Pro-p Groups with Few Normal Subgroups
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1 p. 1/1 Pro-p Groups with Few Normal Subgroups Y. Barnea, N. Gavioli, A. Jaikin-Zapirain, V. Monti, C. M. Scoppola
2 p. 2/1 1. On p-groups of Finite Coclass In this talk p stands for a prime number. For simplicity of the presentation we assume p 2.
3 p. 2/1 1. On p-groups of Finite Coclass In this talk p stands for a prime number. For simplicity of the presentation we assume p 2. We denote the lower central series of a group G by γ 1 (G) = G and γ i+1 (G) = [γ i (G),G]. G is nilpotent of class c if γ c (G) 1 and γ c+1 (G) = 1.
4 p. 2/1 1. On p-groups of Finite Coclass In this talk p stands for a prime number. For simplicity of the presentation we assume p 2. We denote the lower central series of a group G by γ 1 (G) = G and γ i+1 (G) = [γ i (G),G]. G is nilpotent of class c if γ c (G) 1 and γ c+1 (G) = 1. A p-group G has coclass r if G is of class c and G = p c+r.
5 p. 3/1 2. Pro-p Groups of Finite Coclass A pro-p group G has coclass r if it is the inverse limit of finite p-groups of coclass r.
6 p. 3/1 2. Pro-p Groups of Finite Coclass A pro-p group G has coclass r if it is the inverse limit of finite p-groups of coclass r. Equivalently, G/γ n (G) = p n+r for all big enough n.
7 p. 3/1 2. Pro-p Groups of Finite Coclass A pro-p group G has coclass r if it is the inverse limit of finite p-groups of coclass r. Equivalently, G/γ n (G) = p n+r for all big enough n. In particular, G has finite coclass if γ n (G)/γ n+1 (G) = p for all big enough n.
8 p. 3/1 2. Pro-p Groups of Finite Coclass A pro-p group G has coclass r if it is the inverse limit of finite p-groups of coclass r. Equivalently, G/γ n (G) = p n+r for all big enough n. In particular, G has finite coclass if γ n (G)/γ n+1 (G) = p for all big enough n. In 1980 Charles Leedham-Green and Mike Newman came with the five coclass conjectures in decreasing order of difficulty:
9 p. 4/1 3. The Coclass Conjectures Conjecture A. For some function f(p,r), every finite p-group of coclass r has a normal subgroup K of class at most 2 and index at most f(p,r).
10 p. 4/1 3. The Coclass Conjectures Conjecture A. For some function f(p,r), every finite p-group of coclass r has a normal subgroup K of class at most 2 and index at most f(p,r). Conjecture B. For some function g(p, r), every finite p-group of coclass r has derived length at most g(p,r).
11 p. 4/1 3. The Coclass Conjectures Conjecture A. For some function f(p,r), every finite p-group of coclass r has a normal subgroup K of class at most 2 and index at most f(p,r). Conjecture B. For some function g(p, r), every finite p-group of coclass r has derived length at most g(p,r). Conjecture C. Every pro-p group of finite coclass is soluble.
12 p. 4/1 3. The Coclass Conjectures Conjecture A. For some function f(p,r), every finite p-group of coclass r has a normal subgroup K of class at most 2 and index at most f(p,r). Conjecture B. For some function g(p, r), every finite p-group of coclass r has derived length at most g(p,r). Conjecture C. Every pro-p group of finite coclass is soluble. Conjecture D. For fixed p and r there are only finitely many isomorphism classes of infinite pro-p groups of coclass r.
13 p. 4/1 3. The Coclass Conjectures Conjecture A. For some function f(p,r), every finite p-group of coclass r has a normal subgroup K of class at most 2 and index at most f(p,r). Conjecture B. For some function g(p, r), every finite p-group of coclass r has derived length at most g(p,r). Conjecture C. Every pro-p group of finite coclass is soluble. Conjecture D. For fixed p and r there are only finitely many isomorphism classes of infinite pro-p groups of coclass r. Conjecture E. There are only finitely many isomorphism classes of infinite soluble pro-p groups of coclass r.
14 Theorem (Donkin & Leedham-Green, Shalev & Zelmanov) The coclass conjectures are true. p. 5/1
15 p. 5/1 Theorem (Donkin & Leedham-Green, Shalev & Zelmanov) The coclass conjectures are true. The key point (for us) is that pro-p groups of finite coclass are p-adic analytic.
16 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful;
17 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful; 2. G has finite rank;
18 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful; 2. G has finite rank; 3. G has polynomial subgroup growth (PSG);
19 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful; 2. G has finite rank; 3. G has polynomial subgroup growth (PSG); 4. G is a Lie group over the p-adic integers;
20 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful; 2. G has finite rank; 3. G has polynomial subgroup growth (PSG); 4. G is a Lie group over the p-adic integers; 5. G is linear over the p-adics.
