P ROSOLVABLE GROUPS

If G is a prosolvable group then P_G(s) =P

nbn(G)/n^ssatisfies the following properties (which are preserved under inversion):

the sequence {b_n(G)}_nhas polynomial growth;

P_G(s) has an Euler factorization over the prime numbers.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

P ROSOLVABLE GROUPS

If G is a prosolvable group then P_G(s) =P

nb_n(G)/n^ssatisfies the following properties (which are preserved under inversion):

the sequence {bn(G)}nhas polynomial growth;

P_G(s) has an Euler factorization over the prime numbers.

CONSEQUENCES

If G is a ζ-reversible prosolvable group, then

G has Polynomial Subgroup Growth, henceit has finite rank.

Z_G(s) has an Euler factorization over the prime numbers.

P ROSOLVABLE GROUPS

If G is a prosolvable group then PG(s) =P

nbn(G)/n^ssatisfies the following properties (which are preserved under inversion):

the sequence {bn(G)}nhas polynomial growth;

P_G(s) has an Euler factorization over the prime numbers.

CONSEQUENCES

If G is a ζ-reversible prosolvable group, then

G has Polynomial Subgroup Growth, henceit has finite rank.

Z_G(s) has an Euler factorization over the prime numbers.

P_G(s) has an Euler factorization over the primes if and only if G is prosolvable. It is still open the problem of characterizing the profinite groups whose subgroup zeta function has an Euler factorization over the primes. If G is pronilpotent, then Z_G(s) has an Euler factorization.

However there exist non virtually pronilpotent profinite groups whose subgroup zeta function has an Euler factorization.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

PROPOSIZIONE

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.

PROOF

It suffices to prove: ap^m(G) 6= 0 ∀p ∈ π, ∀ m ∈ N.

PROPOSIZIONE

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.

PROOF

It suffices to prove: a_p^m(G) 6= 0 ∀p ∈ π, ∀ m ∈ N. Indeed (a₁,a₂) =1, |G : H₁| = a₁, |G : H₂| = a₂⇒ |G : H₁∩ H₂| = a₁a₂.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

PROPOSIZIONE

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.

PROOF

It suffices to prove: ap^m(G) 6= 0 ∀p ∈ π, ∀ m ∈ N.

Fix p: G contains an open subgroup of index pu, with (u, p) = 1.

PROPOSIZIONE

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.

PROOF

It suffices to prove: a_p^m(G) 6= 0 ∀p ∈ π, ∀ m ∈ N.

Fix p: G contains an open subgroup of index pu, with (u, p) = 1.

apu(G) 6= 0 and P_G(s)Z_G(s) = 1⇒b_pv(G) 6= 0 ∃ v s.t. (v , p) = 1.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

PROPOSIZIONE

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.

PROOF

It suffices to prove: ap^m(G) 6= 0 ∀p ∈ π, ∀ m ∈ N.

Fix p: G contains an open subgroup of index pu, with (u, p) = 1.

apu(G) 6= 0 and P_G(s)Z_G(s) = 1⇒b_pv(G) 6= 0 ∃ v s.t. (v , p) = 1.

P_G(s) =Q

i(1 − c_i/q^s_i)with c_i ≥ 0 and q_i prime-powers. Since b_pv(G) 6= 0 it must be q_i =p for some i.

PROPOSIZIONE

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.

PROOF

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

PROPOSIZIONE

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;

P_G(s) = PH(s) for each H ≤oG.

PRO-p-GROUPS

Assume that G is a ζ-reversible pro-p-group (G must be p-adic analytic):

d (H) 6= 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) 6= d (G) and |G : H| = p^r, then there exist H1and H2with

|G : H₁| = |G : H₂| = p^r and d (H1) <d (G) < d (H2).

G does not contain pro-cyclic open subgroups.

d (G) > 2.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

THEOREM(SKOLEM-MAHLER-LECH)

If {a₀,a₁, ...}is a recurrence sequence, then the set of all k such that a_k =0 is the union of a finite (possibly empty) set and a finite number (possibly zero) of full arithmetical progressions, where a full arithmetic progression is a set of the form {r , r + d , r + 2d , ...} with r ∈ [0, d ).

COROLLARY

Let c1, . . . ,cr, λ₁, . . . , λ_r be algebraic numbers with the property that no quotient λi/λ_j is a non-trivial root of unity. Then the exponential polynomial

ψ(m) = c₁λ^m₁ + · · · +c_rλ^m_r

vanishes for infinitely many integers m only if ψ(m) is identically zero.

