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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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