THREE THEOREMS ON LINEAR GROUPS

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1 THREE THEOREMS ON LINEAR GROUPS BOGDAN NICA INTRODUCTION A group s lnear f t s somorphc to) a subgroup of GL n K), where K s a feld. If we want to specfy the feld, we say that the group s lnear over K. The followng theorems are fundamental, at least from the perspectve of combnatoral group theory. Theorem Mal cev 1940). A fntely generated lnear group s resdually fnte. Theorem Selberg 1960). A fntely generated lnear group over a feld of zero characterstc s vrtually torson-free. A group s resdually fnte f ts elements are dstngushed by the fnte quotents of the group,.e., f each non-trval element of the group remans non-trval n a fnte quotent. A group s vrtually torson-free f some fnte-ndex subgroup s torson-free. As a matter of further termnology, Selberg s theorem s usually referred to as Selberg s lemma, and Mal cev s alternatvely translterated as Maltsev. Resdual fnteness and vrtual torson-freeness are related to a thrd property - roughly speakng, a p-adc refnement of resdual fnteness. A theorem due to Platonov 1968) gves such refned resdual propertes for fntely generated lnear groups. Both Mal cev s theorem and Selberg s lemma are consequences of ths more powerful, but lesser known, theorem of Platonov. Once we have Platonov s theorem and ts proof, we are not too far from our thrd theorem of nterest. In order to formulate t, let us frst observe that every non-trval torson element n a group G gves rse to a non-trval dempotent n the complex group algebra CG. Namely, f g G has order n > 1, then e = 1 n 1 + g gn 1 ) CG satsfes e 2 = e, and e = 0, 1. The Idempotent Conjecture s the bold statement that the converse holds: f G s a torson-free group, then the group algebra CG has no non-trval dempotents. Whle not yet settled n general, ths conjecture s known for many classes of groups. A partcularly mportant partal result s the followng. Theorem Bass 1976). Torson-free lnear groups satsfy the Idempotent Conjecture. 1. VIRTUAL AND RESIDUAL PROPERTIES OF GROUPS Vrtual torson-freeness and resdual fnteness are nstances of the followng termnology. Let P be a group-theoretc property. A group s vrtually P f t has a fnte-ndex subgroup enjoyng P. A group s resdually P f each non-trval element of the group remans non-trval n some quotent group enjoyng P. The vrtually P groups and the resdually P groups contan the P groups. It may certanly happen that a property s vrtually stable e.g., fnteness) or resdually stable e.g., torson-freeness). Besdes vrtual torson-freeness and resdual fnteness, we are nterested n the hybrd noton of vrtual resdual p-fnteness where p s a prme. Ths s obtaned by resdualzng the property of beng a fnte p-group, followed by the vrtual extenson. The noton of vrtual resdual p- fnteness has, n fact, a leadng role n ths account for t relates both to resdual fnteness and to vrtual torson-freeness. Date: December

2 2 BOGDAN NICA Observe the followng. Gong down) If P s nherted by subgroups, then both vrtually P and resdually P are nherted by subgroups. In partcular, vrtual torson-freeness, resdual fnteness, and vrtual resdual p- fnteness are nherted by subgroups. Gong up) Vrtually P passes to fnte-ndex supergroups. In partcular, both vrtual torsonfreeness and vrtual resdual p-fnteness pass to fnte-ndex supergroups. Resdual fnteness passes to fnte-ndex supergroups. Resdual p-fnteness trvally mples resdual fnteness. Gong up, we obtan: Lemma 1.1. Vrtual resdual p-fnteness for some prme p mples resdual fnteness. On the other hand, resdual p-fnteness mposes torson restrctons. Namely, n a resdually p-fnte group, the order of a torson element must be a p-th power. Hence, f a group s resdually p-fnte and resdually q-fnte for two dfferent prmes p and q, then t s torson-free. Vrtualzng ths statement, we obtan: Lemma 1.2. Vrtual resdual p-fnteness and vrtual resdual q-fnteness for two prmes p = q mply vrtual torson-freeness. 