Some Thin Pro-p-Groups

Transcription

1 Joural of Algebra 220, Artcle ID jabr , avalable ole at htt: o Some Th Pro--Grous Sadro Mattare Dartmeto d Matematca Pura ed Alcata, Uersta ` degl Stud d Padoa, a Belzo 7, I-5 Padua, Italy Commucated by Efm Zelmao Receved February 28, 995 A ro--grou s called th f ts lattce of closed ormal subgrous cotas o more tha arwse comarable elemets. A vestgato of certa Ž fte. th ro--grous was made A. Carat et al. Quart. J. Math. Oxford 47 Ž 996., , by studyg the graded Le algebras assocated wth ther lower cetral seres. I artcular, the ma result of A. Carat et al. Quart. J. Math. Oxford 47 Ž 996., asserts that those graded Le algebras are, certa cases, uquely determed by ther quotets of low dmeso. I the reset aer we costruct ro--grous whch have, assocated wth ther lower cetral seres, the graded Le algebras of A. Carat et al. Quart. J. Math. Oxford 47 Ž 996., Academc Press Key Words: th ro--grous; assocated graded Le algebras.. INTRODUCTION I ths aer we costruct some examles of th ro--grous. A grou of -ower order, where s a rme umber, s sad to be th, accordg to Br, f all at-chas Ž subsets of arwse comarable elemets. ts lattce of ormal subgrous have at most elemets. For examle, a -grou of maxmal class s th. Its lattce of ormal subgrous cossts of what we call a damod o to Žthe lattce of subgrous of a elemetary Abela grou of order 2., followed by a cha. It turs out Žsee CMNS for refereces ad further exlaatos. that the lattce of ormal subgrous of a th grou always looks lke a sequece of damods lked by chas. More recsely, the lower cetral factors a th -grou are elemetary of order or 2, ad every ormal subgrou les betwee two cosecutve terms of the lower cetral seres. * Curret address: Dartmeto d Matematca, Uversta ` degl Stud d Treto, va Sommarve 4, I-8050 Povo Ž Treto., Italy. E-mal: mattare@scece.ut.t $0.00 Coyrght 999 by Academc Press All rghts of reroducto ay form reserved. 56

2 SOME THIN PRO--GROUPS 57 The aer CMNS focused o ro--grous, where the defto of th ales after relacg ormal subgrou wth closed ormal subgrou. As a cosequece of N. Blackbur s results o -grous of maxmal class, the legth of the cha betwee the frst two damods Ž frst cha legth what follows. a fte th ro--grou ca oly be a odd umber less tha Žto be recse, sayg that ths legth s l meas that Ž G. Ž G. l2 l s the frst o-cyclc lower cetral factor of G after GG.. I CMNS t was rove that the umber of ossbltes for the frst cha legth of a fte th ro--grou s deedet of Žfor 5., amely, ths legth ca oly be,, or 2. The case 2 occurs for the so-called Nottgham grou, whch s curretly beg vestgated by several eole. The ma result of CMNS says that the remag cases Ž wth a further assumto case of legth. there are, u to somorhsm, three ossbltes all for the graded Le algebra assocated wth the lower cetral seres of the grou Žbut o such uqueess statemet ca be made for the grou tself.. We quote the exact result from CMNS. THEOREM. Let be a odd rme, G a fte th ro--grou, ad L the graded Le algebra oer F assocated wth the lower cetral seres of G. Ž. a If G has frst cha legth, ad 5, the L s uquely determed u to somorhsm. Ž b. If G has frst cha legth,, ad the largest class-four factor grou of G has order 6, the L has oe of two somorhsm tyes. The Le algebras L of the theorem were realzed exlctly CMNS as the graded Le algebras assocated wth the lower cetral seres of certa Le algebras T over the rg of -adc tegers. We observe that such L are th for 5 case Ž. a ad for case Ž. b. However, the hyotheses o the above theorem eed strct equaltes, because the stuato becomes much more comlcated whe the legth of the frst cha reaches ts maxmum value 2 Žths s where the Nottgham grou comes., ad ew somorhsm tyes for L may aear besdes those metoed the theorem. The goal of the reset aer s to costruct ro--grous whch have the Le algebras L of the above theorem as the graded Le algebras assocated wth the lower cetral seres. If the rme s bg eough Žamely, as t turs out, f 7 case Ž a. ad case Ž b.., t s ossble to aly the BakerCambellHausdorff seres to the Le algebras T, makg them to the desred grous. However, we choose a dfferet aroach, whch allows 5 or 7 case Ž. a ad case Ž b.. Furthermore, our aroach, descrbed Secto 2, roduces the ro--grous a rather exlct way, as matrx grous over a -adc feld.