21 p. 6/1 Theorem (Lazard, Lubotzky and Mann) Let G be a pro-p group. The following are equivalent: 1. G is finitely generated and virtually powerful; 2. G has finite rank; 3. G has polynomial subgroup growth (PSG); 4. G is a Lie group over the p-adic integers; 5. G is linear over the p-adics. A point worth noticing: finite rank implies PSG is easy. The other direction is harder.
22 p. 7/1 Facts Let G be a pro-p group of finite coclass.
23 p. 7/1 Facts Let G be a pro-p group of finite coclass. 1. There exists K a normal finite subgroup of G such that G = G/K is just infinite.
24 p. 7/1 Facts Let G be a pro-p group of finite coclass. 1. There exists K a normal finite subgroup of G such that G = G/K is just infinite. 2. Every normal subgroup of big enough index of G is equal γ n ( G) for some n.
25 p. 7/1 Facts Let G be a pro-p group of finite coclass. 1. There exists K a normal finite subgroup of G such that G = G/K is just infinite. 2. Every normal subgroup of big enough index of G is equal γ n ( G) for some n. 3. Thus, there is a constant c such that if N is normal in G, then N contains γ n (G) and N/γ n (G) p c.
26 p. 7/1 Facts Let G be a pro-p group of finite coclass. 1. There exists K a normal finite subgroup of G such that G = G/K is just infinite. 2. Every normal subgroup of big enough index of G is equal γ n ( G) for some n. 3. Thus, there is a constant c such that if N is normal in G, then N contains γ n (G) and N/γ n (G) p c. 4. Pro-p groups of finite coclass are not closed under direct sum.
27 p. 8/1 4. Pro-p Groups of Finite Width Let G be a pro-p group. We say that G has width w if for all n γ n (G)/γ n+1 (G) p w.
28 p. 8/1 4. Pro-p Groups of Finite Width Let G be a pro-p group. We say that G has width w if for all n γ n (G)/γ n+1 (G) p w. Notice: direct sum of groups of finite width has finite width.
29 p. 8/1 4. Pro-p Groups of Finite Width Let G be a pro-p group. We say that G has width w if for all n γ n (G)/γ n+1 (G) p w. Notice: direct sum of groups of finite width has finite width. Examples: 1. Let Z p be the p-adic integers. G n = SL n d (Z p) = ker(sl d (Z p ) SL d (Z p /(p n )). G = G 1 is a pro-p group, G n = γ n (G) and G n /G n+1 = p d2 1.
30 p. 9/1 2. Let F p [[t]] be formal power series over F p. G n = ker(sl d (F p [[t]]) SL d (F p [[t]]/(t n )). G = G 1 is a pro-p group, G n = γ n (G) and G n /G n+1 = p d2 1.
31 p. 9/1 2. Let F p [[t]] be formal power series over F p. G n = ker(sl d (F p [[t]]) SL d (F p [[t]]/(t n )). G = G 1 is a pro-p group, G n = γ n (G) and 3. The Nottingham group G n /G n+1 = p d2 1. J = { t + a 1 t 2 + a 2 t 3 + a i F p }, where the product is by composition. γ n (J)/γ n+1 (J) = { p n 1 mod p 1 p 2 n 1 mod p 1.
32 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. p. 10/1
33 p. 10/1 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. 4. "Bad" examples of index subgroups of the Nottingham group, e.g. the Nottingham group in characteristic 2.
34 p. 10/1 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. 4. "Bad" examples of index subgroups of the Nottingham group, e.g. the Nottingham group in characteristic 2. Still hereditarily just infinite, but their normal subgroup structure is a lot more complex.
35 p. 10/1 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. 4. "Bad" examples of index subgroups of the Nottingham group, e.g. the Nottingham group in characteristic 2. Still hereditarily just infinite, but their normal subgroup structure is a lot more complex. 5. Grigorchuk group: Has finite width, but is not hereditarily just infinite and does not have periodicity in the lower central series.
36 p. 10/1 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. 4. "Bad" examples of index subgroups of the Nottingham group, e.g. the Nottingham group in characteristic 2. Still hereditarily just infinite, but their normal subgroup structure is a lot more complex. 5. Grigorchuk group: Has finite width, but is not hereditarily just infinite and does not have periodicity in the lower central series. Seems hard to work with, e.g. subgroup growth???
