aa r X i v : . [ m a t h . G R ] A ug Expansion in perfect groups
Alireza Salehi Golsefidy ∗ and P´eter P. Varj´u † November 12, 2018
Abstract
Let Γ be a subgroup of GL d ( Z [1 /q ]) generated by a finite symmet-ric set S. For an integer q , denote by π q the projection map Z [1 /q ] → Z [1 /q ] /q Z [1 /q ]. We prove that the Cayley graphs of π q (Γ) with respectto the generating sets π q ( S ) form a family of expanders when q ranges oversquare-free integers with large prime divisors if and only if the connectedcomponent of the Zariski-closure of Γ is perfect, i.e. it has no nontrivialAbelian quotients. Let G be a graph, and for a set of vertices X ⊂ V ( G ), denote by ∂X the set ofedges that connect a vertex in X to one in V ( G ) \ X . Define c ( G ) = min X ⊂ V ( G ) , | X |≤| V ( G ) | / | ∂X || X | , where | X | denotes the cardinality of the set X . A family of graphs is calleda family of expanders, if c ( G ) is bounded away from zero for graphs G thatbelong to the family. Expanders have a wide range of applications in computerscience (see e.g. Hoory, Linial and Widgerson [37] for a survey on expanders)and recently they found remarkable applications in pure mathematics as well.Let S be a symmetric (i.e. closed for taking inverses) subset of GL d ( Q )and let Γ be the group generated by S . For any positive integer q , let π q : Z → Z /q Z be the residue map. If the prime factors of q are large, π q inducesa homomorphism from Γ to GL d ( Z /q Z ). We denote this and all the similarmaps by π q also. In this article, we give a necessary and sufficient conditionunder which the family of Cayley graphs G ( π q (Γ) , π q ( S )) form expanders as q runs through square-free integers with large prime factors. Let us recall that if S ⊂ G is a symmetric set of generators, then the Cayley graph G ( G, S ) of G with respect to the generating set S is defined to be the graph whose vertex setis G , and two vertices x, y ∈ G are connected exactly if y ∈ Sx . ∗ A. S-G. was partially supported by the NSF grant DMS-1001598. † P.P. V. was partially supported by the NSF grant DMS-0835373. heorem 1. Let Γ ⊆ GL d ( Z [1 /q ]) be the group generated by a symmetricset S . Then G ( π q (Γ) , π q ( S )) form a family of expanders when q ranges oversquare-free integers coprime to q if and only if the connected component of theZariski-closure of Γ is perfect. A result of the type of Theorem 1 was proved by Bourgain, Gamburd andSarnak [11]. They proved that such Cayley graphs form expanders if Γ ⊆ SL ( Z )is Zariski-dense. Their main motivation was to formulate and prove an affinesieve theorem. Moreover, they proved the affine sieve theorem for groups moregeneral than SL provided a conjecture of Lubotzky holds (see [11, Conjecture1.5] and [47]) which is a special case of our Theorem 1. Following the ideas in [11]and using Theorem 1, in a forthcoming paper, Salehi Golsefidy and Sarnak [54]get a similar affine sieve theorem whenever the character group of the connectedcomponent of the Zariski-closure of Γ is trivial.Following [11], now there is a rich literature of sieving applications of ex-pander graphs not limited to number theory. We refer to the recent surveys ofKowalski [42] and Lubotzky [48] for more details on these developments. Besidessieving, results similar to the above theorem are useful in studying covers of hy-perbolic 3-manifolds, see the paper of Long, Lubotzky and Reid [46] followingthe work of Lackenby [43].The question about the expanding properties of mod q quotients was firststudied only for “thick” groups, namely lattices in semisimple Lie groups. Thefirst results used the representation theory of the underlying Lie group; property(T) in Kazhdan [38] and Margulis [49] and automorphic forms in e.g. [56],[18], [40] and [20]. Later Sarnak and Xue [55] developed a more elementarymethod. Kelmer and Silberman [39] combined this method with recent advanceson automorphic forms to obtain a very general result on arbitrary arithmeticlattices. The advantage of these results over our method is that they give explicitand very good bounds.Lubotzky was the first person who asked this question for a “thin” group inhis famous 1-2-3 conjecture (see [47]). Shalom [57], [58] obtained the first re-sults which establish the expander property for quotients of certain non-lattices.A few years later Gamburd [27] showed that quotients of Γ < SL ( Z ) are ex-panders if the Hausdorff dimension of the limit set of Γ is larger than 5/6. Thefirst paper which achieved a result which depend merely on the Zariski-closureof Γ was obtained by Bourgain and Gamburd [7]. Their assumptions were thatthe Zariski-closure of Γ is SL , and the modulus q is prime. In the past fouryears, several articles appeared which extended that result, see [7]–[11], and [61].However, there are still interesting questions to explore. For instance, Question 2.
Does the family of Cayley graphs G ( π q (Γ) , π q ( S )) form expandersas q runs through any positive integer with large prime factors if the connected He asked this question for S = (cid:26)(cid:20) ±
30 1 (cid:21) , (cid:20) ± (cid:21)(cid:27) . SL d . Question 3.
If Γ ⊆ GL d ( F p ( t )) is generated by a symmetric set S , then what isthe necessary and sufficient condition such that G ( π p (Γ) , π p ( S )) form expandersas p runs through square-free polynomials with large degree prime factors?Moreover, one might hope that the answer is positive even to the followingvery general question that was communicated to us by Alex Lubotzky: Question 4 (Lubotzky) . Let Γ ⊆ GL n ( A ) is a finitely generated subgroup,where A is an integral domain which is generated by the traces of the elementsof Γ. Is it true that if the Zariski-closure of Γ is semisimple, then the Cayleygraphs G ( π a (Γ) , π a ( S )) form a family of expanders as a ranges through finiteindex ideals of A ?We also mention that there are studies devoted to the problem of expansionwith respect to random generators, see [16] and [17], and in the work of Breuil-lard and Gamburd [14] it is proved that except maybe for a set of primes p ofzero density, SL ( F p ) are expanders with respect to any generators. Let k be a number field and Γ be a finitely generated subgroup of GL n ( k ). Letus also remark that Theorem 1 tells us under what condition the Cayley graphsof the square free quotients of Γ form expanders. To be precise, by the meansof restriction of scalars, we can view Γ as a subgroup Γ ′ of R k/ Q (GL d )( Q ) ⊆ GL rd ( Q ), where r = dim Q k . Now it is easy to see that G ( π q (Γ) , π q ( S )) formexpanders as q runs through square free ideals of O k (the ring of integers in k ) with large prime factors if and only if G ( π q (Γ ′ ) , π q ( S ′ )) form expanders as q runs through square free integers with large prime factors. By Theorem 1, weknow the necessary and sufficient condition under which the latter holds. Inthe following example, we present a finitely generated subgroup Γ of GL d ( Q [ i ])whose Zariski-closure is Zariski connected and perfect but G ( π p (Γ) , π p ( S )) donot form expanders as p runs through prime ideals of Z [ i ]. It shows that ingeneral it is necessary to view Γ as a subgroup of GL rd ( Q ) and then look at itsZariski-closure. Example 5.
Let h be a non-degenerate symplectic form on V = Z m . Let H bethe Heisenberg group associated with h . To be precise, H ( Z ) is the set V × Z endowed with the group law( v, t ) · ( v ′ , t ′ ) := ( v + v ′ , t + t ′ + h ( v, v ′ )) . From the definition it is clear that H is a central extension of the group scheme V associated with V . The action of the symplectic group Sp h,V on V can be3aturally extended to an action on H by acting trivially on the center. Now let G = Sp h,V ⋉ H andΓ := { ( γ, ( v, t )) ∈ G ( Z [ i ]) | γ ∈ Sp h,V ( Z ) , v ∈ V, t ∈ Z [ i ] } . It is easy to see G Q [ i ] := G × Spec( Z ) Spec( Q [ i ]) is a perfect Zariski-connected Q [ i ]-group and Γ is a finitely generated Zariski-dense subgroup of G Q [ i ] . On theother hand, the Zariski-closure of Γ ′ in R Q [ i ] / Q ( G ) is isomorphic to G × G a , whichis not perfect. Thus the Cayley graphs G ( π p (Γ) , π p ( S )) do not form expandersas p runs through primes in Z [ i ].As we have seen in Example 5, in general, the connected component of theZariski-closure G of Γ in GL d ( k ) might be perfect but the connected componentof its Zariski-closure G ′ in R k/ Q ( G )( Q ) ⊆ GL rd ( Q ), where r = dim Q k , mightbe not perfect. However it is easy to see that if G is semisimple, so is G ′ . Hencewe get the following corollary: Corollary 6.
