Powers of the augmentation ideal

Pub . Ma . UAB Nó 17, Feb e , 1980 associa eddimensio n subg ó~ POWERS OF THE AUGMENTATION IDEAL Summa y o alk gi en a Uni e si a Au ónoma de Ba celona, 19 Ap il, 1978, by B . Ha ley . Le Gbe a g oup, R a commu a i e ing wi h 1 (we will in ac only be conce ned wi h he cases when R is a ield o ZL ) . The g oup ing RG is he se o no mal ini e linea combina ions E  a g g  (ag ER), wi h he de ini ions  (Ex g g) +  ( EU g g)  = E(a g+ u g )g and gE G (Ea 9)(EU h) = E ( E au)k . The augmen a ion E : RG-> R gi en by s(Ea g g) = Ea gis a ing homomo phism and i s ke nel o(R, G) is he au~- men a ion ideal o RG . We a e in e es ed in he powe s A (R, G) and he 0n(R, G) = Gn(1 + o n (R, G)) . No e ha Dn(R, G) is he ke nel o he na u al map o G in o he g oup o uni s U(RG/p n (R, G)) o RG/p n (R, G) . This is no in ended o be a de ailedsu eyo his a ea, bu sim- ply a discussion o some sample esul s . I . BACKGROUND Fi s we men ion some connec ions be ween his subjec and o he p oblems . 1 . Conn e c ion wi h ma ix ep esen a ions I is o en use ul o know ha ce ain g oups can be ep esen ed ai h ully by ma iceso e some well beha ed ing . Theo em . I G is a ini ely gene a ednilpo en g oup, hen G can be embedded in some GL(n, 71) (much mo e gene al esul s a e ue) . Indica ion o p oo . We easily educe o he case when G is o sion- ee . Le G ha e nilpo ency class c . Then Dc+1( 2Z , G) =1 (qui e a di i- cul esul ) . Hence G can be embedded in he g oup o uni s o a G/oc+1 (TL , G) = A . The addi i e g oup o A is ini ely gene a ed, and G ope a es ai h ully on his addi i e g oup by igh mul iplica ion . The addi i e o sion subg oup T o Ais ini e, and i is easy o see ha G ope a es ai h ully on A/T . See :P . Hall "The Edmon on No es on Nilpo en G oups" (Queen Ma y College, London) o he de ails . 2 . Connec ion wi h esidual n i lpo encyo g oups Theo em (Qaumslag, Passi, see J . Pu e App . Algeb a 6 (1975)) . Le F be a non-abelian ee g oup, R<1 F . Then F/R' is esidually nilpo en i and only i  n  A (1 , G) = 0, whe e G = F/R . n=1 II . DIMENSION SUBGROUPS Fo any G,  le G 1 =  G and Gn+1  =  [ G n , G],  so  ha G 1 > .G2  >  . . .  is he lowe cen al se ies o G . Then Lemma 2 .1 . Gn < D n (R, G) This ollows .b y induc ion om he iden i y 1 - X -1Y-1 xY = X -1 Y -1 ( (Y-1)(X-1)-(X-1)(Y-1)) . Le  /Go be he o sion subg oup o G/G n . Theo em 2 .2 . I K is a ield o cha ac e is ic ze o, hen D n (K, G) _ _ N/--G n' This is due o Jennings ; a p oo can be ound in P . Hall's Edmon on No es, o D .S . Passman "The Algeb aicS uc u e o G oup Rings" (In e s- cience) . Theo em 2 .3 . (Laza d) I K is a ield o cha ac e is ic p > 0, hen D n (K, G) =  n  Gp J ip 3 > n whe e i H is a g oup, H m = <h m : h c H> . Ano he desc ip ion o D n (K, G) was gi en ea lie by Jennings . Theo em 2 .3 can also be ound in Passman's book . No e ha in Theo em 2 .3, i G has exponen p, hen Dn (K, G) = Gn o all n . Theo em 2 . .2 and 2 .3 show ha o e ields, he dimensionsubg oups can be comple ely desc ibed in g oup heo e ic e ms . Fo he case R = 71 he si ua ion is mo e complica ed . -I was a one ime conjec u ed ha D n (71 , G) = Go o all G (Dimen- sion subg oup conjec u e) and se e al alse p oo s ha e been gi en . Theo em 2 .4 . (i) I F is ee, hen D n (F) = F n o all n . (Magnus, 1937) (ii) I G is any g oup, hen Dn(G) = G n o n = 1,2,3 (I don' know who i s p o ed hese esul s ; qui e a . numbe o p oo s a e now a aila- ble, o example A .H .M . Hoa e, J . LondonMa h . Soc .) iii) I G is a ini e p-g oup, hen Dn(G) = G n o n < p (Mo an ; P oc . Camb idge Philos . Soc .) I ha e w i en D n (G) o D n (7L , G) . E en ually he Dimension Subg oupConjec u e was e u ed by Theo em 2 .5 (Rips, 1972) The e exis s a ini e 2-g oup G o class 3 such ha ID4(G)1 = 2 . (Is ael J . Ma h .) The bes esul o da e is con ained in a ecen and di icul pape o Sjog en (J . Pu e Appd . Algeb a (1979)) . Theo em 2 .6 (Sjog en) The e exis s an (explici ly gi en) unc ion cn such ha i G is any g oup, hen D n (G)/G n has exponen di iding cn . The p oo uses some elemen a y spec al sequence ideas a id some com- plica ed Lie- heo e ic me hods . I ha e a simpli ied e sion o he p oo . We ha e c l = c2 = c 3 = 1, c 4 = 2, so 2 .4(ii) ollows om Sjog