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LDM70 TOPICS ON GROUPS AND THEIR REPRESENTATIONS Prova The Kummerian Property for pro- p -groups CLAUDIO University of QUADRELLI MILANO with Ido Efrat BICOCCA Ben-Gurion Univ. October 10th 2017 Garda Lake - Palazzo Feltrinelli Profinite

  1. LDM70 TOPICS ON GROUPS AND THEIR REPRESENTATIONS Prova The Kummerian Property for pro- p -groups CLAUDIO University of QUADRELLI MILANO with Ido Efrat BICOCCA Ben-Gurion Univ. October 10th 2017 Garda Lake - Palazzo Feltrinelli

  2. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Vladimir Voevodsky 4 Jun. 1966 – 30 Sep. 2017 Fields medalist in 2002 Claudio Quadrelli Universit` a Milano-Bicocca

  3. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Profinite groups A profinite group is a topological group which (1) is compact; (2) is totally disconnected; (3) has a basis { N i � G } of open neighb.hds of 1 Equivalently: G = lim − i G i , with | G i | < ∞ ; or ← G ≃ Gal ( E | F ) for some Galois extension E | F Holy Grail of Galois Theory F s | F )? Which profinite G is Gal (¯ We have some restrictions: if such G is finite, then | G | = 2 (Artin-Schreier Theorem) Claudio Quadrelli Universit` a Milano-Bicocca

  4. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Pro- p -groups G profinite is pro- p if [ G : U ] = p n for every open U ⊆ G ; − i P i with | P i | = p n i equivalently G = lim ← They behave very nicely — almost like finite p -groups Examples finite p -groups Z p = { a 0 + a 1 p + a 2 p 2 + . . . } = � 1 � is the completion of Z w.r.t. the topology induced by { p n Z p , n ≥ 1 } a free pro- p group F is the completion of F abs w.r.t. the topology induced by { U ⊆ F abs | [ F abs : U ] = p n } the Nottingham group F s | F ) G F ( p ), the maximal pro- p quotient of Gal (¯ Claudio Quadrelli Universit` a Milano-Bicocca

  5. � � Profinite groups Representations of dim 1 The Kummerian property Cohomology References p -adic representations of dimension 1 For G pro- p and θ a p -adic (continuous) representation → GL 1 ( Z p ) ≃ Z × θ : G − p let V p denote Z p edowed with the G -action induced by θ , and set N = ker ( θ ). Since either im ( θ ) ≃ Z p or is trivial, the extension � Γ � N � G 1 1 splits, and G ≃ N ⋊ Γ. Claudio Quadrelli Universit` a Milano-Bicocca

  6. Profinite groups Representations of dim 1 The Kummerian property Cohomology References The cyclotomic character For a field F , set F s | α p n = 1 for some n ≥ 0 α ∈ ¯ µ p ∞ = � � Then Aut ( µ p ∞ ) ≃ GL 1 ( Z p ), and the action G F ( p ) � µ p ∞ induces naturally a representation θ F : G F ( p ) − → GL 1 ( Z p ) ≃ Aut ( µ p ∞ ) called the cyclotomic character Idea Study the couples ( G , θ ) where θ “behaves” like the cyclotomic character, to understand better G F ( p ) Claudio Quadrelli Universit` a Milano-Bicocca

  7. Profinite groups Representations of dim 1 The Kummerian property Cohomology References The Kummer subgroup For G endowed with θ : G → GL 1 ( Z p ), the Kummer subgroup is � h − θ ( g ) · ghg − 1 | g ∈ G , h ∈ N � K θ = � G K θ ⊆ N and K θ ⊆ G p [ G , G ] K θ ⊇ [ N , N ], so N / K θ is abelian G / K θ ≃ ( N / K θ ) ⋊ Γ Definition ( G , θ ) is Kummerian if N / K θ is torsion-free — i.e., E.E. Kummer N / K θ ≃ V θ ⊕ . . . ⊕ V θ Claudio Quadrelli Universit` a Milano-Bicocca

  8. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Examples free pro- p -groups are Kummerian for every representation θ If θ ≡ 1 , then ( G , θ ) is Kummerian ⇔ G / [ G , G ] is torsion-free Z / p n is NOT Kummerian for p n � = 2... ... whereas ( Z / 2 , θ ) IS Kummerian for Im ( θ ) {± 1 } (Artin-Schreier Theorem) Proposition For F containing a root of 1 of order p , G F ( p ) with the cyclotomic character IS Kummerian, since by Kummer theory √ � � � p ∞ G F ( p ) / K θ F = Gal F � F � Claudio Quadrelli Universit` a Milano-Bicocca

  9. Profinite groups Representations of dim 1 The Kummerian property Cohomology References The relations of Kummerian groups Given a minimal presentation π � R � F � G � 1 1 (i.e. F is free s.t. F / F p [ F , F ] ≃ G / G p [ G , G ]), set ˆ θ = θ ◦ π , ˆ θ : F → GL 1 ( Z p ) Theorem For im ( θ ) torsion-free the following are equivalent ( G , θ ) is Kummer G / K θ is torsion-free for any minimal presentation G ≃ F / R one has F / K ˆ θ ≃ G / K θ Consequence: ( G , θ ) is Kummer ⇔ R ⊆ K ˆ θ Claudio Quadrelli Universit` a Milano-Bicocca

