MR

85e:5305l

[8]

J.

Milnor,

Growthin

_{finitely generated}

solvable_groups, J. Diff. Geom. 2

₍₁₉₆₈₎

447−449

[9]

A. S.

_Schwarz,

A volume invariant

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Dokl. Ak. Nauk USSR 105

₍₁₉₅₅₎

32−34

[10]

J.

_Wilson,

On

_{exponential growth}

and

_{uniformly exponential growth}

_for

_groups, Proc. Amer.

Math. Soc.60

₍₁₉₇₆₎

22−24

[11]

J.

_Wolf,

Growth

_{offinitely generated}

solvablegroupsandcurvature

_{ofRiemannian manifolds,}

J. Diff. Geom. 2

₍₁₉₆₈₎

434A46