3 58 SANDRO MATTAREI We reverse the order followed the above theorem by costructg the grous for the smler case Ž b. frst, Secto, ad treat the more comlcated grous of Ž. a later, Secto PRELIMINARIES Let F be a fte exteso of the -adc umber feld Q, wth absolute ramfcato dex e, ad let deote the uque valuato of F whch exteds the ormalzed -adc valuato of Q. We fx a elemet F wth Ž. e, ad let R deote the valuato rg of F. The matrx algebra M Ž F. s a comlete ormed algebra Žsee Bo,. 74. wth the usual orm 4 Ž x x max j., j,...,, for x x M F, 0 j but t may be coveet to cosder also dfferet orms, as we see. We cosder the followg R-Le subalgebra of M Ž F. Žad, fact, of M Ž R..: ½ 5 j ½ Ž j. 5 x M F Ž xj. for j e x M F x. e We are terested because the Le Q -algebras T defed CMNS ad metoed the Itroducto have exlct realzatos as matrx Le algebras closely related to. Thus, we costruct a ro--grou out of whch shares wth the same graded Le algebra Žassocated wth sutable fltratos of ad.. A drect method of obtag a grou from s edowg, or some sutable oe Le subalgebra of, wth a grou oerato defed by the BakerCambellHausdorff formula Žsee La, Cha. IV, 2., or DduSMS, Sect However, the BakerCambellHausdorff formula ca be aled to the etre oly f Žwhere deotes the th term of the lower cetral seres for a Le algebra or a dscrete grou, ad ts closure for a ro--grou., ad ths codto s easly see to be equvalet wth e. Sce s a matrx algebra, a more exlct, but equvalet, aroach s costructg the desred grou as exž., where ex deotes the ordary exoetal ower seres. Aga, ths requres the rme to satsfy e, as Proosto 2.2 below. We follow the dfferet aroach of costructg the grou the form. Ths gves the same grou as the exoetal ma whe

4 SOME THIN PRO--GROUPS 59 the latter ales, but has the advatage that t also works wth smaller rmes. To beg wth, followg a exercse La, Cha. III, 2.7. we obta a ew orm o M Ž F. from the orm of a matrx rg M Ž E. 0 over a larger feld E, va a er automorhsm. Let E be a fte-dmesoal exteso of F, wth ramfcato dex over F. Let E wth Ž 2 e, ad let D be the dagoal matrx D dag,,...,.. The the formula x D xd, for x Ž x. M Ž E., 0 j obvously defes a ew orm o M Ž E., ad a ew orm o M Ž F. by w Ž x. restrcto. For x M F t ca be easly checked that x, where ½ 5 j wž x. m Ž xj., j,...,. e Ž. Wth resect to the ew orm or, equvaletly, to the valuato w, has the smler defto ½ 5 Ž e. x M F x x M F 4 wž x.. e For ay o-egatve teger, we defe 4 Ž e. x M Ž F. x, Ž e. the closed ball of radus M Ž F.. I artcular, ote that for all 0. The A 0 s a assocatve R-subalgebra of M Ž F. Ž fact, of M Ž R... The subsets for a o-egatve teger are deals of A as a assocatve algebra, ad form a fltrato of A, sce t follows from ther defto that for all, j 0 Ž j j where j deotes the assocatve roduct of deals.. As a cosequece, the for are also deals of as a Le algebra ad, fact, they form a fltrato of as a Le algebra. Hece we may form a graded assocatve algebra grž A. Ž. 0 assocated wth the fltrato of A, ad ts graded deal gr Ž., regarded as a Le algebra, s the graded Le algebra assocated wth the fltrato of. The set s a subgrou of the grou GL Ž R. of all vertble elemets of M Ž R.. Its subgrous, for a ostve teger, form a cetral fltrato of, accordg to Bo, Ch. II, Sect. 4. Sce the are oe subgrous of -ower dex whch form a base for the eghborhoods of the Hausdorff toologcal grou, ad s comlete Žbeg