PROPOSIZIONE

SKETCH OF THE PROOF

Y

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

ψ_n(m)=

To apply the Skolem-Mahler-Lech Theorem we have to choose n ∈ N such that no nontrivial root of unit can be expressed as the ratio of two elements in the setΛ_n= {αⁿ₁, . . . , αⁿ_s, βⁿ₁, . . . , β_rⁿ, γ_i^n/ni | i ∈ I_n}.

Letebe the order of the subgroup of C^∗generated by the set of roots of unit which can be written as x /y with

x , y ∈ {α₁, . . . , α_s, β₁, . . . , β_r}.

As the γi’s are positive integers, we can choose an integerd such that: x ∈ Λe, (x^d)^m∈ N, m ∈ N ⇒ x^d∈ N.

If we choosen=ed , then there is no pair of elements Λ_nwhose ratio is a root of unit.

ψ_n(m)=

To apply the Skolem-Mahler-Lech Theorem we have to choose n ∈ N such that no nontrivial root of unit can be expressed as the ratio of two elements in the setΛ_n= {αⁿ₁, . . . , αⁿ_s, βⁿ₁, . . . , β_rⁿ, γ_i^n/ni | i ∈ I_n}.

Letebe the order of the subgroup of C^∗generated by the set of roots of unit which can be written as x /y with

x , y ∈ {α1, . . . , α_s, β₁, . . . , β_r}.

As the γ_i’s are positive integers, we can choose an integerd such that: x ∈ Λe, (x^d)^m∈ N, m ∈ N ⇒ x^d∈ N.

If we choosen=ed , then there is no pair of elements Λ_nwhose ratio is a root of unit.

Conclusion: ψ_n(m) = 0 ∀m ∈ N ⇒ Inm=I_n∀m ∈ N ⇒ I is finite

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

X

Let G be a nilpotent torsion-free group.

G may be identified with a subgroup of the upper unitriangular groupTr1(m, Z)for some m.

The setTr₀(m, Q)of upper-triangular matrices over Q with zero diagonal is a Lie algebra with the operation [a, b] = ab − ba.

We may define a map

log:G → Tr₀(m, Q) g 7→ log g =P

n (−1)ⁿ⁻¹

n (g − 1)ⁿ.

G contains a subgroup G₀of finite index such thatL = log G₀satisfies the properties:

L is a Lie subring of Tr₀(m, Q);

(L, +) ∼= Z^r, for some r ;

For almost all primes p, a_pⁿ(G) coincides with the number of Lie subrings of index pⁿin L for all n.

This reduces the study of Z_G(s) to that of thesubring zeta function ζ_L(s) of L.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

The p-adic absolute value of λ ∈ Zpis written |λ| = p^−f where λ =p^fu and u is a p-adic unit.

Fix a prime p and identify L ⊗ Zpwith Z^rpby choosing a Z-basis for L.

ζ_L,p(s) = 1 (1 − 1/p)^r

Z

M

|λ₁₁|^s−1· · · |λ_rr|^s−rd µ

where M is the subset of Tr(r , Zp)defined by a certain collection of bilinear equations, which express the requirement that the linear span of the rows of the matrix (λ_ij)be closed under the Lie bracket.

ζ_L,p(s) is given by what Grunewald and du Sautoy call acone integral.

D = {f_i,gi | 0 ≤ i ≤ l}cone integral data: a family of r -variable polynomials over Q;

V_p(D) := {x ∈ Z^rp| v (f_i(x )) ≤ v (g_i(x )), 1 ≤ i ≤ l} where v is the p-adic valuation.

ZD(s, p) :=R

Vp(D)|f₀(x )||g0(x )|^sd µcone integral defined by D.

THEOREM

There exists D such that for all primes p ζ_L,p(s) = 1

ap,0

ZD(s, p)

where ap,0=ZD(∞,p) is a non-zero constant.

ANDREALUCCHINI THE PROBABILISTIC ZETA FUNCTION

The proofs uses Denef’s method of evaluating p-adic integers by an explicit resolution of singularities.

There exists

V1, . . . ,Vt algebraic varieties defined over Z

P₁(x , y ), . . . , P_t(x , y ) ∈ Q(x, y) rational functions over Q such that for almost all primes p

ZD(s, p) = ap,0+ X

1≤i≤t

bi(p)Pi(p^s,p^−s)

with b_i(p) the number of Fp-points on the reduction modulo p of V_i.

This links the group-theoretic zeta function in an explicit way to the Weil zeta function of certain algebraic varieties.

Nel documento T HE PROBABILISTIC ZETA FUNCTION OF FINITE AND PROFINITE GROUPS (pagine 160-179)