Open Questions We collect here the open questions mentioned in the book. We have main- tained the numeration of the original question so that the reader may consult the context in which the question is posed. The wording of the questions are sometimes modified slightly to make them self-contained. We also mention here those open problems that appeared in the first edition of this book which have been solved or advanced in the meantime; in these cases we provide some relevant information. Open Question 3.5.3 (Inverse problem of Galois Theory) Is every finite group a continuous homomorphic image of the absolute Galois group GQ¯ /Q of the field Q of rational numbers? Question in the First Edition of this Book Let F be a free profinite (or, more generally, pro - C) group on a profinite space X. Is there a canonical way of constructing a basis converging to 1 for F ? NOTE: J-P. Serre has given a negative answer to this question; see Theorem 3.5.13. Open Question 3.7.2 What pro - C groups are pro - C completions of finitely generated abstract groups? Question in the First Edition of this Book Let G be a finitely gener- ated profinite group. Is every subgroup of finite index in G necessarily open? NOTE: This question has been answered positively by N. Nikolov and D. Segal (see Theorem 4.2.2; for a proof see Nikolov and Segal [2007a, 2007b]). Question in the First Edition of this Book Let G be a finitely generated prosolvable group. Are the terms (other than [G, G]) of the derived series of G closed? NOTE: This question has a negative answer; in fact V.A. Roman’kov had already provided a counterexample in 1982 for a more general setting; see Roman’kov [1982]. Open Question 4.8.4 Let G be a finitely generated profinite group and let n be a natural number. Let Gn = xn | x ∈ G be the abstract subgroup of G generated by the n-th powers of its elements. Is Gn closed? L. Ribes, P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 40, DOI 10.1007/978-3-642-01642-4, c Springer-Verlag Berlin Heidelberg 2010 394 Open Questions Open Question 4.8.5b Is a torsion profinite group necessarily of finite exponent? Open Question 6.12.1 Let G be a solvable pro-p group such that Hn(G, Z/pZ) is finite for every n. Is G polycyclic? Open Question 7.10.1 For what finite p-groups G does one have rr(G)= arr(G)? [rr = relation rank as a profinite group; arr = relation rank as an abstract group] Open Question 7.10.4 Let G be a finitely generated pro-p group such that cd(G) > 2 and dim H2(G, Z/pZ)=1,(i.e., relation rank rr(G) is 1). Does G admit a presentation with a single defining relator of the form up? Open Question 7.10.5 Study finitely generated pro-p groups with the fol- lowing property: every closed subgroup of infinite index is free pro-p. Open Question 7.10.6 Let F be a free pro-p group of finite rank. Is vcd(Aut(F )) finite? Question in the First Edition of this Book Does the Grushko-Neumann theorem hold for free profinite products of profinite groups, that is, if G = G1 G2 is the free profinite product of two profinite groups G1 and G2, is d(G)=d(G1)+d(G2)? NOTE: The answer to this is negative. It was answered by A. Lucchini; see Lucchini [2001a, 2001b]. Open Question 9.1.21 Let F be a free pro-p group and let H and K be closed finitely generated subgroups of F . Is there a bound on the rank of H ∩ K intermsoftheranksof H and K? Open Question 9.5.2 Is a general inverse limit of a surjective inverse sys- tem of free profinite groups of finite rank necessarily a free profinite group? Question in the First Edition of this Book For which extension closed varieties C of finite groups is it always true that whenever we are given G1,G2 ∈C, then there is a group G ∈Csuch that G1,G2 ≤ G, G = G1,G2 and d(G)=d(G1)+d(G2)? NOTE: For the class C of all finite solvable groups the answer