en's wo k, and Rips' example shows ha c 4 is bes possible . Also by examining he  unc ion  c n one can eplace he es ic ion n <  p in 2 .4(iii)  by n < p+1 . Some open p oblems P oblem 1 . Dóes he eexis an in ege alued unc ion (c) such ha i G .is a nilpo en g oup o class .c, hen D (c)(G)=l? Rips' example shows ha (c)= c+1 will no do, bu he e is no coun e example known o ule ou he possibili y (c) = c+2 . P oblem 2 . (a weake e sion o P oblem 1) .'I G is nilpo en , is i ue ha  n 1 D n(G) = 1? P oblem3 . Does he eexis a ini e p-g oup (p ~ 2) such ha D n (G) :~ G n o some n? The smalles possibili y is p = 3, n = 5 . III . THE .LIERING AND THE GRADEDRING Le G be a g oup, wi h lowe cen al se ies We w i e Gn /G n+1 addi i ely and o o he abelian g oup and de ine L(G)  6) G n /G n+1 n=1 ( es ic ed di ec sum), and de ine an ope a ion ( , on L(G)by (xGn+1> .yGM+l) = [x, y]Gn+m+1 (x e Gn , y c G m ) and ex ending by addi i i y . This u ns L(G) in o a Lie ing ; he Jacobi iden i y ollows om he Wi iden i y . Now w i ing o = A(a, G), we o o he abelian g oup G(G) = ® on/on+1 , n=0 (a+on+1)(,+o o+1) =as+,n+m+l . We ex end his ope a ion by addi i i y again, and ob ain his ime an associa i e ing, called he associa ed g aded ing o 71 G . We can hink o G(G) as a Lie ing unde usual u - u ope a ion, and hen he map xGn+1 ; (x-1)  + o n+l  (x c  Gn ) 10 ex e .nds o a Lie homomo phism o L(G) in o G(G) . The ke nel o he es- ic ion o 0 o Gn/G n+l is (Dn+1 . (G) n Gn )/Gn+l , and so in his way we ob ain a connec ion be ween dimensionsubg oüps and Lie alneb as . The map 0 also de e mines a homomo phism om L(G) o iz K  ->G(G) ® 2z K, i Kis any ield o cha ac e is ic ze o, and we ha e he ollowing beau- i ul esul o Quillen (J . Algeb a 10 (1968)) . Theo em 3 .1 . Wi h he abo e no a ion, 0 is injec i e and G(G) o K he uni e sal en eloping algeb a o L(G) a K . I is in ac mo e na u al o hese pu poses ing om he cha ac e is ic ze o dimension se ies is Ní~G1 > _'/-G2 hough he wo Lie ings become isomo phic-on enso ing wi h a ield o cha ac e is ic ze o . Howe e using analogy wi h cha ac e is ic p > 0 ; in ol ing he es ic ed uni e sal su eyo hese ma e s, see I .B .S . Passi (J . LondonMa h . Soc . 1979) . IV .POWERS OF THE AUGMENTATION IDEAL He e we con ine ou sel es o he ques ion : n o n = 0, whe e o = o(7L, G)? In dealing wi h his ques ion, i seems n=1 una oidable o conside also he ing ZL/p m 7L . Theo em 4 .1 .  n  o n (G, ZL/p m ZZ) = 0 i and n=1 a nilpo en p-g oup o ini e exponen . o cons uc he Lie he dimension se ies gi es he igh he e we ge a esul like Theo em 3 .1 en eloping algeb a . Fo a mo e de ailed When is i ue ha only i G is esidual ly 1 5 1  i, :~-~ .ii~  aiki~< ü .n1 .11 .'+ . .i1~ws :illh :¢imi i ~d ~J, IIUC . ~ ~s~ . . . il This was p o ed by B . Ha ley (P oc . LondonMa h . Soc . 1969) and al- so by K .41 . G uenbe g(unpublished) . The case m= 1 was done much ea lie by Mal'ce . Theo em 4 .2 . I G is esidually o sion- ee nilpo en , hen o n (7Z , G) = 0 . n=1 This is also due o Ha ley in he abo e pape . The main pa o he p oo consis so adap ing a cons uc ion o P . Hall o ob ain, o a ini ely gene a ed o sion- ee nilpo en g oup G, a 7Z -basiso ZZ G which is closely ela ed o he powe s o o . This can now be done a he be e . Le ~~ n (7Z , G)/o n (7Z , G) deno e he addi i e o sión subg oup o 7Z G/o n (TZ , G) . Theo em 4.3 . Le G be a ini ely gene a ed o sion- ee nilpo en g oup . Then 7Z G has a 7Z -basis B such ha , o each n > 1, -,/ o n (7Z , G) is spanned by a subse o B . (See SymposiaMa h ., 1976) O he p oo s o 4 .2 ha e since been gi en by A .L . Smel'kin using echniques om Lie Algeb as (T ans . Moscow Ma h . Soc . 1973, and also in a ecén issue o Uspekhi Ma . Nauk .) Finally, A .I . Lich man showed ha su icien condi ions gi en by 4 .1 and 4 .2 a e e y nea ly necessa y . I G is a g oup, hen we say ha G is disc imina ed by nilpo en p-g oups o ini eexponen i and only i o each ini e se o elemen s x 1 , . . ., xm e G, he e exis s a p ime p and a homomo phism e o G in o a nilpo en p-g oup o ini e exponen , such ha 6(x i ) # 1  (1 < i< m) . o n (IL , G) = 0 i and only i G is ei he esidually o sion- n=1 ee nilpo en o disc imina ed by nilpo en p-g oups o ini e exponen . Theo em (Is ael J . Ma h . 1977)