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					                   O UTLINE




C OMPARING THE SUBGROUP AND THE
  PROBABILISTIC ZETA FUNCTION

                Andrea Lucchini

               Università di Padova, Italy


             Joint work with Erika Damian


     ESI Programme on Profinite Groups
              December 2008




          A NDREA L UCCHINI    C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




           G a finitely generated profinite group
       an (G) the number of subgroups of index n in G
       bn (G) := |G:H|=n µ(H, G)


µ is the Möbius function of the subgroup lattice of G :

                                            1                       if H = G
             µ(H, G) =
                                    −       H<K ≤G µ(K , G)          otherwise




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



           G a finitely generated profinite group
       an (G) the number of subgroups of index n in G
       bn (G) := |G:H|=n µ(H, G)

µ is the Möbius function of the subgroup lattice of G :

                                            1               if H = G
             µ(H, G) =
                                    −       H<K ≤G µ(K , G)  otherwise

S UBGROUP ZETA FUNCTION
                                                           an (G)
                                     ζG (s) =
                                                             ns
                                                     n∈N


P ROBABILISTIC ZETA FUNCTION
                                                           bn (G)
                                    pG (s) =
                                                             ns
                                                     n∈N
                               A NDREA L UCCHINI       C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




        ˆ
If G = Z, then
                    1
     ζG (s) =     n ns = ζ(s) the                Riemann zeta function
                    µ(n)       −1
    pG (s) =      n ns = (ζ(s))




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




R EMARK
µ(H, G) = 0 ⇒ H is an intersection of maximal subgroups of G.

The probabilistic zeta function pG (s) depends only on the sublattice
generated by the maximal subgroups.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




R EMARK
µ(H, G) = 0 ⇒ H is an intersection of maximal subgroups of G.

The probabilistic zeta function pG (s) depends only on the sublattice
generated by the maximal subgroups.

The probabilistic zeta function encodes information about the lattice
generated by the maximal subgroups of G, just as the Riemann zeta
function encodes information about the primes.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Q UESTION
ζG (s) and pG (s) can be considered as formal Dirichlet series. Do they
converge in some part of the complex plane?

ζG (s) converges in a suitable half plane if and only if G has
Polynomial Subgroup Growth.

The question whether pG (s) converges in a half plane seems to be
related with the probabilistic meaning of pG (s) and with the behaviour
of the maximal subgroup growth of G.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Let ν be the normalized Haar measure on G or on some direct power
Gt ; we define the probability that t random elements generate G as:

         ProbG (t) = ν              (g1 , . . . , gt ) ∈ Gt    g1 , . . . , gt = G


H ALL
If G is finite and t ∈ N, then pG (t) = ProbG (t).




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Let ν be the normalized Haar measure on G or on some direct power
Gt ; we define the probability that t random elements generate G as:

         ProbG (t) = ν              (g1 , . . . , gt ) ∈ Gt    g1 , . . . , gt = G


H ALL
If G is finite and t ∈ N, then pG (t) = ProbG (t).

D EFINITION
G is Positively Finitely Generated when ProbG (t) > 0 for some t ∈ N.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Let ν be the normalized Haar measure on G or on some direct power
Gt ; we define the probability that t random elements generate G as:

         ProbG (t) = ν              (g1 , . . . , gt ) ∈ Gt    g1 , . . . , gt = G


H ALL
If G is finite and t ∈ N, then pG (t) = ProbG (t).

D EFINITION
G is Positively Finitely Generated when ProbG (t) > 0 for some t ∈ N.

T HEOREM (M ANN - S HALEV )
G is PFG if and only if G has polynomial maximal subgroup growth.



                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



Let ν be the normalized Haar measure on G or on some direct power
Gt ; we define the probability that t random elements generate G as:

         ProbG (t) = ν              (g1 , . . . , gt ) ∈ Gt    g1 , . . . , gt = G

H ALL
If G is finite and t ∈ N, then pG (t) = ProbG (t).

D EFINITION
G is Positively Finitely Generated when ProbG (t) > 0 for some t ∈ N.

C ONJECTURES
    pG (s) converges in a suitable half plane iff G is PFG.
    If G is PFG, then pG (t) = ProbG (t) for t ∈ N large enough.

Finitely generated (virtually) prosolvable groups are PFG and satisfy
the previous conjectures.
                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
               T HE SUBGROUP ZETA FUNCTION ζG (s)
           T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                  ζG (s)pG (s) = 1?
                              "N ORMAL " VARIATION




       ˆ
If G = Z, then ζG (s)pG (s) = 1.

P ROBLEM
To study the finitely generated profinite groups G satisfying the
condition
                           ζG (s)pG (s) = 1



D EFINITION
Just for this talk, we will say that G is ζ-reversible if ζG (s)pG (s) = 1.




                                A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




An identity involving the probabilistic zeta functions pH (s) of the open
subgroups H of G can help to understand the meaning of the
condition ζG (s)pG (s) = 1.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




An identity involving the probabilistic zeta functions pH (s) of the open
subgroups H of G can help to understand the meaning of the
condition ζG (s)pG (s) = 1.

If G is a finite group and t ∈ N (and more in general if G has PSG and
t is large enough) then

                                             ProbH (t)
                                                       = 1.
                                             |G : H|t
                                    H≤o G




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




An identity involving the probabilistic zeta functions pH (s) of the open
subgroups H of G can help to understand the meaning of the
condition ζG (s)pG (s) = 1.

If G is a finite group and t ∈ N (and more in general if G has PSG and
t is large enough) then

                                             ProbH (t)
                                                       = 1.
                                             |G : H|t
                                    H≤o G



Independently of the convergency properties of the series pH (s) and
their probabilistic meaning, the following formal identity holds:

                                               pH (s)
                                                       = 1.
                                              |G : H|s
                                     H≤o G



                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




                                       ζG (s)pG (s) = 1


                                 pG (s)                          pH (s)
                                         =1=
                                |G : H|s                        |G : H|s
                       H≤o G                           H≤o G



                                         pG (s) − pH (s)
                                                         =0
                                             |G : H|s
                                H≤o G



C OROLLARY
pG (s) = pH (s) for each open subgroup H of G ⇒ G is ζ-reversible.



                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



E XAMPLES




     If H ∼ G for each H ≤o G, then G is ζ-reversible.
          =




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



E XAMPLES




     If H ∼ G for each H ≤o G, then G is ζ-reversible. This occurs if
          =
     and only if G is abelian and torsion free.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



E XAMPLES



     If H ∼ G for each H ≤o G, then G is ζ-reversible. This occurs if
          =
     and only if G is abelian and torsion free.

     For example if G = Zr then

            ζG (s) = ζ(s)ζ(s − 1) · · · ζ(s − (r − 1)) = (pG (s))−1




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



E XAMPLES


     If H ∼ G for each H ≤o G, then G is ζ-reversible. This occurs if
          =
     and only if G is abelian and torsion free.
     Let d(G) be the smallest cardinality of a generating set of G.
     If G is a pro-p group then

                                                                      pi
                               pG (s) =                        1−
                                                                      ps
                                              0≤i≤d(G)−1


     depends only on d(G). If G is a pro-p group with d(G) = d(H)
     for each H ≤o G, then G is ζ-reversible.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



E XAMPLES
     If H ∼ G for each H ≤o G, then G is ζ-reversible. This occurs if
          =
     and only if G is abelian and torsion free.
     Let d(G) be the smallest cardinality of a generating set of G.
     If G is a pro-p group then

                                                                      pi
                               pG (s) =                        1−
                                                                      ps
                                              0≤i≤d(G)−1


     depends only on d(G). If G is a pro-p group with d(G) = d(H)
     for each H ≤o G, then G is ζ-reversible.

     Non abelian examples. The pro-p group G with the presentation
                                                                                 t
                         x1 , . . . , xr , y | [xi , xj ] = 1, [xi , y ] = xip

     satisfies d(H) = r + 1 ∀ H ≤o G.
                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




N ON PRONILPOTENT EXAMPLES
For any m ∈ Z, m = 0, let Gm be the profinite completion of the
Baumslag-Solitar group Bm = a, b | a−1 ba = bm .




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




N ON PRONILPOTENT EXAMPLES
For any m ∈ Z, m = 0, let Gm be the profinite completion of the
Baumslag-Solitar group Bm = a, b | a−1 ba = bm .

                                 1                          p
    pGm (s) =       p    1−      ps        (p,m)=1   1−     ps

    H ≤o Gm ⇒ H ∼ Gmu for some u ∈ N ⇒ pH (s) = pGm (s).
                =




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




N ON PRONILPOTENT EXAMPLES
For any m ∈ Z, m = 0, let Gm be the profinite completion of the
Baumslag-Solitar group Bm = a, b | a−1 ba = bm .

                                 1                          p
    pGm (s) =       p    1−      ps        (p,m)=1   1−     ps

    H ≤o Gm ⇒ H ∼ Gmu for some u ∈ N ⇒ pH (s) = pGm (s).
                =

                                                                                  p
          ζGm (s) = (pGm (s))−1 = ζ(s)ζ(s − 1)                             1−
                                                                                  ps
                                                                   p|m




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




N ON PRONILPOTENT EXAMPLES
For any m ∈ Z, m = 0, let Gm be the profinite completion of the
Baumslag-Solitar group Bm = a, b | a−1 ba = bm .

                                 1                          p
    pGm (s) =       p    1−      ps        (p,m)=1   1−     ps

    H ≤o Gm ⇒ H ∼ Gmu for some u ∈ N ⇒ pH (s) = pGm (s).
                =

                                                                                  p
          ζGm (s) = (pGm (s))−1 = ζ(s)ζ(s − 1)                             1−
                                                                                  ps
                                                                   p|m



A direct computation of ζGm (s) is due to Gelman (2005).

Gm is virtually pronilpotent if and only if m = ±1.


                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



H OW CAN WE COMPUTE pGm (s)?



  Let {Ni }i be a chain of open normal subgroups of G with i Ni = 1
  and Ni /Ni+1 a chief factor of G/Ni+1 for each i (a chief series of G).

  For each i consider the finite Dirichlet series
                                bi (n)
            pi (s) =                           with    bi (n) =               µ(H, G)
                            n
                                 ns
                                                                    Ni+1 ≤H
                                                                     HNi =G
                                                                    |G:H|=n




                                 A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



H OW CAN WE COMPUTE pGm (s)?

  Let {Ni }i be a chain of open normal subgroups of G with i Ni = 1
  and Ni /Ni+1 a chief factor of G/Ni+1 for each i (a chief series of G).

  For each i consider the finite Dirichlet series
                                bi (n)
            pi (s) =                           with        bi (n) =               µ(H, G)
                            n
                                 ns
                                                                        Ni+1 ≤H
                                                                         HNi =G
                                                                        |G:H|=n



  pG (s) can be written as an infinite formal product of these finite
  series:
                             pG (s) =     pi (s)
                                                       i



                                 A NDREA L UCCHINI         C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



H OW CAN WE COMPUTE pGm (s)?

  Let {Ni }i be a chain of open normal subgroups of G with i Ni = 1
  and Ni /Ni+1 a chief factor of G/Ni+1 for each i (a chief series of G).

  For each i consider the finite Dirichlet series
                                bi (n)
            pi (s) =                           with        bi (n) =               µ(H, G)
                            n
                                 ns
                                                                        Ni+1 ≤H
                                                                         HNi =G
                                                                        |G:H|=n




                                        pG (s) =             pi (s)
                                                       i




                                 A NDREA L UCCHINI         C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                 T HE SUBGROUP ZETA FUNCTION ζG (s)
             T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                    ζG (s)pG (s) = 1?
                                "N ORMAL " VARIATION



H OW CAN WE COMPUTE pGm (s)?
  Let {Ni }i be a chain of open normal subgroups of G with i Ni = 1
  and Ni /Ni+1 a chief factor of G/Ni+1 for each i (a chief series of G).

  For each i consider the finite Dirichlet series
                                 bi (n)
             pi (s) =                           with        bi (n) =               µ(H, G)
                             n
                                  ns
                                                                         Ni+1 ≤H
                                                                          HNi =G
                                                                         |G:H|=n



                                         pG (s) =             pi (s)
                                                        i



  If G is prosolvable, then Ni /Ni+1 is abelian and pi (s) = 1 − ci /qis with
  qi = |Ni /Ni+1 |, ci the number of complements of Ni /Ni+1 in G/Ni+1 .
                                  A NDREA L UCCHINI         C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Let Gm be the profinite completion of the Baumslag-Solitar group Bm .
    H a finite epimorphic image of Gm ⇒ ∃K                              H such that
        K and H/K are cyclic,
        K is complemented in H,
        (|K |, m) = 1.
    p divides m ⇒ in a chief series of Gm there is only one
    complemented p-factor, it is central and the corresponding finite
    Dirichlet series is 1 − 1/ps .
    p does not divide m ⇒ in a chief series of G there are 2
    complemented p-factors, both are cyclic of order p, one has only
    1 complement, the other has p complements, the product of the
    the corresponding finite Dirichlet series is (1 − 1/ps )(1 − p/ps ).
                                 1                          p
    pGm (s) =       p    1−      ps        (p,m)=1   1−     ps




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




A FINAL REMARK ABOUT OUR EXAMPLES
Let G be the group with the following profinite presentation:

                  G = x1 , . . . , xr , y | [xi , xj ] = 1, xiy = xim .


    G is ζ-reversible
                                                                      pr
    ζG (s) = ζ(s)ζ(s − 1) · · · ζ(s − r )               p|m    1−     ps

    If A = Zr ×         (p,m)=1     Zp , then ζG (s) = ζA (s) and pG (s) = pA (s).


All the examples that we have presented can be obtained as
epimorphic images of G, for suitable r and m. Are there different
examples?



                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Q UESTION
Do there exist ζ-reversible groups that are not prosolvable?

    G ζ-reversible ⇒ the coefficients of (pG (s))−1 are non negative.
                                           ci
    G prosolvable ⇒ pG (s) = i (1 − q s ) ⇒
                                                                i


                                                          ci   ci2
                       (pG (s))−1 =                  1+    s + 2s + . . .
                                                          qi  qi
                                                i

                         has non negative coefficients.

Q UESTION
Does there exist a finitely generated non prosolvable group G with the
property that the coefficients of (pG (s))−1 are non negative?


                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




Q UESTION
Do there exist ζ-reversible groups that are not prosolvable?

    G ζ-reversible ⇒ the coefficients of (pG (s))−1 are non negative.
                                           ci
    G prosolvable ⇒ pG (s) = i (1 − q s ) ⇒
                                                                i


                                                          ci   ci2
                       (pG (s))−1 =                  1+    s + 2s + . . .
                                                          qi  qi
                                                i

                         has non negative coefficients.

Q UESTION
Does there exist a finitely generated non prosolvable group G with the
property that the coefficients of (pG (s))−1 are non negative?


                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



P ROSOLVABLE GROUPS



  If G is a finitely generated prosolvable group then the series
  pG (s) = n bn (G)/ns satisfies the following properties (which are
  preserved under inversion):
       the sequence {bn (G)}n has polynomial growth.
       pG (s) has an Euler factorization over the prime numbers:

                                                               bpm (G)
                                   pG (s) =                                   .
                                                     p    m
                                                                pms




                                 A NDREA L UCCHINI       C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



P ROSOLVABLE GROUPS


  If G is a finitely generated prosolvable group then the series
  pG (s) = n bn (G)/ns satisfies the following properties (which are
  preserved under inversion):
       the sequence {bn (G)}n has polynomial growth.
       pG (s) has an Euler factorization over the prime numbers:

                                                               bpm (G)
                                   pG (s) =                                   .
                                                     p    m
                                                                pms

      This is equivalent to say that the sequence {bn (G)}n∈N is
      multiplicative, i.e. brs (G) = br (G)bs (G) whenever (r , s) = 1.



                                 A NDREA L UCCHINI       C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                T HE SUBGROUP ZETA FUNCTION ζG (s)
            T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                   ζG (s)pG (s) = 1?
                               "N ORMAL " VARIATION



P ROSOLVABLE GROUPS

  If G is a finitely generated prosolvable group then the series
  pG (s) = n bn (G)/ns satisfies the following properties (which are
  preserved under inversion):
       the sequence {bn (G)}n has polynomial growth.
       pG (s) has an Euler factorization over the prime numbers:

                                                               bpm (G)
                                   pG (s) =                                   .
                                                     p    m
                                                                pms


  C ONSEQUENCES
  If G is a ζ-reversible prosolvable group, then
       G has Polynomial Subgroup Growth, hence it has finite rank.
       ζG (s) has an Euler factorization over the prime numbers.

                                 A NDREA L UCCHINI       C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




R EMARK
   (Detomi, AL 2004) The probabilistic zeta function pG (s) has an
   Euler factorization if and only if G is prosolvable.
   It is still open the problem of characterizing the finitely generated
   profinite groups G whose subgroup zeta function ζG (s) has an
   Euler factorization.
   If G is pronilpotent, then ζG (s) has an Euler factorization. The
   only other known examples of groups whose subgroup zeta
   function has an Euler factorization come from the profinite
   completions of the Baumslag-Solitar groups described before.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Let G be a ζ-reversible prosolvable group and let π be the set of the
prime divisors of the order of G. For each π-number n, G contains an
open subgroup of index n.

P ROOF
    It suffices to prove: apm (G) = 0 ∀p ∈ π, ∀ m ∈ N.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Let G be a ζ-reversible prosolvable group and let π be the set of the
prime divisors of the order of G. For each π-number n, G contains an
open subgroup of index n.

P ROOF
    It suffices to prove: apm (G) = 0 ∀p ∈ π, ∀ m ∈ N.
    Fix p: G contains an open subgroup of index pu, with (u, p) = 1.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Let G be a ζ-reversible prosolvable group and let π be the set of the
prime divisors of the order of G. For each π-number n, G contains an
open subgroup of index n.

P ROOF
    It suffices to prove: apm (G) = 0 ∀p ∈ π, ∀ m ∈ N.
    Fix p: G contains an open subgroup of index pu, with (u, p) = 1.
    apu (G) = 0 and pG (s)ζG (s) = 1 ⇒ bpv (G) = 0 ∃ v with (v , p) = 1.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Let G be a ζ-reversible prosolvable group and let π be the set of the
prime divisors of the order of G. For each π-number n, G contains an
open subgroup of index n.

P ROOF
    It suffices to prove: apm (G) = 0 ∀p ∈ π, ∀ m ∈ N.
    Fix p: G contains an open subgroup of index pu, with (u, p) = 1.
    apu (G) = 0 and pG (s)ζG (s) = 1 ⇒ bpv (G) = 0 ∃ v with (v , p) = 1.
    pG (s) = i (1 − ci /qis ) with ci ≥ 0 and qi prime-powers. Since
    bpv (G) = 0 it must be qi = p for some i.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Let G be a ζ-reversible prosolvable group and let π be the set of the
prime divisors of the order of G. For each π-number n, G contains an
open subgroup of index n.

P ROOF
    It suffices to prove: apm (G) = 0 ∀p ∈ π, ∀ m ∈ N.
    Fix p: G contains an open subgroup of index pu, with (u, p) = 1.
    apu (G) = 0 and pG (s)ζG (s) = 1 ⇒ bpv (G) = 0 ∃ v with (v , p) = 1.
    pG (s) = i (1 − ci /qis ) with ci ≥ 0 and qi prime-powers. Since
    bpv (G) = 0 it must be qi = p for some i.
    (pG (s))−1 = (1 − ci /ps )−1 ( j=i (1 − cj /qjs ))−1
              = (1 + ci /ps + ci2 /p2s + . . . )( m dm /ms ) with dm ≥ 0.



                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




P ROPOSITION
Assume that G is a ζ-reversible prosolvable group of rank 2. Then
    G is prosupersolvable;
    for each prime divisor p of |G|, G contains a normal subgroup of
    index p;
    pG (s) = pH (s) for each H ≤o G.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION


PRO -p- GROUPS
Assume that G is a ζ-reversible pro-p-group (G must be p-adic
analytic):
                                              pi                                    pi
                   0≤i≤d(G)−1 (1         −    ps )   −  0≤i≤d(H)−1 (1           −   ps )
                                                                                           = 0.
                                              |G :   H|s
        H≤o G

Does this imply d(G) = d(H) for each H ≤o G?

PARTIAL ANSWERS
If d(H) = d(G) for some H ≤o G, then
     d(H) = d(G) for infinitely many open subgroups H of G.
     If r is minimal with respect to the property that there exists H with
     d(H) = d(G) and |G : H| = pr , then there exist H1 and H2 with
     |G : H1 | = |G : H2 | = pr and d(H1 ) < d(G) < d(H2 ).
     G does not contain pro-cyclic open subgroups.
     d(G) > 2.
                               A NDREA L UCCHINI         C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION



If X acts on G, then we may consider the lattice of the open
X -subgroups of G and the Möbius function µX in this lattice.

We may define:
    X
   aG (n) : the number of X -subgroups of G with index n.
    X
   bG (n) := H≤X G,|G:H|=n µX (H, G)
                                       X          X
and the corresponding zeta functions: ζG (s) and pG (s).

"N ORMAL " VARIATION
If X = G then
      X
     ζG (s) = ζG (s) the normal subgroup zeta function
     X
    pG (s) = pG (s) the normal probabilistic zeta function

If H o G µ (H, G)|G : H|−s is absolutely convergent and k ∈ N is
large enough, then pG (k ) gives the probability that the smallest
closed normal subgroup containing k random elements is G.
                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




N ORMAL ζ- REVERSIBLE GROUPS
What can we say about a profinite group G with the property

                                     ζG (s)pG (s) = 1?




                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




N ORMAL ζ- REVERSIBLE GROUPS
What can we say about a profinite group G with the property

                                     ζG (s)pG (s) = 1?

                                                          G         G
                                                         pG (s) − pH (s)
              ζG (s)pG (s) = 1 ⇔                                         =0
                                                             |G : H|s
                                                H   oG




                              A NDREA L UCCHINI      C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




For any finite simple group T , let m(T ) be the largest m such that T m
is an epimorphic image of G :

                       pG (s) = pG,ab (s)pG,nonab (s) with
                                                           
                                                         pi 
         pG,ab (s) =                                  1− s   = pG/G (s)
                             p
                                                         p
                                     0≤i≤m(Cp )−1

                                                                           m(T )
                                                                    1
                  pG,nonab (s) =                           1−
                                                                  |T |s
                                          T non abelian




                                 A NDREA L UCCHINI   C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




P ROPOSITION
If ζG (s)pG (s) = 1 and no nonabelian simple group appears as an
epimorphic image of G, then G is pronilpotent.

P ROOF
    No nonabelian simple group is an epimorphic image of G
                             ⇓




                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




P ROPOSITION
If ζG (s)pG (s) = 1 and no nonabelian simple group appears as an
epimorphic image of G, then G is pronilpotent.

P ROOF
    No nonabelian simple group is an epimorphic image of G
                                ⇓
    pG (s) = pG,ab (s) has an Euler factorization over the primes
                                ⇓




                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




P ROPOSITION
If ζG (s)pG (s) = 1 and no nonabelian simple group appears as an
epimorphic image of G, then G is pronilpotent.

P ROOF
    No nonabelian simple group is an epimorphic image of G
                                ⇓
    pG (s) = pG,ab (s) has an Euler factorization over the primes
                                ⇓
    ζG (s) has an Euler factorization over the primes
                                ⇓




                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




P ROPOSITION
If ζG (s)pG (s) = 1 and no nonabelian simple group appears as an
epimorphic image of G, then G is pronilpotent.

P ROOF
    No nonabelian simple group is an epimorphic image of G
                                ⇓
    pG (s) = pG,ab (s) has an Euler factorization over the primes
                                ⇓
    ζG (s) has an Euler factorization over the primes
                                ⇓
    the normal subgroup growth of G is multiplicative
                                ⇓



                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
             T HE SUBGROUP ZETA FUNCTION ζG (s)
         T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                ζG (s)pG (s) = 1?
                            "N ORMAL " VARIATION




P ROPOSITION
If ζG (s)pG (s) = 1 and no nonabelian simple group appears as an
epimorphic image of G, then G is pronilpotent.

P ROOF
    No nonabelian simple group is an epimorphic image of G
                                ⇓
    pG (s) = pG,ab (s) has an Euler factorization over the primes
                                ⇓
    ζG (s) has an Euler factorization over the primes
                                ⇓
    the normal subgroup growth of G is multiplicative
                                ⇓
    (J.C. Puchta 2001) G is pronilpotent


                              A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




A MORE GENERAL RESULT
Let (pG,ab (s))−1 = n γn /ns . If ζG (s)pG (s) = 1, then γn is the number
of normal subgroups N of G with G/N a nilpotent group of order n.

Q UESTION
Do there exist non pronilpotent groups G satisfying ζG (s)pG (s) = 1?




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




For any set π of prime numbers let Gπ be the largest epimorphic
image G which is a π group.

R EMARK
ζG (s)pG (s) = 1 ⇔ ζGπ (s)pGπ (s) = 1 for each finite set π of primes.


    When we study the groups G with ζG (s)pG (s) = 1, it is not
    restrictive to assume that |G| is divisible only by finitely many
    primes.
    This implies that pG (s) is a finite Dirichlet series and
    consequently that G has polynomial normal subgroup growth.




                               A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
               T HE SUBGROUP ZETA FUNCTION ζG (s)
           T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                  ζG (s)pG (s) = 1?
                              "N ORMAL " VARIATION




P ROPOSITION
Assume that all the nonabelian composition factors of G are
alternating groups. If ζG (s)pG (s) = 1, then G is pronilpotent.

P ROPOSITION
Assume that G is a perfect profinite group. If ζG (s)pG (s) = 1, then
there exists two simple groups S and T and an irreducible T -module
V such that
    S and V T are epimorphic images of G.
    |S| < |T | < |S|2 .
    |S|2 = |T ||V |.

There are only finitely many possibilities for (S, T , V ).



                                A NDREA L UCCHINI     C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
              T HE SUBGROUP ZETA FUNCTION ζG (s)
          T HE PROBABILISTIC ZETA FUNCTION pG (s)
                                 ζG (s)pG (s) = 1?
                             "N ORMAL " VARIATION




                                                           G         G
                                                          pG (s) − pH (s)
               ζG (s)pG (s) = 1 ⇔                                         =0
                                                              |G : H|s
                                                 H   oG



P ROPOSITION
Assume that the following stronger property holds:
                              G         G
                             pG (s) − pH (s)
                                             = 0 for each n ∈ N.
                                 |G : H|s
                |G/H|=n


Then G is pronilpotent.




                               A NDREA L UCCHINI      C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION
                 A PPENDIX




                                            µ(K ,H)
         pH (s)                      K ≤o H |H:K |s
                 =
        |G : H|s                     |G : H|s
H≤o G                   H≤o G
                                     µ(K , H)
                    =
                                     |G : K |s
                        K ≤o H≤o G

                                   K ≤o H   µ(K , H)
                    =
                                     |G : K |s
                        K ≤o G
                                   δK ,G
                    =                      = 1.
                                 |G : K |s
                        K ≤o G




         A NDREA L UCCHINI   C OMPARING THE SUBGROUP AND THE PROBABILISTIC ZETA FUNCTION

				
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