is negative, see Kov´acs and Sim [1991]; for the class C of all finite groups the answer is negative, see Lucchini [2001a, 2001b]. Question in the First Edition of this Book Do all profinite Frobenius groups of the form Zπ C (C is finite cyclic, p |C| for all p ∈ π and C acts fixed-point-free on Zπ) appear as subgroups of free profinite products A B? NOTE: A positive answer is provided in Guralnick and Haran [2010]. Open Questions 395 Open Question 9.5.5 Give (verifiable) sufficient conditions for H to have ∗ bounded generation, where H = H1 H0 H2. Open Question 9.5.6 Give (verifiable) sufficient conditions for Hpˆ to have ∗ bounded generation, where H = H1 H0 H2. Open Question 9.5.7 Let G be a limit pro-p group. Does G satisfy the Howson property? In other words, if H1 and H2 are finitely generated closed subgroups of G, is H1 ∩ H2 a finitely generated pro-p group? Open Question 9.5.8 Are limit pro-p groups residually free pro-p? Open Question 9.5.9 Let H be a finitely generated subgroup of a limit pro-p group G. Is then H a virtual retract of G, i.e., is there an open subgroup M of G and a normal closed subgroup K of M so that M = K H? Open Problem C.3.2 Let G be a finitely generated profinite (respectively, ≥ ≥ pro-p) group with finite def (G) 2(respectively, def p(G) 2). Does G containanopensubgroupU such that there exists a continuous epimorphism U → F onto a free profinite (respectively, pro-p) group F of rank at least 2? Appendix A: Spectral Sequences A.1 Spectral Sequences r,s A bigraded abelian group E is a family E =(E )r,s∈Z of abelian groups. A differential d of E of bidegree (p, q) is a family of homomorphisms d : Er,s → Er+p,s+q such that dd =0. s • • 2,4 •••• E3 • • • ••••d3 • ••••5,2 • E3 • 1,1 • 3,1 ••• E3 E3 • • • • • • • r A spectral sequence consists of a sequence E = {E1, E2, E3,...} of bi- r,s −→ graded abelian groups Et =(Et )r,s∈Z, with differentials dt : Et Et of bidegree (t, −t + 1), such that r,s ∼ r,s −→dt r+t,s−t+1 r−t,s+t−1 −→dt r,s Et+1 = Ker(Et Et )/Im(Et Et ). (1) L. Ribes, P. Zalesskii, Profinite Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics 40, DOI 10.1007/978-3-642-01642-4, c Springer-Verlag Berlin Heidelberg 2010 398 A: Spectral Sequences To simplify the notation, from now on we assume that the isomorphism in (1) is in fact an equality. The bigraded abelian group E2 is called the initial term of the spectral sequence. ∈ r,s Lemma A.1.1 For each r, s Z there exists a series of subgroups of E2 r,s ≤ r,s ≤ r,s ≤···≤ r,s ≤ r,s ≤ r,s r,s 0=B2 B3 B4 C4 C3 C2 = E2 such that r,s r,s r,s ≥ Et = Ct /Bt (t 2). r,s r,s r,s r,s r,s r,s Proof. Set B2 =0andC2 = E2 ;thenE2 = C2 /B2 . Define induc- tively r,s r,s r−t,s+t−1 r−t,s+t−1 r−t,s+t−1 →dt r,s r,s r,s Bt+1/Bt =Im(Et = Ct /Bt Et = Ct /Bt ), and r,s r,s r,s r,s r,s →dt r+t,s−t+1 r+t,s−t+1 r+s,s−t+1 Ct+1/Bt =Ker(Et = Ct /Bt E = Ct /Bt ). Hence r,s ≤ r,s ≤ r,s ≤ r,s ≤ r,s ≤ r,s B2 Bt Bt+1 Ct+1 Ct C2 , and r,s r,s r,s r,s r,s r,s r,s Et+1 =(Ct+1/Bt )/(Bt+1/Bt )=Ct+1/Bt+1. r,s r,s Let Ct , Bt be as in Lemma A.1.1. Define r,s r,s r,s r,s C∞ = Ct ,B∞ = Bt t t and r,s r,s r,s E∞ = C∞ /B∞ . r,s The bigraded abelian group E∞ =(E∞ )r,s∈Z, is completely determined by the spectral sequence. We think of the terms Et of the spectral sequence as approximating E∞. A filtered abelian group with filtration F consists of an abelian group A together with a family of subgroups F n(A)ofA,(n ∈ Z), such that A ≥···≥F n(A) ≥ F n+1(A) ≥···. We always assume that a filtration satisfies the additional condition: F r(A)=A and F r(A)=0. (2) r r To each filtered abelian group A we associate a grading in the following manner A.2 Positive Spectral Sequences 399 Gr(A)=F r(A)/F r+1(A)(r ∈ Z). A filtered graded abelian group with filtration F , consists of a family H = n n (H )n∈Z, of filtered groups H .