Let Γ ⊆ GL d ( k ) be the group generated by a symmetric set S , where k is a number field. If the Zariski-closure of Γ is semisimple, then G ( π q (Γ) , π q ( S )) form a family of expanders when q ranges over square-free ide-als of the ring of integers O k in k with large prime factors. Similarly to most of the previous works done on this problem [7]–[11] and [61],first we prove escape from proper subgroups and then we show the occurrence ofthe flattening phenomenon . Proposition 7 (Escape from subgroups) . Let G be a Zariski-connected, perfectalgebraic group defined over Q . Let S ⊂ G ( Q ) be finite and symmetric. Assumethat S generates a subgroup Γ < G ( Q ) which is Zariski dense in G .Then there is a constant δ depending only on S , and there is a symmetricset S ′ ⊂ Γ such that the following holds. For any square-free integer q whichis relatively prime to the denominators of the entries of S , and for any propersubgroup H < π q (Γ) and for any even integer l ≥ log q , we have π q [ χ ( l ) S ′ ]( H ) ≪ [ π q (Γ) : H ] − δ . The notation used in this and the next proposition is explained in detail inSection 2. Now we only mention that χ S ′ is the normalized counting (probabil-ity) measure on S ′ and χ ( l ) S ′ is the l -fold convolution of χ S ′ with itself. Proposition 8 ( l -flattening) . Let G be a Zariski-connected, perfect algebraicgroup defined over Q . Let Γ < G ( Q ) be a finitely generated Zariski-dense sub-group. Then for any ε > , there is some δ > depending only on ε and G suchthat the following holds. Let q be a square-free integer which is relatively primeto the entries of the elements of Γ and let µ and ν be probability measures on π q (Γ) such that µ satisfies k µ k > | π q (Γ) | − / ε and µ ( gH ) < [ π q (Γ) : H ] − ε or any g ∈ π q (Γ) and for any proper subgroup H < π q (Γ) . Then k µ ∗ ν k < k µ k / δ k ν k / . We deduce Theorem 1 from the above propositions. The method to provespectral gap using analogues of these propositions was discovered by Bourgainand Gamburd [7] building on ideas that go back to Sarnak and Xue [55]. Verybriefly it goes as follows:By Proposition 7, we can bound π q [ χ ( l ) S ′ ]( gH ) which is the probability thatthe random walk after l ≈ log q steps is in a coset of a proper subgroup H .In particular, taking H = { } , we get k χ ( l ) S ′ k ≤ | π q (Γ) | δ . Now we can applyProposition 8 and iterate it to get improved bounds. Finally the representationtheory of G ( Z /q Z ) gives a lower bound for the multiplicity of the eigenvalues ofthe adjacency matrix of the Cayley graph G ( π q (Γ) , π q ( S )). Then we can use atrace formula to deduce an upper bound for the eigenvalues.The papers [7]–[11], [61] and the current work are all based on the abovestrategy. The difference between the proofs is in the way the analogues of thesetwo propositions are proved.We divided the proof of Proposition 7 into two parts. First, mainly usingNori’s result, we lift up the problem to Γ, and show that “small” lifts of acertain large subgroup of H is inside a proper algebraic subgroup H . (The ideaof using Nori’s theorem in this context is not new, it goes back to the paperof Bourgain and Gamburd [9].) Then we give a geometric description for beingin a proper algebraic subgroup in the spirit of Chevalley’s theorem. To thisend, we construct finitely many irreducible representations ρ i of the semisimplequotient of G . Then for any i we also give an algebraic family { φ i,v } v of affinetransformation lifts of ρ i to G . And we show that a proper algebraic subgroup H either fixes a line via ρ i or fixes a point via φ i,v for some i and v . In thesecond step, using some ideas of Tits, we construct certain “ping-pong players”,and show that, in the process of the random walk with respect to this set ofgenerators, the probability of fixing either a particular line or a point in thesefinitely many algebraic families of affine representations is exponentially small.In order to prove Proposition 8, first we prove a triple product theoremsimilar to Helfgott’s result [35], [36]. I.e. we show that if A ⊂ G ( Z /q Z ) issuitably distributed among proper subgroup cosets (to be made precise, seeProposition 26) then | A.A.A | ≥ | A | δ . Then the proposition can be deducedfrom the Balog-Szemer´edi-Gowers Theorem just as in [7].We comment on the new ideas of the current work compared to the previousresults, especially to [61], where Theorem 1 was proved for G = R k/ Q (SL d ). Wealso indicate which of these ideas are relevant also when the Zariski closure G of Γ is semisimple, since this case is of special interest.Compared to [61], the ”ping-pong argument” used in the current work ismore flexible. In [61], the argument applies only for representations that areboth proximal and irreducible. Whereas in the current paper we give a moreself-contained argument that needs only irreducibility. This is significant be-cause even in the semisimple case, it could be difficult to construct suitable5epresentations with both properties. Moreover, we do not rely on the result ofGoldsheid and Margulis on proximality of Zariski dense subgroups. This allowsus to work both in the Archimedean and the non-Archimedean setting whichis needed to prove Theorem 1 for S -integers. (The theorem of Goldsheid andMargulis does not hold over p-adic fields.)When G is not semi-simple further new ideas are needed. The unipotentradical is in the kernel of any irreducible representation. In order to detectproper subgroups which surject onto the semisimple factor of G , we introducealgebraic families of affine representations.We also need to use more complicated constructions when we use Nori’sresult to lift subgroups of finite groups to algebraic subgroups. On the otherhand we eliminate the use of the quantitative Nullstellensatz which was a toolin [61].It was proved in [61] (see Proposition B in Section 4) that if the triple producttheorem holds for a family of simple groups (which also satisfy some additional,more technical properties), then the triple product theorem is also true for theirdirect product. The proof of this result in [61] is closely related to the proofof the square-free sum-product theorem proved in [11]. The triple producttheorem for finite simple groups of Lie type of bounded rank was achievedin a recent breakthrough of Breuillard, Green, Tao [15] and of Pyber, Szab´o[52] independently. (See Theorem C in Section 4.) These results are used asblack boxes in our paper. When G is semisimple, then Proposition 8 almostimmediately follows from these results. The new contribution of the currentwork in the proof of Proposition 8 is when G is not semisimple. To this end, wehave to deal with certain semidirect products and one can see some similaritieswith the work of Alon, Lubotzky and Wigderson [3].We note that the all the constants appearing in the paper are effective.However, an explicit computation would be tedious especially since some ofour references are non-explicit, too. In particular the paper of Nori [51] usesnon-effective techniques, but it can be made effective using some results ofcomputational algebraic geometry. We discuss this briefly in the Appendix.All of our arguments are constructive, and the constants could be computedin a straightforward way, except for some of the proofs in Section 3.2. At thoseplaces, we prove the existence of certain objects by nonconstructive means.However, these objects can be found by an algorithm simply by checking count-able many possibilities. The existence of the object implies that the algorithmterminates in finite time.For example Proposition 21 claims the existence of a finite subset of Γ andcertain subsets of vector spaces with certain properties. It is easy to see that wecan choose those sets to be bounded by rational hyperplanes, so the data whoseexistence is claimed in the proposition can be found within a countable set.Since it is a finite computation to check the required properties, one can alwaysfind a suitable subset of Γ and the accompanied data by finite computation.The other place is the proof of Proposition 20, where we show that theintersection of a collection of sets parametrized by an integer k is empty forsome k . Although the proof does not give a clue how large k needs to be, but6e can always compute it by computing the intersection of the sets for every k until it becomes empty.The organization of the paper is as follows. In Section 2 we introduce somenotation. Section 3 is devoted to the proof of Proposition 7. In Section 4 weprove Proposition 8. In Section 5, we finish the proof of Theorem 1. Finally inthe appendix the effectiveness of Nori’s results [51] is showed. Acknowledgment.
We would like to thank Peter Sarnak and Jean Bourgainfor their interest and many insightful conversations. We are very grateful toBrian Conrad for his help in the proof of Theorem 40. We are also in debt toAlex Lubotzky for his interest and permission to include Question 4. We alsowish to thank the referee for her or his suggestions and careful reading of thepaper.
We introduce some notation that will be used throughout the paper. We useVinogradov’s notation x ≪ y as a shorthand for | x | < Cy with some constant C . Let G be a group. The unit element of any multiplicatively written group isdenoted by 1. For given subsets A and B , we denote their product-set by A.B = { gh | g ∈ A, h ∈ B } , while the k -fold iterated product-set of A is denoted by Q k A . We write e A forthe set of inverses of all elements of A . We say that A is symmetric if A = e A .The number of elements of a set A is denoted by | A | . The index of a subgroup H of G is denoted by [ G : H ] and we write H . L H if [ H : H ∩ H ] ≤ L for some subgroups H , H < G . We denote the center of G by Z ( G ). If ρ isa representation of G , then we denote the underlying vector space by W ρ andwe denote by ( W ρ ) G the set of points fixed by all elements of G . Occasionally(especially when a ring structure is present) we write groups additively, then wewrite A + B = { g + h | g ∈ A, h ∈ B } for the sum-set of A and B , P k A for the k -fold iterated sum-set of A and 0 forthe unit element.If µ and ν are complex valued functions on G , we define their convolutionby ( µ ∗ ν )( g ) = X h ∈ G µ ( gh − ) ν ( h ) , and we define e µ by the formula e µ ( g ) = µ ( g − ) .
7e write µ ( k ) for the k -fold convolution of µ with itself. As measures andfunctions are essentially the same on discrete sets, we use these notions inter-changeably, we will also use the notation µ ( A ) = X g ∈ A µ ( g ) . A probability measure is a nonnegative measure with total mass 1. Finally, thenormalized counting measure on a finite set A is the probability measure χ A ( B ) = | A ∩ B || A | . In this section we prove Proposition 7. Some ideas are taken from [61] butthere are substantial new difficulties especially due to the lack of proximalityof the adjoint representation and because we also consider groups that are notsemisimple.We begin this section by recalling some results from the literature on thesubgroup structure of π q (Γ). Then in Section 3.1, in order to solve the problem ofescaping from proper subgroups of π q (Γ), we lift it up to a problem on escapingfrom certain proper subgroups of Γ. For that purpose, we consider “small”lifts of elements of H in Γ; namely, for a square-free integer q and a subgroup H < G q , we write L δ ( H ) := { h ∈ Γ | π q ( h ) ∈ H, k h k S < [ G q : H ] δ } , where k h k S = max p ∈S∪{∞} k h k p and k h k p is the operator norm on Q dp . Inthe next section, we give a geometric description of the set L δ ( H ) in terms ofits action in an irreducible representation. Then in Section 3.2, we present anargument to show that only a small fraction of the elements of Γ satisfy thisgeometric property. Finally, we combine these two results to get Proposition 7.Let G ⊆ GL d be a Zariski-connected Q -group. Then it is well-known that itsunipotent radical U is also defined over Q (e.g. see [59, Proposition 14.4.5]) andit has a Levi subgroup defined over Q , i.e. a reductive subgroup L defined over Q such that G is Q -isomorphic to L ⋉ U . If G is a perfect group, i.e. G = [ G , G ],then clearly L is a semisimple group. We say that G is simply-connected if L is simply-connected. If L is a simply-connected Q -group, then we can write L as product of absolutely almost simple groups. The absolute Galois grouppermutes these factors. So there are number fields κ i and absolutely almostsimple κ i -groups L i such that L ≃ m Y i =1 R κ i / Q ( L i )as Q -groups. By a result of Bruhat-Tits [60, Section 3.9], for large enough p , L ( Z p ) is a hyper-special parahoric subgroup and so L ( F p ) is a product of8uasi-simple groups. We also have that U ( F p ) is a finite p -group, and, for largeenough p , G ( F p ) ≃ L ( F p ) ⋉ U ( F p ) and is a perfect group. Again as part ofBruhat-Tits theory, we know that L ( F p ) is generated by order p elements. (Tobe precise, we know that, for large enough p , the special fiber of L over F p is aconnected, simply connected, semisimple F p -group.) Thus, for large enough p , G ( F p ) = G ( F p ) + , where G + is the subgroup generated by p -elements, for anysubgroup G of GL d ( F p ). As part of Nori’s Strong Approximation [51, Theorem5.4], we have that, Theorem A.