37 p. 10/1 All of the examples till now are hereditarily just infinite and have periodicity in the lower central series. 4. "Bad" examples of index subgroups of the Nottingham group, e.g. the Nottingham group in characteristic 2. Still hereditarily just infinite, but their normal subgroup structure is a lot more complex. 5. Grigorchuk group: Has finite width, but is not hereditarily just infinite and does not have periodicity in the lower central series. Seems hard to work with, e.g. subgroup growth??? Goal: Find a good definition to avoid all the more difficult examples.
38 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n.
39 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n. A Pro-p group G has Constant Normal Subgroup Growth (CNSG) if there exists c such that a n (G) c for all n.
40 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n. A Pro-p group G has Constant Normal Subgroup Growth (CNSG) if there exists c such that a n (G) c for all n. Our first 3 examples (usually) have CNSG the Grigorchuk group and the other "bad" examples do not.
41 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n. A Pro-p group G has Constant Normal Subgroup Growth (CNSG) if there exists c such that a n (G) c for all n. Our first 3 examples (usually) have CNSG the Grigorchuk group and the other "bad" examples do not. Lemma: A pro-p group with CNSG has finite normal rank.
42 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n. A Pro-p group G has Constant Normal Subgroup Growth (CNSG) if there exists c such that a n (G) c for all n. Our first 3 examples (usually) have CNSG the Grigorchuk group and the other "bad" examples do not. Lemma: A pro-p group with CNSG has finite normal rank. Problem 1: A pro-p group with finite normal rank has PNSG. What about the other direction?
43 p. 11/1 5. Few Normal Subgroups A Pro-p group G has Polynomial Normal Subgroup Growth (PNSG) if there exists c such that a n (G) n c for all n, where a n (G) is the number of normal subgroups of index n. A Pro-p group G has Constant Normal Subgroup Growth (CNSG) if there exists c such that a n (G) c for all n. Our first 3 examples (usually) have CNSG the Grigorchuk group and the other "bad" examples do not. Lemma: A pro-p group with CNSG has finite normal rank. Problem 1: A pro-p group with finite normal rank has PNSG. What about the other direction? There is a soluble counter example, what about just infinite?
44 p. 12/1 Finite obliquity: For all n there are a "few" normal subgroups which neither contain γ n (G) or contained in it. Formal definition is somewhat technical.
45 p. 12/1 Finite obliquity: For all n there are a "few" normal subgroups which neither contain γ n (G) or contained in it. Formal definition is somewhat technical. A group G is called r-sandwich if there is r such that for all normal subgroup N of G there exists i such that γ i (G) N γ i+r (G).
46 p. 12/1 Finite obliquity: For all n there are a "few" normal subgroups which neither contain γ n (G) or contained in it. Formal definition is somewhat technical. A group G is called r-sandwich if there is r such that for all normal subgroup N of G there exists i such that γ i (G) N γ i+r (G). Theorem 1: Let G be a non-nilpotent pro-p group. Then G has finite obliquity if and only if it is sandwich. Moreover, in such a case, G is just infinite of finite width and has CNSG.
47 Suggested: CNSG is equivalent to finite obliquity. p. 13/1
48 p. 13/1 Suggested: CNSG is equivalent to finite obliquity. Not true: CNSG closed under taking a direct sum with a finite p-group, finite obliquity is not.
49 p. 13/1 Suggested: CNSG is equivalent to finite obliquity. Not true: CNSG closed under taking a direct sum with a finite p-group, finite obliquity is not. Theorem 2: Let G be a non-nilpotent pro-p group with CNSG. Then G has a maximal finite normal subgroup K and G/K is just infinite. Moreover, G has finite width.
50 p. 13/1 Suggested: CNSG is equivalent to finite obliquity. Not true: CNSG closed under taking a direct sum with a finite p-group, finite obliquity is not. Theorem 2: Let G be a non-nilpotent pro-p group with CNSG. Then G has a maximal finite normal subgroup K and G/K is just infinite. Moreover, G has finite width. Problem 2: Suppose G is hereditarily just infinite pro-p group with CNSG. Is it sandwich?
51 p. 14/1 6. Periodicity A period on a pro-p group G is a map τ : M G, where M is an open normal subgroup of G such that
52 p. 14/1 6. Periodicity A period on a pro-p group G is a map τ : M G, where M is an open normal subgroup of G such that 1. τ(m) is an open subgroup of G;
53 p. 14/1 6. Periodicity A period on a pro-p group G is a map τ : M G, where M is an open normal subgroup of G such that 1. τ(m) is an open subgroup of G; 2. for every open normal subgroup H of G contained in τ(m) we have that τ 1 (H) is an open normal subgroup of G and [G : H] > [G : τ 1 (H)].