2. PLATONOV S THEOREM In lght of Lemmas 1.1 and 1.2, we see that Mal cev s theorem and Selberg s lemma are consequences of the followng: Theorem Platonov 1968). Let G be a fntely generated lnear group over a feld K. If char K = 0, then G s vrtually resdually p-fnte for all but fntely many prmes p. If char K = p, then G s vrtually resdually p-fnte. Actually, the zero characterstc part of Platonov s theorem had been proved slghtly earler by Kargapolov 1967) and, ndependently, Merzlyakov 1967). Example 2.1. SL n Z), where n 2, s a fntely generated lnear group over Q. Reducton modulo a postve nteger N defnes a group homomorphsm SL n Z) SL n Z/N), whose kernel ΓN) := ker SL n Z) SL n Z/N) ) = { X SL n Z) : X 1 n mod N } s the prncpal congruence subgroup of level N. The prncpal congruence subgroups are fntendex, normal subgroups of SL n Z). They are organzed accordng to the dvsblty of ther levels: ΓM) ΓN) f and only f M N, that s, to contan s to dvde. Hence the prme stratum {Γp) : p prme}, and each descendng chan {Γp k ) : k 1} correspondng to fxed prme p, stand out. Elements of SL n Z) can be dstngushed both along the prme stratum, p Γp) = {1 n }, as well as along each descendng p-chan, k Γp k ) = {1 n }. We thus have two ways of seeng that SL n Z) s resdually fnte. There s no prme p for whch SL n Z) s resdually p-fnte, smply because ) has order 6. However, SL n Z) s vrtually resdually p-fnte for each prme p. The reason s that Γp) s resdually p-fnte, and ths s easly seen by notng that each successve quotent Γp k )/Γp k+1 ) n the descendng chan {Γp k ) : k 1} s a p-group: for X Γp k ) we have p ) X p p = 1 n + X 1 n ) Γp k+1 ). =1 Example 2.2. SL n F p [t]), where n 2, s lnear over F p t) and fntely generated for n 3 though not for n = 2). A smlar argument to the one of the prevous example, ths tme nvolvng the prncpal congruence subgroups correspondng to the descendng chan of deals t k ) for k 1, shows that SL n F p [t]) s vrtually resdually p-fnte. On the other hand, SL n F p [t]) contans a

3 THREE THEOREMS ON LINEAR GROUPS 3 copy of the nfnte torson group F p [t], +), and ths prevents SL n F p [t]) from beng vrtually torson-free. Consequently, SL n F p [t]) cannot be vrtually resdually q-fnte for any prme q = p. Platonov s theorem mples the followng p-adc refnement of Mal cev s theorem. Corollary 2.3. A fntely generated lnear group s vrtually resdually p-fnte for some prme p. Ths corollary, combned wth Example 2.2, leads us to a smple example of a fntely generated group whch s non-lnear but resdually fnte: SL n F p [t]) SL n F q [t]), where p and q are dfferent prmes, and n PROOF OF PLATONOV S THEOREM Let G be a fntely generated lnear group over a feld K, say G GL n K). In K, consder the subrng A generated by the multplcatve dentty 1 and the matrx entres of a fnte, symmetrc set of generators for G. Thus A s a fntely generated doman, and G s a subgroup of GL n A). Platonov s theorem s then a consequence of the followng: Theorem 3.1. Let A be a fntely generated doman. If char A = 0, then GL n A) s vrtually resdually p-fnte for all but fntely many prmes p. If char A = p, then GL n A) s vrtually resdually p-fnte. Here, and for the remander of the secton, rngs are commutatve and untal. The proof of Theorem 3.1 s a straghtforward varaton on the example of SL n Z), as soon as we know the followng facts: Lemma 3.2. Let A be a fntely generated doman. Then the followng hold:. A s noetheran.. k I k = 0 for any deal I = A.. If A s a feld, then A s fnte. v. The ntersecton of all maxmal deals of A s 0. v. If char A = 0, then only fntely many prmes p = p 1 are nvertble n A. Let us postpone the proof of