  10. Profinite groups Representations of dim 1 The Kummerian property Cohomology References The group H 1 ( G , V ) Let V be a (continuous) G -module. A 1-cocycle is a (continuous) map c : G → V such that c ( g 1 g 2 ) = c ( g 1 ) + g 1 . c ( g 2 ) ∀ g 1 , g 2 ∈ G Definition H 1 ( G , V ) := Z 1 ( G , V ) / B 1 ( G , V ), where Z 1 ( G , V ) = { c : G → V | c is a 1-cocycle } B 1 ( G , V ) = { g �→ ( g − 1) . v , v ∈ V } A morphism V → W induces H 1 ( G , V ) → H 1 ( G , W ) If G � V is trivial then H 1 ( G , V ) = Hom G ( G , V ) If G is pro- p then H 1 ( G , Z / p ) = ( G / G p [ G , G ]) ∗ Claudio Quadrelli Universit` a Milano-Bicocca

  11. Profinite groups Representations of dim 1 The Kummerian property Cohomology References The Kummerian property and H 1 ( G , V θ ) Theorem For im ( θ ) torsion-free the following are equivalent (1) ( G , θ ) is Kummerian c : G → V θ ker ( c ) ( ker ( c ) = c − 1 (0)) (2) K θ = � (3) The projection V θ → Z / p induces an epimorphism � � H 1 ( G , Z / p ) H 1 ( G , V θ ) (4) For { x i , i ∈ I } minimally generating G and v i ∈ Z p , there is a unique 1-cocycle c : G → V θ s.t. c ( x i ) = v i for all i ∈ I The equivalence (3) ⇔ (4) is due to J. Labute (1967) 1 . 1 N.D. Tˆ an deserves my gratitude for putting this result under my attention and therefore inspiring some of these results Claudio Quadrelli Universit` a Milano-Bicocca

  12. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Hunting “bad” relations 1 Given a pro- p group G with minimal presentation � R � F � G � 1 1 and a representation θ , ( G , θ ) is Kummerian if and only if � R ⊆ ker ( c ) c : F → V ˆ θ Recipe: take { x i , i ∈ I } minimally generating F and G . (1) Let ˆ θ : F → GL 1 ( Z p ) be arbitrary s.t. R ⊆ ker (ˆ θ ) (2) Assign “smart” values v i ∈ Z p to each x i , and compute the 1-cocycle c : F → V ˆ θ induced by such values (3) If there are some r ∈ R s.t. c ( r ) � = 0, then ( G , θ ) is NOT Kummerian for any θ : G → GL 1 ( Z p ) Claudio Quadrelli Universit` a Milano-Bicocca

  13. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Hunting “bad” relations 2 Examples Set G = � x 1 , x 2 , x 3 | R � and let θ : G → GL 1 ( Z p ) be arbitrary. Assume r = x q 1 [ x 2 , x 3 ] ∈ R , whith q = p n (1) θ ( r ) = 1 ⇒ θ ( x q 1 [ x 2 , x 3 ]) = θ ( x q 1 ) = 1 ⇒ θ ( x 1 ) = 1 (2) Set v 1 = c ( x 1 ) = 1, v 2 = c ( x 2 ) = v 3 = c ( x 3 ) = 0 (3) c ( r ) = c ( x q 1 ) + θ ( x q 1 ) · c ([ x 2 , x 3 ]) = q · c ( x 1 ) + 0 = q � = 0 Assume r 1 = x q 1 1 [ x 1 , x 3 ] ∈ R and r 2 = x q 2 2 [ x 2 , x 3 ] ∈ R , q 1 � = q 2 (1) θ ( r i ) = 1 ⇒ θ ( x 1 ) = θ ( x 2 ) = 1 (2) Set v 1 = c ( x 1 ) = v 2 = c ( x 2 ) = 1 and v 3 = c ( x 3 ) = 0 1 ) · c ([ x 2 , x 3 ]) = q i + ( θ ( x 3 ) − 1 − 1), and if (3) c ( r i ) = c ( x q i i ) + θ ( x q c ( r 1 ) = c ( r 2 ) = 0 then q 1 = q 2 , a contradiction In both cases ( G , θ ) is NOT Kummer Claudio Quadrelli Universit` a Milano-Bicocca

  14. Profinite groups Representations of dim 1 The Kummerian property Cohomology References More “bad” relations Consider a pro- p group G with minimal presentation � 1 , � R � F � G 1 with { x 1 , x 2 , . . . } minimally generating F , and assume that r ∈ R with... r = x q 1 s , 0 � = q ∈ p Z p and s ∈ � x 2 , x 3 , . . . � =: F 1 , or r = x q 1 st , q ∈ p Z k p Z p , k ≥ 2, s ∈ F 1 ∩ [ F , F ], t ∈ γ k +1 ( F ) Then 2 ( G , θ ) is NOT Kummerian for any θ : G → GL 1 ( Z p ). 2 Similar results have been obtained recently and independently by J. Min´ aˇ c, M. Rogelstad and N.D. Tˆ an too, employing the equivalence (3) ⇔ (4) as well Claudio Quadrelli Universit` a Milano-Bicocca

  15. Profinite groups Representations of dim 1 The Kummerian property Cohomology References Further research Theorem (Weigel-Q., 2017) Given F containing a root of 1 of order p , for every i ≥ 1, V ⊗ i θ F → Z / p induces an epimorphism H i ( G F ( p ) , V ⊗ i � � H i ( G F ( p ) , Z / p ) θ F ) This is a consequence of Rost-Voevodsky theorem. What can we say on the structure of G ? (Group-theoretic translation in case i = 2?) Also, the Bogomolov Conjecture (1995) can be translated as follows: Given F containing a root of 1 of order p , is K θ F free? Claudio Quadrelli Universit` a Milano-Bicocca

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