54 p. 14/1 6. Periodicity A period on a pro-p group G is a map τ : M G, where M is an open normal subgroup of G such that 1. τ(m) is an open subgroup of G; 2. for every open normal subgroup H of G contained in τ(m) we have that τ 1 (H) is an open normal subgroup of G and [G : H] > [G : τ 1 (H)]. We say that a period is uniform if there is a constant c such that for all H as above, [G : H] = p c [G : τ 1 (H)].
55 Proposition: If G admits a period it admits a uniform period. p. 15/1
56 p. 15/1 Proposition: If G admits a period it admits a uniform period. Theorem 3: Suppose G is a non-abelian just infinite pro-p group which admits a period. Then G is sandwich, in particular it has CNSG. Moreover, there is d such for all big enough n, a p n(g) = a p n+d (G).
57 Theorem 4: Suppose G is one of the known examples of hereditarily just infinite pro-p groups with CNSG. Then G has a period. p. 16/1
58 p. 16/1 Theorem 4: Suppose G is one of the known examples of hereditarily just infinite pro-p groups with CNSG. Then G has a period. Thus G is sandwich.
59 p. 16/1 Theorem 4: Suppose G is one of the known examples of hereditarily just infinite pro-p groups with CNSG. Then G has a period. Thus G is sandwich. Moreover, every subgroup of finite index of G has all of the above properties too.
60 p. 16/1 Theorem 4: Suppose G is one of the known examples of hereditarily just infinite pro-p groups with CNSG. Then G has a period. Thus G is sandwich. Moreover, every subgroup of finite index of G has all of the above properties too. In addition, Branch groups and all the other known examples of hereditarily just infinite pro-p groups are all not CNSG.
61 p. 17/1 7. Conjectures? Do I dare to conjecture that if G is hereditarily just infinite pro-p group, then:
62 p. 17/1 7. Conjectures? Do I dare to conjecture that if G is hereditarily just infinite pro-p group, then: 1. If G has CNSG, then G has a period and, in particular, there exists d such that for all big enough n we have a p n(g) = a p n+d (G).
63 p. 17/1 7. Conjectures? Do I dare to conjecture that if G is hereditarily just infinite pro-p group, then: 1. If G has CNSG, then G has a period and, in particular, there exists d such that for all big enough n we have a p n(g) = a p n+d (G). 2. If G has finite obliquity or CNSG or a period, then every subgroup of finite index of G has finite obliquity or CNSG or a period respectively.
64 p. 17/1 7. Conjectures? Do I dare to conjecture that if G is hereditarily just infinite pro-p group, then: 1. If G has CNSG, then G has a period and, in particular, there exists d such that for all big enough n we have a p n(g) = a p n+d (G). 2. If G has finite obliquity or CNSG or a period, then every subgroup of finite index of G has finite obliquity or CNSG or a period respectively. 3. If G has few normal subgroups, then there exists a constant c such that for all n, a n (G) n c log n.
65 p. 18/1 8. The Nottingham Group Open subgroups of the Nottingham group have a period:
66 p. 18/1 8. The Nottingham Group Open subgroups of the Nottingham group have a period: Let H be an open subgroup of J. Then H contains some } J k = {t + a k t k+1 + a k+1 t k+2 + a i F p.
67 p. 18/1 8. The Nottingham Group Open subgroups of the Nottingham group have a period: Let H be an open subgroup of J. Then H contains some } J k = {t + a k t k+1 + a k+1 t k+2 + a i F p. It is easy to see that there exists m such that for all N normal subgroups of H of big enough index there exists n such that J n+p m N J n.
68 p. 18/1 8. The Nottingham Group Open subgroups of the Nottingham group have a period: Let H be an open subgroup of J. Then H contains some } J k = {t + a k t k+1 + a k+1 t k+2 + a i F p. It is easy to see that there exists m such that for all N normal subgroups of H of big enough index there exists n such that J n+p m N J n. We define the period on J k by τ m (t(1 + f(t))) = t(1 + t pm f(t)).
69 p. 19/1 Lemma: Let φ = a(t) J and ψ = t + s(t) J k. Then φψφ 1 t + s(a(t)) a (t) mod t 2k+2.
70 p. 19/1 Lemma: Let φ = a(t) J and ψ = t + s(t) J k. Then φψφ 1 t + s(a(t)) a (t) mod t 2k+2. Corollary: For k p m, the map τ m induces a J-isomorphism from J k /J k+p m onto J k+p m/j k+2p m.
71 p. 19/1 Lemma: Let φ = a(t) J and ψ = t + s(t) J k. Then φψφ 1 t + s(a(t)) a (t) mod t 2k+2. Corollary: For k p m, the map τ m induces a J-isomorphism from J k /J k+p m onto J k+p m/j k+2p m. The fact that τ m is a period follows from the sandwich property on the previous slide.
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