Lemma 3.2 for the moment, and focus nstead on dervng Theorem 3.1. The prncpal congruence subgroup of GL n A) correspondng to an deal I of A s defned by ΓI) = ker GL n A) GL n A/I) ). If π s a maxmal deal then A/π s a fnte feld, by part ) of Lemma 3.2, so Γπ) has fnte ndex n GL n A). Also π Γπ) = {1 n } as π runs over the maxmal deals of A, by part v) of Lemma 3.2. Ths shows that GL n A) s resdually fnte, thereby provng Mal cev s theorem. For each k 1, the quotent π k /π k+1 s naturally an A/π-module. It nherts fnte generaton from the fnte generaton of the A-module π k, the latter due to A beng noetheran. As A/π s fnte, π k /π k+1 s fnte as well. It follows that the rng A/π k s fnte, and so Γπ k ) has fnte ndex n GL n A). Furthermore, k Γπ k ) = {1 n } by part ) of Lemma 3.2, whch shows once agan that GL n A) s resdually fnte. Now let p denote the characterstc of A/π, so p = p 1 π. Then Γπ k )/Γπ k+1 ) s a p-group: for X Γπ k ) we have p X p = 1 n + =1 p ) X 1 n ) Γπ k+1 ). To conclude, GL n A) s vrtually resdually p-fnte for each prme p not nvertble n A. By part v) of Lemma 3.2, ths happens for all but fntely many prmes p n the zero characterstc case. In characterstc p, there s only such prme, namely p tself. Theorem 3.1 s proved. We now return to the proof of the lemma.

4 4 BOGDAN NICA Proof of Lemma 3.2. The frst two ponts are standard: ) follows from the Hlbert Bass Theorem, and ) s the Krull Intersecton Theorem for domans. ) We clam the followng: f F Fu) s a feld extenson wth Fu) fntely generated as a rng, then F Fu) s a fnte extenson and F s fntely generated as a rng. We use the clam as follows. Let F be the prme feld of A and let a 1,..., a k be generators of A as a rng. Thus A = Fa 1,..., a k ). Gong down the chan A = Fa 1,..., a k ) Fa 1,..., a k 1 )... F we obtan that F A s a fnte extenson, and that F s fntely generated as a rng. Then F s a fnte feld, as Q s not fntely generated as a rng, and so A s fnte. Now let us prove the clam. Assume that u s transcendental over F,.e., Fu) s the feld of ratonal functons n u. Let P 1 /Q 1,..., P k /Q k generate Fu) as a rng, where P, Q F[u]. The multplcatve nverse of 1 + u Q s a polynomal expresson n the P /Q s, whch can be wrtten as R/ Q s. Therefore Qs = 1 + u Q )R n F[u]. But ths s mpossble, snce Q s s relatvely prme to 1 + u Q. Thus u s algebrac over F. Let X d + α 1 X d α d be the mnmal polynomal of u over F. Let also a 1,..., a k be rng generators of Fu) = F[u]. We may wrte each a as 0 m d 1 β,m u m, wth β,m F. We clam that the α j s and the β,m s are rng generators of F. Let c F. Then c s a polynomal n a 1,..., a k over F, hence a polynomal n u over the subrng of F generated by the β,m s, hence a polynomal n u of degree less than d over the subrng of F generated by the α j s and the β,m s. By the lnear ndependence of {1, u,..., u d 1 }, the latter polynomal s actually of degree 0. Hence c ends up n the subrng of F generated by the α j s and the β,m s. v) Let a = 0 n A. To fnd a maxmal deal of A not contanng a, we rely on the basc avodance: maxmal deals do not contan nvertble elements. Consder the localzaton A = A[1/a]. Let π be a maxmal deal n A, so a / π. The restrcton π = π A s an deal n A, and a / π. We show that π s maxmal. The embeddng A A nduces an embeddng A/π A /π. As A /π s a feld whch s fntely generated as a rng, t follows from ) that A /π s fnte feld. Therefore the subrng A/π s a fnte doman, hence a feld as well. v) We shall use Noether s Normalzaton Theorem: f R s a fntely generated algebra over a feld F R, then there are elements x 1,..., x k R algebracally ndependent over F such that R s ntegral over F[x 1,..., x k ]. In our case, Z s a subrng of A, and A s an ntegral doman whch s fntely generated as a Z-algebra. Extendng to ratonal scalars, we have that A Q = Q Z A s a fntely generated Q-algebra. By the Normalzaton Theorem, there exst elements x 1,..., x k n A Q whch are algebracally ndependent over Q, and such that A Q s ntegral over Q[x 1,..., x k ]. Up to replacng each x by an ntegral multple of tself, we may assume that x 1,..., x k are n A. There s some postve m Z such that each rng generator of A s ntegral over Z[1/m][x 1,..., x k ]. Thus A[1/m] s ntegral over the subrng Z[1/m][x 1,..., x k ]. If a prme p s nvertble n A, then t s also nvertble n A[1/m] whle at the same tme p Z[1/m][x 1,..., x k ]. Now we use the followng general fact. Let R be a rng whch s ntegral over a subrng S. If s S s nvertble n R, then s s already nvertble n S. The proof s easy. Let r R wth rs = 1. We have r d + s 1 r d s d 1 r + s d = 0 for some s S, snce r s ntegral over S. Multplyng through by s d 1 yelds r S. Returnng to our proof, we nfer that p s nvertble n Z[1/m][x 1,..., x k ]. By the algebrac ndependence of x 1,..., x k, t follows that p s actually nvertble n Z[1/m]. But only fntely many prmes have ths property, namely the prme factors of m. 4. THE IDEMPOTENT CONJECTURE FOR LINEAR GROUPS Our approach to Bass s theorem reles on the followng crteron of Formanek [2], whose proof s postponed for the next secton.

5 THREE THEOREMS ON LINEAR GROUPS 5 Theorem 4.1 Formanek 1973). Let G be a torson-free group wth the property that, for nfntely many prmes p, G has no p-self-smlar elements. Then the Idempotent Conjecture holds for G. Gven a group G, we say that a non-trval element g G s self-smlar f g s conjugate n G to a proper power g N, where N 2. Clearly, torson elements are self-smlar. It turns out that the converse holds for lnear groups n postve characterstc. Lemma 4.2. In a lnear group over a feld of postve characterstc, every self-smlar element s torson. Proof. Let char K = p, and consder the relaton g N = x 1 gx n GL n K), where N 2. Wthout loss of generalty, K s algebracally closed and g s n Jordan normal form. Each Jordan block s of the form λ 1 k + k, where k s the k k-matrx wth 1 s on the super-dagonal and 0 s everywhere else. Snce λ 1 k + k ) ps = λ ps 1 k + ps k, and ps k = 0 for large enough s, t follows that g ps s dagonal for large enough s. Thus, up to replacng g by g ps, we may assume that g s dagonal. So let g have λ 1,..., λ n K along the dagonal, and wrte out the relaton gx = xg N n matrx form: x j λ ) = x j λ N j ). Compare the -th row on the two sdes. At least one of x 1, x 2,..., x n s non-zero, hence λ = λ σ) N for some σ) {1,..., n}. Snce σs = σ s+t for some postve ntegers s and t, t follows that λ = λσ Ns+t s+t ) = ) λσ Ns N t s ) = λ Nt for each. We conclude that g Nt 1 = 1 n GL n K). In characterstc zero, a lnear group may contan self-smlar elements of nfnte order. A smple example n, say, GL 2 R) s provded by ) , whch s conjugated nto ts N-th power by 1 0 ) 0 N. Furthermore, t can be checked that the entre subgroup generated by these two matrces s torson-free. The analogue of Lemma 4.2 n characterstc zero nvolves the followng refned noton of selfsmlarty. Gven a group G and a prme p, let us say that a non-trval element g G s p-selfsmlar f g s conjugate n G to a proper p-th power g pk, where k 1. Lemma 4.3. In a fntely generated lnear group over a feld of characterstc zero, the followng holds for all but fntely many prmes p: every p-self-smlar element s torson. Proof. The characterstc zero case of Platonov s theorem reduces the clam to showng that, n a vrtually resdually p-fnte group, every p-self-smlar element s torson. Ths easly follows from the observaton that a resdually p-fnte group has no p-self-smlar elements. The upshot of Lemmas 4.2 and 4.3 s that a fntely generated, torson-free lnear group comfortably meets the requrement of Formanek s crteron, and so t satsfes the Idempotent Conjecture. The theorem of Bass follows. 5. PROOF OF FORMANEK S CRITERION The proof of Theorem 4.1 uses tracal methods. Let us frst recall that a trace on a K-algebra A s a K-lnear map τ : A K wth the property that τab) = τba) for all a, b A. In short, traces are lnear functonals whch vansh on commutators. The ersatz commutatvty afforded by a trace s extremely valuable n a noncommutatve world. On a group algebra KG, the standard trace tr : KG K s the lnear functonal whch records the coeffcent of the dentty element: tr a g g ) = a 1