Let G ⊆ GL d be a Zariski-connected, perfect, simply-connected Q -group. Let Γ be a finitely generated Zariski-dense subgroup of G ( Q ) ; thenthere is a finite set S of primes such that,1. The closure of Γ in Q p ∈P\S G ( Z p ) is an open subgroup.2. There is a constant p depending on Γ such that for any square free integer q with prime factors larger than p , π q (Γ) = G ( Z /q Z ) .3. There is a constant p depending on G and its embedding into GL d , suchthat for any square free integer q with prime factors larger than p , Y p | q π p : G ( Z /q Z ) ∼ −→ Y p | q G ( Z /p Z ) . Moreover, for p > p , G ( F p ) = L ( F p ) ⋉ U ( F p ) , L ( F p ) is a product ofquasi-simple finite groups whose number of factors is bounded in terms of dim L . If Γ is a finitely generated subgroup of G ( Q ), ι : e G → G is a Q -isogeny,and e G = e L ⋉ U is simply connected, then it is easy to see that Γ ∩ ι ( e G ) isa finite index subgroup of Γ. Furthermore, for large enough p , ι ( e G ( Z p )) ⊆ G ( Z p ) and the pre-image of the first congruence subgroup of G ( Z p ) is the firstcongruence subgroup of e G ( Z p ). In particular, there is a square free number q such that, for any q , π q (Γ) ≃ π ( q,q ) (Γ) × Q p | q/ ( q,q ) π p (Γ); moreover π p (Γ) = ι ( e L ( F p )) ⋉ U ( F p ) and ι ( e L ( F p )) is a product of quasi-simple finite groups if p is large enough. Let us also add that U / [ U , U ] is a commutative unipotent Q -group, and so it is a Q -vector group, i.e. it is Q -isomorphic to G Ma forsome M (the logarithm and exponential maps give Q -isomorphisms between acommutative unipotent Q -group and its Lie algebra). Thus, for large enough p ,( U / [ U , U ])( F p ) is an F p -vector space. As [ U , U ] is an F p -split unipotent algebraicgroup, ( U / [ U , U ])( F p ) = U ( F p ) / [ U , U ]( F p ). The next lemma, shows that, forlarge enough p , we have [ U , U ]( F p ) = [ U ( F p ) , U ( F p )], and so overall we havethat, for large enough p , U ( F p ) / [ U ( F p ) , U ( F p )] is an F p -vector space. Let γ k ( U )be the k -th lower central series, i.e. γ ( U ) = U and γ i +1 ( U ) = [ U , γ i ( U )]. It iswell-known that, if U is defined over Q , then all of the lower central series arealso defined over Q . Lemma 9.
Let U be a unipotent Q -algebraic group. Then, for any k and largeenough p , γ k ( U )( F p ) = γ k ( U ( F p )) . roof. As U has a Q -structure, there is a lattice Γ U in U ( R ). In particular, itis a finitely generated, Zariski-dense subgroup of U ( Q ). Thus γ k (Γ U ) is Zariski-dense in γ k ( U ), for any k . By Nori’s result, γ k (Γ U ) modulo p is the the fullgroup γ k ( U )( F p ), for large enough p , which finishes the proof.For the rest of this section, S is a finite set of primes such that Γ ⊆ GL d ( Z S ), q will be a square-free integer, and we assume that it has no prime divisor lessthan a constant which depends on Γ. We write G q = π q (Γ). For future referencewe record the properties of G q that we deduced above: For any square-freeinteger q with sufficiently large prime divisors, and for any sufficiently largeprime p , we have1. G q = Q p | q G p ,2. G p = G ( F p ) + = L p ⋉ U p , where(a) L p is a product of quasi-simple finite groups of Lie type over finitefields which are extensions of F p ; moreover the number of the quasi-simple factors has an upper-bound independent of p ,(b) U p is a k -step nilpotent p -group, where k is independent of p , and U p / [ U p , U p ] is isomorphic to an F p -vector space; moreover log p | U p | is bounded independently of p .Let us also recall from Nori’s paper [51, Theorem B and C] that for largeenough p , any subgroup H of GL d ( F p ) satisfies the following properties:3. There is a Zariski-connected algebraic subgroup H of GL d defined over F p such that H ( F p ) + = H + .4. There is a commutative subgroup F of H such that H + · F is a normalsubgroup of H and [ H : H + · F ] < C , where C just depends on d the sizeof the matrices.5. There is a correspondence between p -elements of H and nilpotent elementsof h ( F p ), where h is the Lie algebra of H ; moreover h ( F p ) is generated byits nilpotent elements. In this section we describe in geometric terms the set L δ ( H ) defined above.In fact what we show is that there is a subgroup H ♯ of small index, such that L δ ( H ♯ ) is contained in a certain proper algebraic subgroup of G . It is well-knownby Chevalley’s theorem, that then there is a representation of G in which L δ ( H ♯ )fixes a line that is not fixed by the whole group. For technical reasons we needthat the representations come from a fixed finite family; therefore we constructthem explicitly. In addition, the methods of the next section would require thatthe representations are irreducible, which is not possible to fulfill since G is notnecessarily semisimple. For this reason, besides the irreducible representations10e also consider homomorphisms into the group of affine transformations. Un-fortunately, a finite family of such homomorphism is not rich enough to captureall possible subgroups. We need to consider uncountable families, where the lin-ear part of the action is the same and the translation part can be parametrizedby elements of an affine space. The precise formulation is contained in the nextproposition that we will use later as a black box in the paper. Proposition 10.
Let G be a Zariski-connected perfect Q -group, and let Γ be aZariski-dense, finitely generated subgroup of G ( Q ) . Then there are non-trivialirreducible representations ρ i for ≤ i ≤ m of G and morphisms ϕ i : G × V i → Aff( W ρ i ) with the following properties:1. For any i , V i is a (possibly -dimensional) affine space. V i , ρ i and ϕ i aredefined over a local field K i ; and ρ i (Γ) is an unbounded subset of GL( W ρ i ) ,where W ρ i = W ρ i ( K i ) .2. For any i and = v ∈ V i ( K i ) , ϕ i,v = ϕ i ( · , v ) is a group homomorphismto Aff( W ρ i ) whose linear parts are ρ i , and G ( K i ) does not fix any point of W ρ i via this action.3. There are positive constants C and δ such that the following holds. Let q = p · · · p n be a square-free number, such that each p i is a sufficientlylarge prime, and let H be a proper subgroup of π q (Γ) . Then there is asubgroup H ♯ of index at most C n in H that satisfies one of the followingtwo conditions:(a) For some ≤ i ≤ m , there is w = 0 in W ρ i such that ρ i ( h )([ w ]) =[ w ] , for any h ∈ L δ ( H ♯ ) .(b) For some < i ≤ m , there is v ∈ V i ( K i ) and w ∈ W ρ i such that k v k = 1 and φ i,v ( h )( w ) = w , for any h ∈ L δ ( H ♯ ) . We will easily deduce Proposition 10 from the following more technical ver-sion.
Proposition 11.
Let G and Γ be as in the setting of Proposition 10 and G = L ⋉ U , where L is a semisimple group and U is a unipotent group. Then thereare finitely many representations ρ , . . . , ρ m ′ , ψ , . . . , ψ k of G with the followingproperties:1. For any i , U ⊆ ker( ρ i ) and the restriction of ρ i to L is a non-trivialirreducible representation.2. For any i , there is a sub-representation W (1) i of W ψ i such that(a) U acts trivially on W (1) i and W ψ i / W (1) i .(b) W (1) i is a non-trivial irreducible representation of L that we denoteby ρ m ′ + i . c) W ψ i = W (1) i ⊕ W (2) i , where W (2) i = W L ψ i is the set of L -invariantvectors.3. For any i , there are local fields K i such that ρ i is defined over K i and ψ i are defined over K i + m ′ ; moreover ρ i (pr(Γ)) , where pr is the projection to L , is an unbounded subset of ρ i ( L ( K i )) .4. There are positive constants C and δ such that the following holds. Let q = p · · · p n be a square-free number, such that each p i is a sufficientlylarge prime, and let H be a proper subgroup of π q (Γ) . Then there is asubgroup H ♯ of index at most C n in H that satisfies one of the followingtwo conditions:(a) For some i , there is w = 0 in W ρ i = W ρ i ( K i ) such that ρ i ( h )([ w ]) =[ w ] , for any h ∈ L δ ( H ♯ ) .(b) For some i , there is a vector = w ∈ W ψ i such that ψ i ( h )( w ) = w ,for any h ∈ L δ ( H ♯ ) . Moreover there is no nonzero vector w ′ in W ψ i such that ψ i ( G )( w ′ ) = w ′ .Proof of Proposition 10 assuming Proposition 11. For any 1 ≤ i ≤ m ′ , we let ρ i be the same as in Proposition 11. For these representations, we take V i = 0and let ϕ i ( g,
0) = ρ i ( g ). For 1 ≤ i ≤ k , we let ρ m ′ + i be the representation of L on W (1) i and V m ′ + i = W (2) i . For any w ∈ W (1) i , let ϕ m ′ + i ( g, v )( w ) := ψ i ( g )( w + v ) − v. Notice that, since g acts trivially on W ψ i / W (1) i , ϕ m ′ + i ( g, v ) ∈ Aff( W (1) i ), forany i .With these choices, in order to complete the proof, it is enough to make thefollowing observations. If for some m ′ < i ≤ m and 0 = v ∈ V i , G fixes a point w ∈ W ρ i , then by definition, w + v is fixed by G . Therefore w + v = 0, andso w = v = 0, which is a contradiction. On the other hand, if ψ i ( h )( w ) = w ,where w = w + v , w ∈ W (1) i and v ∈ W (2) i = V i , then ϕ i,v ( h )( w ) = w . We prove Proposition 11 in two steps. First we show that for an appropriatechoice of δ and H ♯ , the Zariski-closure of the group generated by L δ ( H ♯ ) is aproper subgroup of G . Then, for any proper closed subgroup of G , we constructthe desired representations. We start with some auxiliary lemmata describingthe normal subgroups of G p . Lemma 12.
Let L = Q mi =1 L ( i ) , where L ( i ) are quasi-simple groups. Then anynormal subgroup H of L is of the form Q i ∈ I L ( i ) × Z , where I is a subset of { , . . . , m } and Z ≤ Q i I Z ( L ( i ) ) .Proof. For any i , either pr i ( H ), the projection onto L ( i ) , is central or pr i ( H ) = L ( i ) . If ( s , . . . , s m ) is in H , then, for any g ∈ L ( i ) , (1 , . . . , , [ g, s i ] , , . . . , N . On the other hand, the group generated by [ g, s i ] is a normal12ubgroup of L ( i ) . If s i is not central, then the above group cannot be central as Z (cid:0) L ( i ) /Z ( L ( i ) ) (cid:1) = { } , and so it is the full group L ( i ) . The rest of the argumentis straightforward. Lemma 13.
Let L be a direct product of quasi-simple finite groups which actson a finite nilpotent group U . Then any normal subgroup H of L ⋉ U is of theform ( H ∩ L ) ⋉ ( H ∩ U ) if the prime factors of | U | are larger than an absoluteconstant depending only on the size of the center of L . Moreover H ∩ L actstrivially on U/H ∩ U .Proof. Passing to
H/H ∩ U E L ⋉ ( U/H ∩ U ), we can and will assume that H ∩ U = { } . Thus projection to L induces an embedding and we get a map ϕ : pr( H ) → U , where pr : L ⋉ U → L is the projection map, such that H = { ( s, ϕ ( s )) | s ∈ pr( H ) } . One can easily check that ϕ is a 1-cocycle, i.e. ϕ ( s s ) = s − ϕ ( s ) s · ϕ ( s ).Furthermore, for any u ∈ U and s ∈ L , we have(1 , u − )( s, ϕ ( s ))(1 , u ) = ( s, s − u − s · ϕ ( s ) · u ) ∈ H ;thus ϕ ( s ) = s − u − s · ϕ ( s ) · u , for any u ∈ U and s ∈ L . In particular, setting s = s and u = ϕ ( s ) − , and then using the cocycle relation, we have ϕ ( s ) · ϕ ( s ) = s − ϕ ( s ) s · ϕ ( s ) = ϕ ( s s ) . Since ϕ (1) = 1, by the above equation, we have that ϕ ( s − ) = ϕ ( s ) − . There-fore, by the above discussion, θ ( s ) = ϕ ( s − ) defines a homomorphism frompr( H ) to U . On the other hand, pr( H ) is a normal subgroup of L , so byLemma 12, if the prime factors of | U | are larger than the size of the center of L , then θ is trivial, which finishes the proof of the first part. For the secondpart, it is enough to notice that sus − u − is in H ∩ U , for any s ∈ H ∩ L and u ∈ U . Corollary 14.
Let L be a product of quasi-simple finite groups, which acts ona finite nilpotent group U . Assume that G = L ⋉ U is a perfect group. If H is a normal subgroup of G and the projection of H onto L is surjective, then H = G .Proof. By Lemma 13, H = L ⋉ U ′ , for a normal subgroup U ′ of U , and L actstrivially on U/U ′ . Thus L ⋉ U/U ′ ≃ L × U/U ′ is not a perfect group unless U = U ′ . On the other hand, any quotient of G is perfect, which finishes theproof.In the next step, for any proper subgroup H of G q , we will find anothersubgroup containing H which is of product form and is of comparable size. Lemma 15.
Let H be a proper subgroup of G = Q p ∈ Σ G p , where0. Σ is a finite set of primes larger than , . G p = L p ⋉ U p , where L p = Q i L ( i ) p , L ( i ) p are quasi-simple groups of Lietype over a finite field of characteristic p , and U p is a p -group,2. G p is perfect,3. | G p | ≤ p k for any p ∈ Σ , with a fixed k independent of p .Then there is a positive number δ depending on k , such that Y p ∈ Σ [ G p : π p ( H )] ≥ [ G : H ] δ , where π p is the projection onto G p .Proof. We prove this by induction on | G | . Let Σ ′ = { p ∈ Σ | π p ( H ) = G p } . IfΣ ′ is empty, then [ G p : π p ( H )] ≥ p for any p , as π p ( H ) is a proper subgroup of G p and G p is generated by p elements. Therefore Y p ∈ Σ [ G p : π p ( H )] ≥ Y p ∈ Σ p ≥ Y p ∈ Σ | G p | /k ≥ [ G : H ] /k . So we shall assume that Σ ′ is non-empty. For any p , H p := G p ∩ H is a normalsubgroup of π p ( H ); in particular, H p is a normal subgroup of G p when p ∈ Σ ′ .Let G ′ p = G p if p Σ ′ , G p /H p if p ∈ Σ ′ , and H ′ = H/ Y p ∈ Σ ′ H p ⊆ Y p ∈ Σ G ′ p =: G ′ . If H p is not trivial for some p ∈ Σ ′ , then | G ′ | < | G | . Moreover, by Lemma 13and Corollary 14, we can apply the induction hypothesis, and we get that[ G : H ] δ = [ G ′ : H ′ ] δ ≤ Y p ∈ Σ [ G ′ p : π p ( H ′ )] = Y p ∈ Σ [ G p : π p ( H )] , and we are done. Thus, without loss of generality, we can and will assume that H p is trivial for any p ∈ Σ ′ .Let p ∈ Σ ′ . Since H p is trivial and π p ( H ) = G p , there is an epimorphismfrom H ′ := π Σ \{ p } ( H ) to G p and H ′ is isomorphic to H . Let N be the kernelof this epimorphism. By the induction hypothesis, we have that Y p ∈ Σ \{ p } [ G p : π p ( N )] ≥ [ Y p ∈ Σ \{ p } G p : N ] δ . On the other hand, | H | = | H ′ | = | G p | · | N | ; so Y p ∈ Σ \{ p } [ G p : π p ( N )] ≥ [ G : H ] δ . (1)14y (1), it is clear that, if we have π p ( N ) = π p ( H ) for all p , then we are done.Thus we can assume that π p ( N ) = π p ( H ) for some p .Since H ′ /N ≃ G p , it is clear that, for any p = p , π p ( H ) /π p ( N ) is ahomomorphic image of G p . Thus, by the Jordan-H¨older Theorem, either G p and π p ( H ) have some composition factor in common or π p ( H ) = π p ( N ). Onthe other hand, the composition factors of G p are the cyclic group of order p and L ( i ) p . In particular, if p and p ′ are distinct primes and either p or p ′ is largerthan 7, then G p and G p ′ do not have any composition factor in common. So π p ( H ) = π p ( N ) if p ∈ Σ ′ \ { p } . So, for any p ∈ Σ ′ , there is p ∈ Σ \ Σ ′ suchthat | L ( i ) p | divides | G p | , for some i . In particular, p divides | G p | . Thus we havethat Y p ∈ Σ ′ p | Y p ∈ Σ \ Σ ′ | G p | . Therefore Q p ∈ Σ ′ p ≤ Q p ∈ Σ \ Σ ′ p k . Now it is straightforward to show that Y p ∈ Σ [ G p : π p ( H )] ≥ [ G : H ] k ( k +1) , and we are done.After these preparations we are ready to make the first step towards theproof of Proposition 11. Recall that we try to describe the set L δ ( H ) in ageometric way, which set is the set of ”small lifts” of H to G ( Z ). The nextProposition shows that for every H < G q , we can find a subgroup H ♯ for which L δ ( H ♯ ) is contained in a proper algebraic subgroup of G . Proposition 16.
There are positive numbers δ and C for which the followingholds:Take any proper subgroup H of G q , where q is a square free integer with largeprime factors. Then one can find a proper algebraic subgroup H < G definedover Q , and a subgroup H ♯ of H , such that1. L δ ( H ♯ ) ⊂ H ( Q ) ,2. [ H : H ♯ ] ≤ C n , where n is the number of prime factors of q .Proof. We begin the proof by constructing H ♯ . Let q ′′ be the product of primefactors of q such that G p = π p ( H ). By Nori’s result (see 4. on page 10),for large enough p , there is a commutative subgroup F p of π p ( H ) such that H ♯p := π p ( H ) + · F p is a normal subgroup of π p ( H ) and [ π p ( H ) : H ♯p ] < C , where C just depends on the size of the matrices. Let H ♯q ′′ = Q p | q ′′ H ♯p and H ♯ = { h ∈ H | π q ′′ ( h ) ∈ H ♯q ′′ } . It is clear that [ H : H ♯ ] ≤ [ Q p | q ′′ π p ( H ) : H ♯q ′′ ] ≤ C n , where n is the numberof prime factors of q . We will see that without loss of generality we can replace15 by q ′′ and H by H ♯q ′′ . By Lemma 15 and the discussions in the beginning ofSection 3, we have that [ G q : H ] δ ≤ [ G q ′′ : Y p | q ′′ π p ( H )] . Hence [ G q : H ♯ ] δ ≤ [ G q ′′ : H ♯q ′′ ] . Therefore L δ ( H ♯ ) ⊆ L δ/δ ( H ♯q ′′ ). So, without loss of generality, we assume that1. H = Q p | q H p , where H p is a proper subgroup of G p .2. H ♯ = Q p | q H ♯p .Consider an embedding G ⊆ GL d defined over Q . Below we will show thatfor each prime p | q there is a polynomial P p ∈ ( Z /p Z )[ x , , . . . , x d,d ] of degreeat most 4 such that all elements g ∈ H ♯p satisfy this equation, i.e. P p ( g ) = 0,but not all elements of G p do. Now we show that this easily implies the firstpart of the proposition with some δ >
0. To do this, we first show that there isa polynomial P ∈ Q [ x , , . . . , x d,d ] which vanishes on L δ ( H ♯ ) but not on all of G . Consider the usual degree 4 monomial map: Ψ : GL d → A ( d ). Denote by D the dimension of the linear subspace of A ( d ) spanned by Ψ( G ). We needto show that Ψ( L δ ( H ♯ )) spans a subspace of dimension lower than D . Supposethe contrary, and let g , . . . , g D ∈ Ψ( L δ ( H ♯ )) which are linearly independent.We can consider the g i as column vectors, and form a (cid:0) d +44 (cid:1) × D matrix. Byindependence this has a nonzero D × D subdeterminant, whose entries are allless than 1 D ! q D ( |S| +1) in the k · k S -norm, if we choose δ sufficiently small. Recall that S is the setof primes which occur in the denominators of elements of Γ. Now, the valueof this subdeterminant is a nonzero rational number less than q / ( |S| +1) , whosedenominator is less than q |S| / ( |S| +1) . This implies that the projection mod p ofthis determinant is still nonzero for some p | q , which contradicts the existenceof P p to be demonstrated below.So far we showed that for sufficiently small δ , L δ ( H ♯ ) is contained in aproper subvariety X ⊆ G . By [24, Proposition 3.2], there is an integer N suchthat if A ⊆ G ( Q ) is a finite symmetric set generating a Zariski dense subgroupof G , then Q N A * X ( Q ). This implies that L δ/N ( H ♯ ) is contained in a properalgebraic subgroup. We note that the proof of [24, Proposition 3.2] gives that N depends only on the dimension, the degree and the number of irreduciblecomponents of X . The proof in [24, Proposition 3.2] is based on the idea, thatby Zariski density, one can find an element g ∈ A such that X ∩ g X is either oflower dimension or contains less components of maximal dimension than X . N is the number of iterations we need to make to get a trivial intersection. It is16lear that we can keep track of the dimension, the number of components andthe degree of the varieties that arise this way. Hence the procedure terminatesin N steps controlled by those parameters only.It remains to show our claim about the existence of the polynomials P p . Inwhat follows, let g be the Lie algebra of G and g ∗ its dual.We consider the adjoint representation of G and its dual on these spaces,respectively. For large enough p , these actions reduce to the action of G ( F p ) on g ( F p ) and g ∗ ( F p ); moreover g ∗ ( F p ) = g ( F p ) ∗ . There is a natural non-degeneratebilinear form on g ⊗ g ∗ , which is the linear extension of the following map h v ⊗ f , v ⊗ f i := f ( v ) f ( v ) . (It is worth mentioning that this bilinear map is G -invariant.) It is also well-known that g ⊗ g ∗ is isomorphic to End( g ) as a G -module, where G acts onEnd( g ) via the conjugation by the adjoint representation. We denote both ofthese representations by ρ .Clearly, we can assume that any prime divisor p | q is sufficiently large, hence h· , ·i induces a non-degenerate bilinear form on g ( F p ) ⊗ g ∗ ( F p ).For any x and y in g ⊗ g ∗ , let η x,y be a polynomial in d variables withcoefficients in Z [1 /q ], which is defined as follows, η x,y ( g ) := h ρ ( g )( x ) , y i , We show that for a prime divisor p of q , we can find some g ∈ S and x and y in g ⊗ g ∗ such that η x,y ( g ) = 0 modulo p , for any h ∈ H ♯p , and η x,y ( g ) = 1.Since P p := η x,y is of degree at most 4, this proves our claim, and hence theproposition.Since h· , ·i is a non-degenerate bilinear form on g ( F p ) ⊗ g ( F p ) ∗ , it is enoughto show that there is a proper subspace of g ( F p ) ⊗ g ( F p ) ∗ which is H ♯p -invariant,but not G ( F p ) + -invariant.If H + p is not a normal subgroup of G ( F p ) + , then clearly, by Nori’s result(see 3. and 5. on page 10 in Section 3), h ( F p ) is H ♯p -invariant, but not G ( F p ) + -invariant. So h ( F p ) ⊗ g ∗ ( F p ) has the desired property.Now, let us assume that H + p is a normal subgroup. Since it is a propernormal subgroup of G ( F p ) + , by Corollary 14, the projection pr( H + p ) of H + p to L ( F p ) + is a proper normal subgroup. On the other hand, we know that L ( F p ) + ≃ m Y i =1 Y p ∈P ( k i ) , p | p L i ( F p ) + , and L i ( F p ) + is quasi-simple, for any i and p . Hence, by Lemma 12, there is anon-empty subset I of possible indices ( i, p ) such thatpr( H + p ) ⊆ Y I L i ( F p ) + × Y I c Z ( L i ( F p ) + ) , where I c is the complement of I in the set of all the possible indices. Thus wehave that pr( H + p ) = Q I L i ( F p ) + as H + p is generated by p -elements. We also17otice that g = l ⊕ u , where l is the Lie algebra of L and u is the Lie algebra of U . Moreover g ( F p ) = m M i =1 M p ∈P ( k i ) , p | p l i ( F p ) ⊕ u ( F p ) , where l i is the Lie algebra of L i . For any possible indices ( i, p ), let F i, p bethe projection of F p (defined at the beginning of this proof) to L i ( F p ) + . Wenotice that l i ( F p ) is an irreducible L i ( F p ) + -module. Therefore, if l i ( F p ) is notan irreducible F i, p -module, for some ( i, p ), then one can easily get a propersubspace of g ( F p ) which is invariant under H ♯p , but not under G ( F p ) + and finishthe argument as before. So, without loss of generality, we assume that l i ( F p )is an irreducible F i, p -module. Since F i, p is a commutative group, the F p -span E i, p of its image in End( l i ( F p )) is a field extension of F p of degree dim F p l i ( F p ).(In particular, by the Double Centralizer Theorem, the centralizer of E i, p inEnd( l i ( F p )) is itself.) Now we consider the subspace W of End( g ( F p )) consistingof elements x with the following properties,1 . x ( u ( F p )) ⊆ u ( F p )2 . x ( l i ( F p )) ⊆ l i ( F p ) ⊕ u ( F p ) if ( i, p ) ∈ I ,3 . ∃ y ∈ E i, p , ∀ l i ∈ l i ( F p ) : x ( l i ) − y ( l i ) ∈ u ( F p ) if ( i, p ) I. We claim that W is H ♯p -invariant, but not G ( F p ) + -invariant. Let g ∈ G ( F p ) + and x ∈ W ; then Ad( g ) − x Ad( g ) clearly satisfies the first and second conditions.It is straightforward to check that Ad( g ) − x Ad( g ) satisfies the third conditionfor all x if and only if Ad( g i, p ) − E i, p Ad( g i, p ) = E i, p , (2)for any ( i, p ) I , where g i, p is the projection of g onto L i ( F p ) + . So clearly W is H ♯p -invariant. On the other hand, any element in N ( E i, p ) = { x ∈ GL( l i ( F p )) | x − E i, p x = E i, p } induces a Galois element and E i, p is a maximal subfield of End( l i ( F p )). There-fore [ N ( E i, p ) : E ∗ i, p ] ≤ dim F p l i ( F p ). Thus G ( F p ) + cannot leave W invariant if p is large enough, as we wished.The next proposition is the source of the desired representations claimed inProposition 11. Proposition 17.
Let G = L ⋉ U be a perfect group, where L is a semisim-ple group and U is a unipotent group. There are finitely many representations ρ , . . . , ρ m ′ , ψ , . . . , ψ k of G with the following properties:1. For any i , U ⊆ ker( ρ i ) and the restriction of ρ i to L is a non-trivialirreducible representation. . For any i , there is a sub-representation W (1) i of W ψ i such that(a) U acts trivially on W (1) i and W ψ i / W (1) i .(b) W (1) i is a non-trivial irreducible representation of L that we denoteby ρ m ′ + i .(c) W ψ i = W (1) i ⊕ W (2) i , where W (2) i = W L ψ i is the set of L -invariantvectors.3. For any proper subgroup H of G , one of the following holds:(a) For some i , there is w = 0 in W ρ i such that ρ i ( H )([ w ]) = [ w ] .(b) For some i , there is = w ∈ W ψ i such that ψ i ( H )( w ) = w . Moreoverthere is no non-zero vector w ′ in W ψ i such that ψ i ( G )( w ′ ) = w ′ .Proof. We divide the argument into several cases. First we consider the case,where the projection of H onto L is not surjective. In the second case we willassume that the group generated by its projection to U is a proper subgroupof U , and the third case finishes the argument. In each step, we introduce onlyfinitely many representations which satisfy the desired properties.In the first case, without loss of generality, we can assume that G = L is asemisimple group and H is a proper subgroup. If H is not a normal subgroup,then, in the representation ∧ dim h Ad, [ w ] = ∧ dim h h ∈ W is H -invariant, but itis not L -invariant. Since L is semisimple, we can take a decomposition of W into irreducible components. Since [ w ] is not L -invariant, its projection to oneof the non-trivial irreducible components is not zero, and so this representationsatisfies the condition 3(a).Notice that this process introduced only finitely many representations whichsatisfy the properties of ρ i .If H is a proper normal subgroup and L = Q m i =1 L i , where L i is an absolutelyalmost simple group, then there is a proper subset I of indices such that H ⊆ Y i ∈ I L i × Z ( Y i I L i ) . Consider the action of L on the Lie algebra l i of L i via the adjoint representationof L i , for any i I . Clearly any line in this representation is fixed by H andnot by G , which finishes the proof of the first case.We notice that if H is a proper subgroup of G , then H [ U , U ] is also a propersubgroup of G . So, without loss of generality, we assume that U is a vectorgroup, i.e. isomorphic to G k a , for some k .Now we assume that the projection of H onto L is surjective, but the groupgenerated by its projection to U is a proper subgroup. So, without loss ofgenerality, we assume that H = L ⋉ U ′ , where U ′ is a proper subgroup of U . Weconsider f W = U as an L -space (and f W ′ = U ′ is a proper L -subspace), and takeits decomposition into homogeneous subspaces, i.e. f W = f W ⊕ f W ⊕ · · · ⊕ f W n , L ( f W i , f W j ) = 0 if i = j , and f W i ≃ W m i i , where W i is an irreducible L -space. Here Hom L ( f W i , f W j ) denotes the space of L -equivariant linear mapsfrom f W i to f W j . Since W ′ is a proper L -subspace, its projection to at least oneof the homogeneous spaces is proper. Therefore, without loss of generality, weassume that f W ≃ W m , where W is an irreducible L -space. We call a map f from f W to W an affine map if f ( t ˜ w + t ˜ w ) = t f ( ˜ w ) + t f ( ˜ w ) , for any ˜ w and ˜ w in f W and t + t = 1. Let Aff( f W , W ) be the set of all affinemaps. If f is an affine map, then there are f lin ∈ Hom( f W , W ) and w ∈ W suchthat f ( x ) = f lin ( x ) + w, for any x ∈ f W ; f lin is called the linear part of f . Let Aff L ( f W , W ) be the set ofaffine maps whose linear part is in Hom L ( f W , W ). ThereforeAff L ( f W , W ) = Con( f W , W ) ⊕ Hom L ( f W , W ) , where Con( f W , W ) is the space of the constant functions. We claim that therepresentation ψ of G = L ⋉ f W on Aff L ( f W , W ), defined by ψ ( l, ˜ w )( f )( x ) := l · f ( l − · x − ˜ w ) , satisfies our desired conditions. Alternatively, we can say that if f ( x ) = f lin ( x )+ w, then ψ ( l, ˜ w )( f )( x ) = f lin ( x ) + l · w − l · f lin ( ˜ w ) . Both of the subspaces Con( f W , W ) and Hom L ( f W , W ) are L -invariant. Moreover, L acts trivially on Hom L ( f W , W ) and Con( f W , W ) is isomorphic to W as an L -space, and, in particular, it is irreducible. It is also clear that f W acts triviallyon Con( f W , W ) and Aff L ( f W , W ) / Con( f W , W ).Now since f W ′ is a proper L -subspace of f W , there is 0 = f ∈ Hom L ( f W , W )such that f W ′ ⊆ ker( f ). Thus, by the definition of ψ , ψ ( l, ˜ w ′ )( f ) = f, for any( l, ˜ w ′ ) ∈ L ⋉ f W ′ . Now assume that, for some w ∈ W and f lin ∈ Hom L ( f W , W ), f w ( x ) = f lin ( x ) + w is G -invariant. Then f w is a constant function as it isinvariant under any translation, i.e. f lin = 0. Hence w is L -invariant and so w is also 0. Therefore this representation satisfies all the desired properties.Now, we assume that the projection of H to L is surjective and the groupgenerated by the projection of H to U generates U . By Corollary 14, H is nota normal subgroup of G . (In fact, Corollary 14 was stated for finite groups,however, the proof works verbatim for algebraic groups, as well.) Hence, again,in the representation ˜ ρ = ∧ dim h Ad, [ w ] = ∧ dim h h ∈ f W is H -invariant, but it isnot G -invariant. We claim that H does not have any character, and therefore w is fixed by H . Let χ be a character of H ; then H ∩ U ⊆ ker( χ ) as U ∩ H is a20nipotent group. So χ factors through a character of H / ( H ∩ U ) ≃ L . Since L is semisimple, χ is trivial.Since U is a unipotent and normal subgroup of G , f W ⊇ (˜ ρ ( U ) − f W ) ⊇ · · · ⊇ (˜ ρ ( U ) − n +1 ( f W ) = 0 , and for any i , (˜ ρ ( U ) − i ( f W ) is a G -space. Let k + 1 be the smallest possibleinteger such that the projection of w to f W / h (˜ ρ ( U ) − k +1 · f W i is not G -invariant.Notice that k is definitely positive as G = H · U , w is H -invariant and U actstrivially on f W / h (˜ ρ ( U ) − · f W i . After going to the quotient space, we can andwill assume that h (˜ ρ ( U ) − k +1 · f W i = 0, i.e. U acts trivially on h (˜ ρ ( U ) − k · f W i .Let c W be the subspace of f W such that( f W / h (˜ ρ ( U ) − k · f W i ) G = c W / h (˜ ρ ( U ) − k · f W i , i.e. c W = { w ∈ f W | ∀ g ∈ G , ˜ ρ ( g )( w ) − w ∈ h (˜ ρ ( U ) − k · f W i} . By the the aboveargument, w ∈ c W .Now take a decomposition W ⊕· · ·⊕ W n of h (˜ ρ ( U ) − k · f W i into irreducible L -spaces. Since L is a semisimple group and it acts trivially on c W / h (˜ ρ ( U ) − k · f W i ,there is a subspace W such that c W = W ⊕ W ⊕ · · · ⊕ W n and L acts triviallyon W . We claim that the projection of w to one of the non-trivial irreduciblecomponents is non-zero. Otherwise, w is L -invariant and so it is also invariantby the image of the projection of H onto U . By our assumptions on H , weconclude that w is G -invariant, which is a contradiction. So, for some i , wehave that the projection of w to W = c W / ⊕ j = i W j is not G -invariant. Now let W (1) = ⊕ j W j / ⊕ j = i W j and W (2) = W L / W G .Clearly W / W G = W (1) ⊕ W (2) , U acts trivially on W (1) and W / W (1) , and W (1) is an irreducible L -space; more-over H fixes w = w + w , where w is the image of w in W / W G , w ∈ W (1) and w ∈ W (2) . Moreover, if w ′ ∈ ( W / W G ) G , then g · w ′ − w ′ ∈ W G ⊆ W L = W (2) .On the other hand, G acts trivially on W / W (1) , and so g · w ′ − w ′ ∈ W (1) .Therefore overall, we have that w ′ ∈ W G , i.e. w ′ = 0 as we wished. Remark 18.
From the proof of Proposition 17, it is clear that, if G is definedover Q , then there is a number field κ such that all the desired representations ρ i and ψ i are defined over κ . Lemma 19.
Let Γ be a Zariski-dense subgroup of G ⊆ GL d , a Zariski-connected Q -group, such that Γ ⊆ G ( Z S ) . Let ρ be a non-trivial representation of G whichis defined over a number field κ . Then there are p ∈ S ∪ {∞} and a place p of κ such that,1. p divides p , i.e. Q p is a subfield of κ p . . ρ (Γ) is an unbounded subset of ρ ( G ( κ p )) .Proof. If this is not the case, then ρ (Γ) is bounded in ρ ( G ( κ p )), for any p ∈ P ( κ ).Hence ρ (Γ) is a finite group. On the other hand, ρ (Γ) is Zariski-dense in ρ ( G ).Moreover we know that ρ ( G ) is Zariski-connected. Thus ρ is trivial which is acontradiction. Proof of Proposition 11.
This is a direct consequence of Proposition 16, Propo-sition 17, Remark 18 and Lemma 19.
We recall the notation from Proposition 10. G is a Zariski-connected perfect Q -group, and Γ is a Zariski dense subgroup of G ( Q ). We are given finitely manyirreducible representations ρ i for 1 ≤ i ≤ m each defined over a local field K i ,and ρ i (Γ) is unbounded. Furthermore, for each i we are given an affine space V i and a morphism ϕ i : G × V i → Aff( W ρ i ). Often we think about ϕ i ashomomorphisms from G to Aff( W ρ i ) parametrized by the elements of V i . Thenwe also write ϕ i ( · , v ) = ϕ i,v ( · ). We also recall that for any i and 0 = v ∈ V i ( K i ), ϕ i,v : G ( K i ) → Aff( W ρ i ) is a homomorphism whose linear part is ρ i , and nopoint of W ρ i is fixed under the action of G ( K i ). Our aim in this section is toprove that if we modify our generating set in an appropriate way, then only asmall fraction of our group satisfy a condition like 3.(a) or 3.(b) in Proposition10. Let A be a subset of a group that generates freely a subgroup. A reducedword over A is a product of the form g · · · g l , where g i ∈ A or g − i ∈ A and g i g i +1 = 1 for any i. We write B l ( A ) (or simply B l ) for the set of reduced wordsover A of length l . Proposition 20.
Let notation be as above. Then there is a set A ⊂ Γ generatingfreely a subgroup Γ ′ , which satisfies the following properties. Write S ′ = A ∪ e A .Then for any i and for any vector w ∈ W ρ i , we have |{ g ∈ B l | ρ i ( g )[ w ] = [ w ] }| < | B l | − c , furthermore, for any i , v ∈ V i ( K i ) with k v k = 1 and for w ∈ W ρ i we have |{ g ∈ B l | ϕ i,v ( g ) w = w }| < | B l | − c , where c is a constant depending on S and on the representations. The rest of this section is devoted to the proof of this proposition and inSection 3.3 we combine it with Proposition 10 to get Proposition 7. First weconstruct a desirable set of generators.
Proposition 21.
Let notation be as above. Then there is a symmetric set S ′ ⊂ Γ , and a number R > and for each g ∈ S ′ and ≤ i ≤ m there are twosets K ( i ) g ⊂ U ( i ) g ⊂ W ρ i such that the following hold. . For each i and g we have ρ i ( g )( U ( i ) g ) ⊂ K ( i ) g .2. For each i , any vector w = 0 ∈ W ρ i is contained in U ( i ) g for at least twoelements g ∈ S ′ .3. For each i and for any elements g , g ∈ S ′ we have K ( i ) g ⊂ U ( i ) g unless g g = 1 .4. For each i and for any elements g , g ∈ S ′ we have K ( i ) g ∩ K ( i ) g = ∅ unless g = g .5. For each i , v ∈ V i ( K i ) with k v k = 1 and g ∈ S ′ we have that if w ∈ U ( i ) g and k w k > R , then ϕ i,v ( g ) w ∈ K ( i ) g and k ϕ i,v ( g ) w k > k w k .6. For each i , v ∈ V i ( K i ) with k v k = 1 and w ∈ W ρ i , there are at least twoelements g , g ∈ S ′ such that ϕ i,v ( g ) w = w and ϕ i,v ( g ) w = w . The construction of the set S ′ relies on the notion of quasi-projective trans-formation introduced by Furstenberg [26] and further studied by Goldsheid andMargulis [29] and Abels, Margulis, and Soifer [1]. We use a slightly differentnotion also used by Cano and Seade [19], which suits better our purposes. Let b ∈ Mat d ( K ) be a not necessarily invertible linear transformation, where K isa local field. Write V + ( b ) = Im( b ) and V − ( b ) = Ker( b ). Denote by P ( K d )the projective space, and in general we denote by P ( · ) the projectivization of aconcept. Then P ( b ) : P ( K d ) \ P ( V − ( b )) → P ( K d ) is a partially defined map onthe projective space, and we call it a quasi-projective transformation.Consider a sequence { g i } ∞ i =1 ⊂ GL d ( K ). It is easy to see (see e.g. [19,Proposition 2.1]) that it contains a subsequence, still denoted by { g i } ∞ i =1 suchthat lim i →∞ g i / k g i k = b (3)uniformly for some linear transformation b : K d → K d . Here and everywherebelow k·k denotes a fixed submultiplicative matrix-norm. Moreover, this impliesthat lim i →∞ P ( g i ) = P ( b )uniformly on compact subsets of P ( K d ) \ P ( V − ( b )). Let Γ ≤ GL d ( K ) be a groupand denote by Γ the set of maps b for which (3) holds for some sequence { g i } ∞ i =1 ⊆ Γ. The following lemma, crucial for us, is statement b, in [1, Lemma4.3]. For completeness we give the proof.
Lemma 22.
Let { g i } ∞ i =1 , { h i } ∞ i =1 ⊆ GL d ( K ) be two sequences such that lim i →∞ g i / k g i k = b and lim i →∞ h i / k h i k = b (4) for some linear transformations b , b . If b b = 0 , then lim i →∞ g i h i / k g i h i k = λb b for some nonzero λ ∈ K . roof. Since the convergences in (4) are uniform, we havelim i →∞ g i k g i k ◦ h i k h i k = b b . Observe the following: If { γ i } ∞ i =1 ⊆ GL d ( K ) and γ i /λ i converge to a nonzerolinear transformation for a sequence of scalars { λ i } ∞ i =1 ⊂ K , then γ i / k γ i k isconvergent, too. This proves the lemma.This lemma implies that if b , b ∈ Γ, and b b = 0, then λb b ∈ Γ. Thisproperty is crucial for us. Denote by r the minimum of the ranks of the elementsin Γ. If b ∈ Γ is of rank r and if V + ( b ) ∩ V − ( b ) = { } for some b ∈ Γ, then b b = 0, whence V + ( b ) ⊂ V − ( b ).We will use the following lemma to construct the first element of S ′ . Thislemma is a variant of [1, Lemma 5.15]. Lemma 23.
Let G be a Zariski-connected algebraic group, and let ρ , . . . , ρ m be irreducible representations defined over local fields K i . Let Γ ≤ G ( Q ) beZariski-dense. For each ≤ i ≤ m denote by r i the minimal rank of an elementin ρ i (Γ) .Then for each ≤ i ≤ m there is a b i ∈ ρ i (Γ) and there is a sequence ofelements { h j } ∞ j =1 ⊆ Γ such that the following hold.1. For each ≤ i ≤ m , V + ( b i ) ∩ V − ( b i ) = { } and dim V + ( b i ) = r i .2. For each ≤ i ≤ m , we have lim j →∞ ρ i ( h j ) / k ρ i ( h j ) k = b i . Proof.
Let 1 ≤ k ≤ m and assume that { h j } ∞ j =1 ⊆ Γ is a sequence such thatfor each i we have ρ i ( h j ) / k ρ i ( h j ) k → b i for some linear transformation b i , and1. holds for i < k . We show below that we can replace { h j } ∞ j =1 with anothersequence such that 1. holds for i = k as well. Then the Lemma follows byinduction.Let { h ′ j } ∞ j =1 ⊆ Γ be a sequence such that ρ k ( h ′ j ) / k ρ k ( h ′ j ) k → b ′ k , where b ′ k is a linear transformation of rank r k . By taking a subsequence we can assumethat ρ i ( h ′ j ) / k ρ i ( h ′ j ) k → b ′ i for some linear transformation b ′ i for all 1 ≤ i ≤ m .Take two elements g , g ∈ Γ. We consider the sequence { e h j = g h ′ j g h j } ∞ j =1 By Lemma 22 we get that for all 1 ≤ i ≤ m we have ρ i ( e h j ) / k ρ i ( e h j ) k → λ i ρ i ( g ) b ′ i ρ i ( g ) b i provided ρ i ( g ) b ′ i ρ i ( g ) b i = 0. Then for each i , there is a nonempty Zariski-opensubset X i of G ( K i ) such that for g ∈ X i we have ρ i ( g )( V + ( b i )) * V − ( b ′ i ) . ρ i . (A more detailed argument for a similar statement will be given in theproof of Lemma 24). Now take g ∈ Γ ∩ T X i . Then ρ i ( g ) b ′ i ρ i ( g ) b i = 0 nomatter how we choose g . Similarly, there is a nonempty Zariski-open subset X ′ i of G ( K i ) such that for g ∈ X ′ i , we have ρ i ( g )( V + ( b ′ i ρ i ( g ) b i )) * V − ( b ′ i ρ i ( g ) b i ) . (5)Take g ∈ Γ ∩ T X ′ i . For i ≤ k , the rank of ρ i ( g ) b ′ i ρ i ( g ) b i is r i , and then (5)implies that V + ( ρ i ( g ) b ′ i ρ i ( g ) b i ) ∩ V − ( ρ i ( g ) b ′ i ρ i ( g ) b i ) = { } by the remarks after Lemma 22, which we wanted to show.Let { g i } ni =1 ⊆ GL d ( K ) be a sequence such thatlim i →∞ g i / k g i k = b and lim i →∞ g − i / k g − i k = e b for some non-invertible b, e b ∈ Mat d ( K ). Let w ∈ V + ( b ) and assume to thecontrary that w / ∈ V − ( e b ). Then there is some vector u ∈ K d such thatlim i →∞ g i ( u ) / k g i k = w. By uniform convergence, we then havelim i →∞ g − i ( g i ( u ) / k g i k ) k g − i k = u . for some nonzero u ∈ K d . This implies that k g i k · k g − i k is bounded whichcontradicts to the non-invertibility of b . Therefore we can conclude that V + ( b ) ⊂ V − ( e b ) and V + ( e b ) ⊂ V − ( b ).In the proof of Proposition 21 we will use Lemma 23 to produce an element g ∈ Γ with certain nice properties, and then we will define A to be a set ofappropriate conjugates of it, whom we will find using the following two lemmata. Lemma 24.
Let G be a Zariski-connected algebraic group defined over a localfield K , and let ρ be an irreducible representation of it. Let V +1 , V − , V +2 , V − ⊆ W ρ be subspaces such that V +1 ∩ V − = V +2 ∩ V − = { } and V +1 ⊆ V − and V +2 ⊆ V − . Let M be an integer and denote by X ⊆ G ( K ) M the set of M -tuples ( g , . . . , g M ) such that the following hold. If we have ρ ( g α )( V + i ) ⊆ ρ ( g β )( V − j ) for some ≤ i, j ≤ and ≤ α, β ≤ M , then α = β and i + j = 3 . Then X isa nonempty Zariski-open set.Proof. Let v , . . . , v r be a basis for V +1 and let ψ , . . . , ψ r be a basis for the spaceof functionals vanishing on V − . Then the condition ρ ( g )( V +1 ) ⊆ ρ ( g )( V − ) isequivalent to the equations h ρ ( g ) v j , ρ ( g ) ∗ ψ i i = 0 for 1 ≤ i, j, ≤ r . The other25onditions can be described in terms of algebraic equations similarly, whencethe Zariski-openness follows.It is clear that there is an M -tuple ( g , . . . , g M ) for which the single condition ρ ( g )( V +1 ) * ρ ( g )( V − ) is satisfied. For example we can take g = 1 pick avector w ∈ V +1 and choose g in such a way that ρ ( g ) w / ∈ V − , the existenceof g follows from irreducibility. It is a similar argument to show that theother constraints can be satisfied, so X , being the intersection of finitely manynonempty Zariski-open sets, is nonempty. Lemma 25.
Let G be a Zariski-connected algebraic group, and let ρ be anirreducible representation of it. Let V be an affine space and let ϕ : G × V → Aff( W ρ ) be a morphism such that the linear part of ϕ v is ρ . Assume thatfor some = v ∈ V and w ∈ W ρ there is an element g ∈ G ( K ) such that ϕ v ( g ) w = w .Then for M ≥ W ρ ) + dim( V ) + 1 , there is a nonempty Zariski-openset X ⊂ G ( K ) M such that if ( g , . . . , g M ) ∈ X then the following hold.1. Let w ∈ W ρ and W ( W ρ be a proper linear subspace. Then for any setof indices I ⊂ { , . . . , M } with | I | = 2 dim( W ρ ) − , there is some i ∈ I such that ρ ( g i ) w / ∈ W .2. Let v ∈ V ( K ) , w ∈ W ρ and W ⊂ W ρ be an affine subspace, then for anyset of indices I ⊂ { , . . . , M } with | I | = 2 dim( W ρ ) + dim( V ) + 1 , there issome i ∈ I such that ϕ v ( g i ) w / ∈ W .Proof. We only show that property 1. can be satisfied, 2. is similar, and thenwe can take the intersection of the two sets. Moreover, it is enough to showthat 1. can be satisfied for the index set I = { , . . . , W ρ ) − } . Considerthe algebraic variety P ( W ρ ) × P ( W ∗ ρ ) × G ( K ) M . Consider also the subvariety Y = { ([ w ] , [ ψ ] , g , . . . , g M ) : h ρ ( g i )( w ) , ψ i = 0 for 1 ≤ i ≤ W ρ ) − } . By the irreducibility of ρ it follows that for ([ w ] , [ ψ ]) ∈ P ( W ρ ) × P ( W ∗ ρ ) fixed,the variety Y w,ψ = { g ∈ G ( K ) : h ρ ( g )( w ) , ψ i = 0 } is a proper subvariety of G ( K ), hence dim( Y w,ϕ ) ≤ dim( G ) −
1. This impliesthat the fiber of Y over ([ w ] , [ ψ ]) is of codimension at least 2 dim( W ρ ) − G ( K ) M . Now let Z be the Zariski-closure of the image of Y under the projectionmap P ( W ρ ) × P ( W ∗ ρ ) × G ( K ) M → G ( K ) M . Then dim( Z ) ≤ dim( Y ), hence Z is a proper subvariety, and by constructionits complement satisfies 1. for I = { , . . . , W ρ ) − } .26 roof of Proposition 21. If ρ is a representation of G , then write e ρ for the rep-resentation that associates the transpose inverse of ρ ( g ) for every g ∈ G . ApplyLemma 23 to the representations ρ , . . . , ρ m , e ρ , . . . , e ρ m . We get a sequence { h i } ∞ i =1 ⊂ Γ and linear transformations b (1)1 , . . . , b (1) m , b (2)1 , . . . , b (2) m with the fol-lowing properties. For each 1 ≤ i ≤ m , we havelim j →∞ ρ i ( h j ) / k ρ i ( h j ) k = b (1) i and lim j →∞ ρ i ( h − j ) / k ρ i ( h − j ) k = b (2) i . Furthermore we have that dim( V + ( b (1) i )) = dim( V + ( b (2) i )) = r i and V + ( b ( j ) i ) ∩ V − ( b ( j ) i ) = { } . Here and everywhere below r i denotes the minimal rankof the elements of ρ i (Γ). By the remarks preceding Lemma 24 we see that V + ( b ( j ) i ) ⊂ V − ( b (3 − j ) i ) for j = 1 ,
2. Let d be the maximum of the dimen-sions of the representation spaces W ρ i and parameter spaces V i . Apply Lemma24 with M = 3 d + 2 for each ρ i and for the subspaces V + j = V + ( b ( j ) i ) and V − j = V − ( b ( j ) i ), j = 1 ,
2. This way we get Zariski-open subsets X i ⊂ G ( K i ) M .Also apply Lemma 25 for the representations ρ i and for the morphisms ϕ i , thisgives Zariski-open subsets X ′ i ⊂ G ( K i ) M . Since Γ is Zariski dense, we get ele-ments g , . . . , g M ∈ Γ such that ( g , . . . , g M ) ∈ X i ∩ X ′ i for all i , hence they havethe following properties. Recall that if c , c ∈ ρ i (Γ), and dim( V + ( c )) = r i ,then either V + ( c ) ∩ V − ( c ) = { } or V + ( c ) ⊂ V − ( c ). For each i we have ρ i ( g α )( V + ( b ( j ) i )) ∩ ρ i ( g β )( V − ( b ( k ) i )) = { } for every 1 ≤ j, k ≤ ≤ α, β ≤ M , except for α = β and i + j = 3. Using1. in Lemma 25 with W = V − ( b (1) i ), we also have that ρ i ( g α )( V − ( b (1) i )) ∩ . . . ∩ ρ i ( g α d − )( V − ( b (1) i )) = { } for any 1 ≤ α < . . . < α d − ≤ M .We show that if we set A = { g h j g − , . . . , g M h j g − M } and j is large enoughthen we can choose the sets K ( i ) g and U ( i ) g in such a way that the propositionholds. At this point we fix i , and omit the corresponding indices everywhere.For a set X ⊂ P ( K d ) denote by B ε ( X ) the set of points which are of distanceat most ε from X with respect to any fixed metric which induces the standardtopology on P ( K d ). Let ε > V , . . . V l areamong the subspaces ρ ( g k )( V ± ( b ( j ) )), then B ε ( P ( V )) ∩ . . . ∩ B ε ( P ( V l )) = ∅ only if P ( V ) ∩ . . . ∩ P ( V l ) = ∅ . For g = g k h j g − k ∈ A define K g = { w ∈ K d \{ }| [ w ] ∈ B ε ( P ( ρ ( g k )( V + ( b (1) )))) } and U g = { w ∈ K d \{ }| w ] / ∈ B ε ( P ( ρ ( g k )( V − ( b (1) )))) } . We define K g and U g in a similar manner for g ∈ e A , but we use b (2) instead of b (1) . Properties 2., 3. and 4. can be deduced immediately from the properties27f the spaces ρ ( g k )( V + ( b (1) )) and ρ ( g k )( V − ( b (1) )) provided ε is small enough.We remark that property 4. follows from property 3., since by construction K g ∩ U g − = ∅ .Property 1. holds if j is large enough, since P ( ρ ( g k h j g − k )) converges to P ( ρ ( g k ) b (1) ρ ( g − k )) uniformly on compact subsets of P ( W ρ ) \ P ( V − ( b (1) )). Forproperty 5, we note that there is a constant c > ε such that k ρ ( g k h j g − k )( w ) k > c k ρ ( g k h j g − k ) k · k w k for w ∈ U g . Since lim j →∞ k ρ ( g k h j g − k ) k = ∞ , we have c k ρ ( g k h j g − k ) k > j . Then for k w k > R large and v ∈ V ( K ), k v k = 1, the translation com-ponent of ϕ v ( g k h j g − k ) is negligible compared to the linear part, and property5 follows. Here we used that the unit ball in V ( K ) is compact, hence for j fixed, the translation part of ϕ v ( g k h j g − k ) is bounded in v . Finally, denote by W ⊂ W ρ the possibly empty affine subspace that consist of the fixed points of ϕ v ( h j ). Then the set of fixed points of ϕ v ( g k h j g − k ) is g k ( W ). From this we seethat property 6. follows from part 2. of Lemma 25. Proof of Proposition 20.
Let A ⊂ Γ be the set that we constructed in Proposi-tion 21, and let K ( i ) g and U ( i ) g be the corresponding sets. In what follows we fix i and omit the corresponding indices.We deal with the two parts separately, first we estimate the size of the set { g ∈ B l | ρ ( g )[ w ] = [ w ] } . For 0 ≤ k < l denote by X k the set of those reducedwords g l · · · g ∈ B l for which ρ ( g k · · · g ) w ∈ U g k +1 , and k is the smallest indexwith this property. We write X l for those words which are not contained in anyof the X k with k < l . Let g l · · · g ∈ X k . We remark that by the properties ofthe sets U g and K g , we have ρ ( g k +1 · · · g ) w ∈ K g k +1 ⊂ U g k +2 . In fact, by induction we can conclude that ρ ( g j · · · g ) w ∈ K g j for j > k . Assumefurther that ρ ( g l · · · g )[ w ] = [ w ]. Then we also have ρ ( g − j +1 · · · g − l )[ w ] = ρ ( g j · · · g )[ w ] ∈ P ( K g j )for j > k . Since the sets K g are disjoint, we see that [ w ] determines g j uniquelyfor j > k . Indeed, once g l , . . . , g j +1 are known, they determine which of thesets P ( K g j ) does ρ ( g − j +1 · · · g − l )[ w ] belong to. On the other hand we know thatfor j ≤ k , we have ρ ( g j − · · · g ) w / ∈ U g j . Since ρ ( g j − · · · g ) w is covered by atleast two of the sets U g , we have at most | S ′ | − g j . Thereforewe have |{ g ∈ B l | ρ ( g )[ w ] = [ w ] } ∩ X k | ≤ ( | S ′ | − k , from where the first part of the proposition follows easily.Now we give an estimate for |{ g ∈ B l | ϕ v ( g ) w = w }| . We show that thereis an integer k such that for any v ∈ V ( K ) with k v k = 1, w ∈ W ρ and g ∈ S ′ there is a reduced word ω ∈ B k of length k with the following properties.The first letter of ω is not g , and we have k ϕ v ( ω ) w k > R , k ϕ v ( ω ) w k > k w k ϕ v ( ω ) w ∈ K g ′ , where g ′ is the last letter of ω . If | w | > R , this is easy,since there are at least two letters g ′ ∈ S ′ such that w ