Zeta function of a group

Zeta function of a group

Let

G

be a

finitely generated group

and let

𝒳

be a
family of finite index

subgroups

of

G

. Define

a

n

(

𝒳

)

=

|

{

H

∈

𝒳

∣

|

G

:

H

|

=

n

}

|

.

Note that these numbers are finite since a finitely generated group
has only finitely many subgroups of a given index. We define the

zeta function

of the family

𝒳

to be the formal

Dirichlet series

ζ

𝒳

⁢

(

s

)

=

∑

n

=

1

∞

a

n

⁢

(

𝒳

)

⁢

n

-

s

.

Two important special cases are the zeta function counting all
subgroups and the zeta function counting

normal subgroups

. Let

𝒮

⁢

(

G

)

and

𝒩

⁢

(

G

)

be the families of all finite
index subgroups of

G

and of all finite index normal subgroups of

G

, respectively. We write

a

n

⩽

⁢

(

G

)

=

a

n

⁢

(

𝒮

⁢

(

G

)

)

and

a

n

⊴

⁢

(

G

)

=

a

n

⁢

(

𝒩

⁢

(

G

)

)

and define

ζ

G

⩽

(

s

)

=

ζ

𝒮

⁢

(

G

)

(

s

)

=

∑

H

⩽

f

G

|

G

:

H

|

-

s

,

and

ζ

G

⊴

(

s

)

=

ζ

𝒩

⁢

(

G

)

(

s

)

=

∑

N

⊴

f

G

|

G

:

N

|

-

s

.

If, in

addition

,

G

is

nilpotent

, then

ζ

G

⩽

has a
decomposition as a formal

Euler product

ζ

G

⩽

⁢

(

s

)

=

∏

p

⁢

prime

ζ

G

,

p

⩽

⁢

(

s

)

,

where

ζ

G

,

p

⩽

⁢

(

s

)

=

∑

i

=

0

∞

a

p

i

⩽

⁢

(

G

)

⁢

p

-

i

⁢

s

.

An analogous result holds for the normal zeta function

ζ

G

⊴

. The result for both

ζ

G

⩽

and

ζ

G

⊴

can be proved using

properties

of the

profinite
completion

of

G

. However, a simpler proof for the normal zeta
function is provided by the fact that a

finite nilpotent group

decomposes into a

direct product

of its

Sylow subgroups

. These results
allow the zeta functions to be expressed in terms of

p

-adic
integrals, which can in turn be used to prove (using some high-powered
machinery) that

ζ

G

,

p

⩽

⁢

(

s

)

and

ζ

G

,

p

⊴

⁢

(

s

)

are

rational functions

in

p

and

p

-

s

.

In the case when

G

is a

𝒯

-group, that is,

G

is

finitely generated

,

torsion free

, and nilpotent, define

α

G

⩽

to be the abscissa of convergence of

ζ

G

⩽

. That is,

α

G

⩽

is the smallest

α

∈

ℝ

such that

ζ

G

⩽

defines a

holomorphic function

in the

right half-plane

{

z

∈

ℂ

⁢

∣

ℜ

⁡

(

z

)

>

⁢

α

}

. It can then be shown that

α

G

⩽

⩽

h

⁢

(

G

)

, where

h

⁢

(

G

)

is the

Hirsch number

of

G

. Therefore, if

G

is a

𝒯

-group,

ζ

G

⩽

defines a holomorphic function in some right half-plane.

References

1

F. J. Grunewald, D. Segal, and G. C. Smith,

Subgroups of finite index in

nilpotent groups

, Invent. math.

93

(1988), 185–223.

2

M. P. F. du Sautoy,

Zeta functions of groups

: the quest for order versus the flight
from ennui

, Groups St. Andrews 2001 in Oxford. Vol. I, London Math. Soc.
Lecture Note Ser., vol. 304, Cambridge Univ. Press, 2003, pp. 150–189.

Title

Zeta function of a group

Canonical

name

ZetaFunctionOfAGroup

Date of creation

2013-03-22 15:16:00

Last modified on

2013-03-22 15:16:00

Owner

avf (9497)

Last modified by

avf (9497)

Numerical id

7

Author

avf (9497)

Entry type

Definition

Classification

msc 20E07

Classification

msc 20F69

Classification

msc 20F18

Related topic

Group

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