5 60 SANDRO MATTAREI a closed subset of M Ž F.., s a ro--grou. Deotg by gr the graded Le Z-algebra assocated wth ths fltrato of, we quote the followg result from Bo, Cha. II, Sect. 5, Proosto. PROPOSITION 2.. The bjecto ge by x x from oto duces a somorhsm of graded Le Z-algebras g: gr gr. Note, however, that the fltratos of ad are ot the lower cetral seres of ad. Note also that grž., ad hece grž., are Le algebras over the resdue feld RR of F ad, hece, over the rme feld F. If, addto, we have Ž e. Ž., whch s equvalet to e, the Bo, Cha. II, Sect. 8, Proosto 4 also ales ad gves the followg result. PROPOSITION 2.2. If e, the ower seres ex defes a aalytc bjecto ex:, wth aalytc erse defed by the ower seres log. Furthermore, exž x. x mod, f x, for, ad hece ex duces the same graded Le F-algebra somorhsm g: gr gr of Proosto 2.. The secod statemet of the above roosto follows from the fact that m x x m Ž m.ž. m x x, m! m! Ž. for x ad m. If L Ž res., G. s a Le subalgebra of Ž res., subgrou of., ts Le subalgebras L L Ž res., subgrous G G. form a cetral fltrato of L Ž res., G.. Because of the caocal somorhsms L L Ž L., the graded Le algebra grž L. as- socated wth ths cetral fltrato ca ad s caocally detfed wth a graded Le subalgebra of grž.. Smlarly, the graded Le algebra grž G. s detfed wth a graded Le subalgebra of grž.. We defe the Le subalgebra 4 x trž x. 0, of, ad the closed subgrou of, hece a ro--subgrou, 4 S x detž x.. As a examle of the otato defed above, deotes.

6 SOME THIN PRO--GROUPS 6 Ž. PROPOSITION 2.. We hae ggr gr S. Remark 2.4. If e we have ex S, accordg to Proosto 2.2 ad the well-kow formula detž ex x. exž tr x.. Hece Proosto 2. follows mmedately ths case. Proof. For x we have detž x. detž D xd. trž D xd. trž x. mod 2, the valuato rg S of E. If detž x. the we have trž x. Ž 2. Ž S R S. R, for. Hece, we obta trž x y. 0by takg, for examle, y dagž,0,...,0.. Ths shows that S. I the other drecto, assumg that trž x. 0 yelds that detž. Ž x S. R, for. Therefore, we obta detžž x.ž y.. by takg, for examle, y dagžž.,,...,.. Ths shows that ad cocludes the roof. S, Now assume that s odd. Let a a be a feld automorhsm of F, of order or 2, let F0 be ts fxed subfeld, ad let R0 F0 R. A F0-algebra automorhsm x x of A s duced the obvous way, that t s, x xj f x x j. Let x x deote matrx trasosto. If J t GL F s a o-sgular matrx such that J J s a scalar matrx Ž artcular, f J s symmetrc, skew symmetrc, or Hermta., the the ma t x x J x J s a F-semlear voluto Žor volutory at-automorhsm. of M Ž F.. The t 4 4 x x x 0 x xj Jx 0 s a Le F -subalgebra of, ad 0 4 t 4 O x x x x x J x J s a closed subgrou of. We also ut, ad SO S O. It wll ot be ecessary to assume that. However, some arts of the roof of the followg roosto we eed the hyothess, where s the Le subalgebra of all dagoal matrces of. I artcular, ths forces J to be a moomal matrx Ž other words, a matrx whch every row ad every colum cotas exactly oe o-zero etry.. From

7 62 SANDRO MATTAREI t follows that Ž. for all, because Ž., where Ž. deotes the largest teger whch does ot exceed Ž.. PROPOSITION 2.5. Let be odd. Let J GL Ž F. be a o-sgular matrx t such that J J s a scalar matrx. The we hae ggr Ž. grž O.. If, addto,, the we also hae ggr Ž. grž SO.. Remark 2.6. If e we have ex O, accordg to Proosto 2.2, ad because t s well kow that t t xjjx 0 f ad oly f ex x J ex x J. Hece Proosto 2.5 follows mmedately ths case. Proof. Frst of all, we have to rove that O, for all. I fact, we rove the stroger statemet that O 2, ad smlarly the other arts of the roof. Let x such that x O. The x satsfes Ž x. Ž x., whch becomes x x x x 0, ad also yelds x Ž x. xž x.. We have to fd y 2 such that x y, that s, such that Ž x y. Ž x y. 0. We take y x x x x x x. The y satsfes our requremets, ad our clam s roved. We observe assg that the elemet x y that we have costructed s gve by the sum of a seres startg as x y x x 2 2 x 2. I case e, ths should be comared wth the elemet 2 log x x x 2 x, whch reresets the same elemet as x y grž., accordg to Remark 2.6. Thus the formal 2 ower seres X 2 X Ž X. whch defes x y lays the role of a substtute for the formal ower seres logž X.. Now suose, addto, that detž x.. As the roof of Ž 2 Proosto 2., we have tr x S. R, whece trž x y. Ž 2 S. R. Thus, the dagoal matrx d dagž, 0,..., 0. belogs to. Because of our hyothess, the matrx d 2 s dagoal ad belogs to. It follows that z Ž 2.Žd d By takg traces Ž x y. Ž x y. we see that, whece trž x y z. 0. We coclude that SO 2. Note that the argumet of ths aragrah s suerfluous f the feld automorhsm a a s the detty, because trž x y. tržž 2.Ž x x.. Ž 2.ŽtrŽ x. trž x.. 0 that case.

8 SOME THIN PRO--GROUPS 6 Now we rove that O 2. Let x, whece x x 0. We fd y such that Ž x.ž 2 y. O, that s, such that Ž y. Ž x. Ž x.ž y.. If we take to accout that x, ths becomes Ž y. Ž x 2.Ž y., As the frst art of the roof, the trck s choosg y to be -varat. Hece we take 2 2 y x, as defed by evaluatg for X x 2 the ower seres 2 2 Ž X. Ý 4 ž / X, 0 whch has coeffcets Z, for odd. It s clear that the seres coverges, that y, ad that y y. The fact that Ž y. Ž 2 x 2.Ž y. roves our clam. As we dd the frst art of ths roof, t may be structve to comare the frst few terms of the seres exaso of the elemet Ž x.ž y. 2 O that we have costructed, amely, x y x x 2 2 x 2 wth the elemet ex x x x 2 x 6...O, whch oly exsts f e. I ths case, the role of a substtute for the formal ower seres exž X. s layed by the seres Ž X. Ž X 2. 2 Ž X. 2 Ž X. 2. Aother ossble substtute could have bee the seres X Ž X 2. 2, whch s the formal verse of the seres X 2 2 X Ž X. used the frst art of the roof as a substtute for logž X.. Suose, addto, that trž x. 0. Oce aga, as the roof of Ž 2 Proosto 2., we have det x S. R, whece detžž.. Ž 2 x y S. R, ad the dagoal matrx d dagž. 2,,..., belogs to. Aga because of our hyothess, the matrx d s dagoal ad belogs to 2. I artcular, d ad d commute. It follows that z ŽŽ d.ž d.. 2 O 2. By takg determats Ž y. Ž x. Ž x.ž y. we see that, whece det z. Therefore detžž x.ž y.ž z.., ad we obta that SO 2, whch cocludes the roof. We observe that, aga, the argumet of ths aragrah s suerfluous f the feld automorhsm a a s the detty, because t ca be easly show that detžž x.ž y.. that case. Remark 2.7. The hyothess Proosto 2.5 ca sometmes be weakeed. For examle, t ca be droed Žthus allowg J ot to be a moomal matrx. f does ot dvde. The roof ths case would requre oly mor modfcatos: relacg d dagž, 0,..., 0. wth the

9 64 SANDRO MATTAREI scalar matrx d dagž,...,., ad later relacg d dagž,,...,. wth the scalar matrx d dagžž.,..., Ž.., otg that the ower seres Ž X. whch defes Ž. has coeffcets Z f does ot dvde. Although the subgrous S form a cetral seres of S, t s ot always true that they form ts lower cetral seres, ad smlarly for O ad SO Žsee, for examle, the dscusso Chater V of L-GPK.. For stace, the subgrous S do ot form the lower cetral seres of S whe dvdes. However, f we hae to kow that for all Ž as the examles to be cosdered the followg sectos., the Proosto 2., ad a alcato of the stadard Lemma 2.8 to the ro- grou S, allows us to coclude that Ž S. S for all. Smlar assertos hold for O ad SO, f Proosto 2.5 ales. LEMMA 2.8. Let G be a Hausdorff toologcal grou wth a cetral fltrato G G G2 such that the subgrous G form a fudametal set of oe eghborhoods of. Suose that G G, G G2 for all. The Ž G. G for all. Proof. We have G G, GG G, GG, GG 2 G, GG, ad a easy ducto shows that G G, GG k for all k 2. It follows that G G, GG G, G k k, the toologcal closure of G, G, ad the roof s cocluded by a further ducto o.. FIRST CHAIN LENGTH Let be a odd rme, ad let T be the Le algebra over Z freely geerated by x, y, ad c as a Z -module, ad wth Le multlcato defed by yx c, cx y, cy x, where s a ut Z. It s easy to see that the terms of the lower cetral seres of T are Ž T. T, Ž T. ² c: 2 T, ad are gve geeral by the recursve rule Ž T. Ž T. 2 for 2. I artcular, the lower cetral factors of T are vector saces over F, ad ther dmesos are erodc wth erod Ž 2,.. It was show Secto 4 of CMNS that the graded Le F-algebra L assocated wth the lower cetral seres of T s a th Le algebra. Ideed, L s ether of the two graded Le algebras referred to as CMNS, Theorem 2Ž b. Ž ad quoted the Itroducto., deedg o the choce of. More recsely, the two somorhsm tyes for L are dstgushed by the codto that s a square of a elemet

10 SOME THIN PRO--GROUPS 65 of Q, or ot Žsee CMNS, Secto 4.. I ths secto we costruct a ro--grou G, for ay odd rme, such that the graded Le algebra assocated wth the lower cetral seres of G s somorhc wth L. Our frst task wll be embeddg T to a matrx rg. Deotg by a square root of Q or a quadratc exteso of Q, the Le algebra T T Q Ž. Q Ž. Z, s easly see to be slt smle of tye A.A Chevalley bass e, f, h, wth ef h, he 2 e, hf 2f, s gve, for examle, by ž / 2 e x y, f x y, h c. By vertg the relatos above, ad detfyg T wth ŽQ Ž.. Q Ž. 2 va we obta 0 0 e, f, h, 0 0 Wth ths detfcato, we have x Ž e f., 2 2 y e f, 2 2 c h. 2 2 ½ 5 a b T T Z a, b, c Z Z Z. c a We dstgush two cases, accordg as Q, or ot. We suose frst that Q, whece Z, because s a ut Z. The we are the stuato studed Secto 2, wth F Q ad 2, ad we have ½ 5 a b a, b, c Z TZ T. c a Also, the descrto of the lower cetral seres of T gve earler shows that for all. A alcato of Proosto 2. ad

11 66 SANDRO MATTAREI Lemma 2.8 shows that the lower cetral seres of the ro--grou a b S ½ a, b, c, d Z, a d ad bc c d 5 s gve by Ž S. S for all, ad that the assocated graded Le algebra grž S. s somorhc to L grž.. Note that f, accordg to Remark 2.4, we also have S exž T., but for the exoetal ma does ot coverge o all of T. Thus, for ay odd we have costructed a th ro--grou whose assocated graded Le algebra belogs to oe of the somorhsm tyes of CMNS, Theorem 2Ž b., whch we have quoted the Itroducto. Ths grou S s the bary -adc grou M descrbed 0,, Hu, Ka. III, Sect. 7. We fally ote that the lotet matrces e ad f geerate T as a Le Z -algebra. It follows that a ar of toologcal geerators for the ro-- grou S s gve by e exž e. ad f exž f.. Now we tur to the other somorhsm tye of L, startg aga wth the Le Z -algebra T costructed above, but ths tme assumg that Q. Sce s a ut of Z, the quadratc exteso Q Ž. of Q s uram- fed, ad the -adc orm o Q exteds uquely to a orm o Q Ž., gve by a b a b max a, b, for all a, b Q. Regardg T as a Le subalgebra of the ormed Le Q Ž. Q Ž. -alge bra A M ŽQ Ž.. the way show earler, we see that 2 T ½ a b c 5 a, b, c Z Ž b c. a ½ 5, Z, 0, where overlg deotes the o-detty Galos automorhsm of Q Ž., whch seds to. Wth the otato defed Secto 2, before Proosto 2.5, wth Ž. F Q, ts automorhsm as above, 2, ad J,we

12 SOME THIN PRO--GROUPS 67 ca easly see that T. As we dd the revous case for, wthout dog ay calculato we may coclude that ts lower cetral seres s gve by for all. It s straghtfor- ward to check that ½ 5 SO, Z,. Proosto 2.5 ad Lemma 2.8 the show that the lower cetral seres of the ro--grou SO s gve by Ž SO. SO for all, ad that the assocated graded Le algebra grž SO. s somorhc wth L grž..if, we also have SO exž T., accordg to Remark 2.6. Thus, for ay odd rme we have costructed a ro--grou wth assocated graded Le algebra belogg to the other somorhsm tye of CMNS, Theorem 2Ž b.. We fd a ar of geerators of SO as a ro--grou, but we are ot as lucky as the revous case, because lotet matrces geeratg T do ot exst the reset case. I fact, T does ot cota ay o-zero lotet matrx. For, a ar of Ž toologcal. geerators for SO ca be obtaed by alyg the ower seres ex to a ar of geerators of T as a Le Z -algebra. Hece, for, the ro--grou SO s geerated by ex ad ex. A ar of geerators for SO vald for all odd ca be obtaed a smlar way by relacg ex wth the seres Ž X. Ž X 2. 2, whch we have already used the roof of Proosto 2.5, ad are 2 2 Ž. ad Ž., where of course Ž. 2, beg defed by evaluato of the ower seres Ž X. 2, deotes the square root of Ž. whch s mod. We observe that SO s the fxed ot subgrou of the automorhsm of the 4-geerated ro--grou ½ 5 H,,, Z,

13 68 SANDRO MATTAREI Ž t. defed by y y J y J. The grou H also has a automorhsm of order 2 gve by y y, whose fxed ot subgrou s the grou S of the case Q, whch we examed earler. We coclude ths secto by observg that ŽQ Ž.,. ca be somet- rcally embedded as a ormed Q -subalgebra of ŽM Ž Q.,. 2 0 va the ma a b a b, b a where 0 s the stadard orm o M Ž Q. 2 as defed Secto 2. It follows that our grou H ca also be realzed as a closed subgrou of GL Ž Z FIRST CHAIN LENGTH I ths secto we costruct, for ay 5, a th ro--grou G whose assocated graded Le algebra L has the somorhsm tye referred to CMNS, Theorem 2Ž. a, whch we have quoted the Itroducto. Ths graded Le algebra L ca also be obtaed as the graded Le algebra assocated wth the lower cetral seres of a Le Z -algebra T defed as follows. Let 5, ad let T be a free Z -module of rak 8, wth bass x, y, c, d,, s, t, w. Accordg to CMNS, the alteratg multlcato defed o T by yx c, cx d, cy 0, dx, dy 0, x s, y t, sx 0, sy 2w, tx w, ty 0, wx x, wy 2y satsfes the Jacob detty, ad therefore makes T to a Le Z -algebra. The terms of the lower cetral seres of T are Ž T. T, Ž T. 2 ² c, d,, s, t, w: T, Ž T. ² d,, s, t, w: T, Ž T. ², s, t, w: 4 T, Ž T. ² s, t, w: T, Ž T. ² w: 5 6 T, ad are gve geeral by the recursve rule Ž T. Ž T. for 6. I artcular, the lower 6

14 SOME THIN PRO--GROUPS 69 cetral factors of T are vector saces over F, ad ther dmesos are erodc wth erod Ž 2,,,, 2,.. It was show Secto of CMNS that the graded Le F-algebra L assocated wth the lower cetral seres of T was a th Le algebra, ad was the graded Le algebra of CMNS, Theorem 2Ž. a. The feld Q, where, s a totally ramfed exteso of Q, ad T T Q Q Z s a slt smle Le algebra of tye A 2. I fact, ² d, w: s a Carta subalgebra of T Q, ad f, form a base for the corresodg root system, a Chevalley bass s e y, e s, 2 e x, e c t, e c t, e x, h w, h Ž w d.. 2 Here the choce of sgs volved the costructo of a Chevalley bass has bee the same as for the Chevalley bass of ŽQ. gve by e E 2, e E 2, e E 2, e E 2, e E, e E, h E E 22, h E22 E, where Ej deotes the matrx whch as as ts Ž, j. th etry, ad 0 elsewhere. Therefore, we detfy TQ wth ŽQ. va the formulas above. By vertg the formulas whch gve a Chevalley bass of TQ we obta x e e, y e, c e e, d Ž h 2h., ž e e /, s 2 e, t ž e e /, w h.

15 70 SANDRO MATTAREI Thus, uder our detfcato, T becomes a Le Z -subalgebra of ŽQ. ad s geerated, as a Z -submodule, by the matrces x 0 0 0, y 0 0 0, 0 0 c, d, , s 2 0, t 0, w. 0 0 We may ote that x ad t c, where overlg deotes the o-detty Galos automorhsm of Q Ž., whch seds to. Clearly T s cotaed, f we adot the otato of Secto 2, wth F Q ad. Moreover, a geerc elemet of T has the form d w y c t s x, for some coeffcets,,,,,,, Z Therefore, ts ma trx form s

16 SOME THIN PRO--GROUPS 7 We deduce that T cossts of all matrces Ž., wth,,,, Z,, ad 0. If we take the matrx J to be 0 J 0, a straghtforward calculato shows that T. From our kowledge of the lower cetral seres of T we easly coclude that the lower cetral seres of s gve by for all. A exlct matrx descrto of the ro--grou SO s oly a matter of comutato, but s ot esecally leasat, ad we do ot gve t here. We oly ote that SO s toologcally geerated by the matrces exž x. 6 ad exž y.. Accordg to Proosto 2.5 ad Lemma 2.8, the lower cetral seres of SO s gve by Ž SO. SO for all, ad that the assocated graded Le algebra grž SO. s somorhc wth L grž..if e 7, we also have SO exž T., accordg to Remark 2.6. Hece, for ay rme 5 we have costructed a ro--grou wth assocated graded Le algebra as CMNS, Theorem 2Ž. a. ACKNOWLEDGMENT I am grateful to the referee for hs costructve crtcsm whch resulted a mrovemet of the aer.

17 72 SANDRO MATTAREI REFERENCES Bo N. Bourbak, Le Grous ad Le Algebras Chas., Srger-Verlag, New York, Berl, ad Hedelberg, 989. Br R. Bradl, The Dlworth umber of subgrou lattces, Arch. Math. Ž Basel. 50 Ž 988., CMNS A. Carat, S. Mattare, M. F. Newma, ad C. M. Scoola, Th grous of rme-ower order ad th Le algebras, Quart. J. Math. Oxford 47 Ž 996., DduSMS J. D. Dxo, M. P. F. du Sautoy, A. Ma, ad D. Segal, Aalytc ro- Grous, Cambrdge Uv. Press, Cambrdge, U.K., 99. Hu B. Huert, Edlche Grue I, Srger-Verlag, Berl, Hedelberg, ad New York, 967. La M. Lazard, Groues aalytques -adques, Ist. Hautes Etudes Sc. Publ. Math. 26 Ž 965., L-GPK C. R. Leedham-Gree, W. Pleske, ad G. Klaas, Pro--grous of fte wdth, to aear.