6 6 BOGDAN NICA In general, traces on KG are n bjectve correspondence wth maps G K whch are constant on conjugacy classes. The characterstc map 1 C : G K of a conjugacy class C G defnes the trace τ C a g g ) = a g g C so tr = τ {1} wth ths notaton. The traces τ C, where C runs over the conjugacy classes of G, provde a natural bass for the K-lnear space formed by the traces of KG. Another dstngushed trace s the augmentaton map ɛ : KG K gven by ɛ a g g ) = a g. Ths s the trace on KG defned by the constant map 1 : G K. The augmentaton map s n fact a untal K-algebra homomorphsm, hence ɛ s a trace whch s {0, 1}-valued on dempotents. Understandng the range of the standard trace on dempotents s much more dffcult. The followng theorem addresses ths problem n the case of complex group algebras. Theorem 5.1 Kaplansky 1969). Let e be an dempotent n CG. Then tre) [0, 1]. Furthermore, tre) = 0 f and only f e = 0, and tre) = 1 f and only f e = 1. Now let us return to the proof of Formanek s crteron. It conssts of two steps. Postve characterstc clam) Fx a prme p. If G has no p-self-smlar elements and K s a feld of characterstc p, then the standard trace s {0, 1}-valued on the dempotents of KG. It s a famlar fact that the dentty a + b) p = a p + b p holds n any commutatve K-algebra. Its noncommutatve generalzaton, somewhat lesser known, says that, n a K-algebra, a + b) p a p b p s a sum of commutators. Indeed, we may assume that we are n the free K-algebra on a and b. We expand a + b) p nto monomals of degree p n a and b, and we let the cyclc group of order p act on these monomals by cyclc permutatons. We see orbts of sze p, except for a p and b p, whch are fxed by the acton. Now we observe that the sum of monomals correspondng to each orbt of sze p s a sum of commutators. Ths follows from the dentty x 1 x 2... x p 1 x p + x 2 x 3... x p x x p x 1... x p 2 x p 1 = p x 1 x 2... x p 1 x p [x 1, x 2... x p ] [x 1 x 2, x 3... x p ] [x 1... x p 1, x p ]. Next, let us terate: we show by nducton that a + b) pk a pk b pk s a sum of commutators for every postve nteger k. For the nducton step we wrte p a + b) pk+1 = a pk + b pk + [u, v ]) = a p k+1 + b pk+1 + [u, v ] p + [u j, v j ] and [u, v] p = uv) p vu) p + [y l, z l ] = [ uv) p 1 u, v ] + [y l, z l ]. In partcular, a trace τ on a K-algebra has the property that τ a + b) pk ) = τ a p k ) + τ b p k ) for every postve nteger k. For a basc trace τ C, where C = {1}, and an dempotent e KG, we obtan τ C e) = τ C e pk ) = τ C e g g ) p k ) = τ C eg g) pk ) = e pk g 1 C g p k ) for each postve nteger k. The hypothess that G has no p-self-smlar elements mples that, for each g n the support of e, there s at most one k so that g pk C. Thus, takng k large enough, we see that τ C e) = 0. Usng the relaton ɛ = tr + C ={1} τ C, we conclude that tr s {0, 1}-valued on the dempotents of KG. Zero characterstc clam) Assume that, for nfntely many prmes p, the followng holds: the standard trace s {0, 1}-valued on the dempotents of KG, whenever K s a feld of characterstc p. Then the standard trace s {0, 1}-valued on the dempotents of CG. Argung by contradcton, we assume that e s an dempotent n CG wth e 1 = tre) / {0, 1}. Let A C be the subrng generated by the support of e together wth 1/e 1 and 1/1 e 1 ), and

7 THREE THEOREMS ON LINEAR GROUPS 7 vew e as an dempotent n the group rng AG. By part v) of Lemma 3.2, for all but fntely many prmes p there s a quotent map A K, a a, onto a feld of characterstc p. Note that e 1 = 0, 1 n K, snce e 1 and 1 e 1 are nvertble n A. The nduced rng homomorphsm AG KG sends e to an dempotent e n KG wth tre) = 0, 1, thereby contradctng our hypothess. The proof of Theorem 4.1 s concluded by nvokng Kaplansky s theorem. NOTES Platonov s theorem. Besdes the Russan orgnal [4], the only other source n the lterature for Platonov s theorem appears to be the presentaton by Wehrfrtz n Infnte lnear groups. An account of the group-theoretc propertes of nfnte groups of matrces Sprnger 1973). The proof presented heren seems consderably smpler. It s manly nfluenced by the dscusson of Mal cev s theorem n lecture notes by Stallngs Commutatve rngs and groups, UC Berkeley 2000), and t has a certan degree of smlarty wth Platonov s own arguments n [4]. Selberg s lemma. It s mportant to note that Selberg s lemma s just a mnor step n Selberg s paper [5], whose true mportance s that t started the rch stream of rgdty results for lattces n hgher rank. An alternatve road to Selberg s lemma s to use valuatons. Ths s the approach taken by Cassels n Local felds Cambrdge Unversty Press 1986), and by Ratclffe n Foundatons of hyperbolc manfolds 2nd edton, Sprnger 2006). The Idempotent Conjecture. The Idempotent Conjecture s usually attrbuted to Kaplansky, but a reference seems elusve. What Kaplansky dd state on more than one occason Problem 1, p.122 n Felds and rngs, The Unversty of Chcago Press 1969; Problem 6, p.448 n Amer. Math. Monthly 1970) s a problem nowadays referred to as the Zero-Dvsor Conjecture: f G s a torson-free group and K s a feld, then the group algebra KG has no zero-dvsors,.e., ab = 0 whenever a, b = 0 n KG. The Zero-Dvsor Conjecture over the complex feld, whch clearly mples the Idempotent Conjecture, s stll not settled for the class of torson-free) lnear groups. Kaplansky s theorem. We refer to Burger and Valette J. Le Theory 1998) for a proof, as well as for a nce complementary readng. The man nsght of Kaplansky s analytc proof s to pass from the group algebra CG to a completon afforded by the regular representaton on l 2 G. One can use the weak completon, that s the von Neumann algebra LG, or the norm completon, the so-called reduced C -algebra C r G. Kaplansky s proof, whle remarkable n tself, s perhaps more mportant for suggestng what came to be known as the Kadson Conjecture: for every torson-free group G, the reduced C -algebra C r G has no non-trval dempotents. At the tme of wrtng, the Kadson Conjecture for the class of torson-free) lnear groups s stll open. Bass s theorem. As we have seen, the step from Formanek s crteron to the theorem of Bass s rather short, and t uses results on lnear groups whch were known - certanly on the eastern sde of the Iron Curtan, but probably also on ts western sde - at the tme of [2]. Ascrbng the theorem to Bass and Formanek s therefore not entrely unwarranted. The hard facts, however, are that Bass actually proves much more n [1] whereas Formanek states less n [2]. REFERENCES [1] H. Bass: Euler characterstcs and characters of dscrete groups, Invent. Math ), [2] E. Formanek: Idempotents n Noetheran group rngs, Canad. J. Math ), [3] A.I. Mal cev: On somorphc matrx representatons of nfnte groups of matrces Russan), Mat. Sb ), & Amer. Math. Soc. Transl. 2) ), 1 18 [4] V.P. Platonov: A certan problem for fntely generated groups Russan), Dokl. Akad. Nauk BSSR ), [5] A. Selberg: On dscontnuous groups n hgher-dmensonal symmetrc spaces, n Contrbutons to Functon Theory, Tata Insttute of Fundamental Research, Bombay 1960), MATHEMATISCHES INSTITUT, GEORG-AUGUST UNIVERSITÄT GÖTTINGEN E-mal address: bogdan.nca@gmal.com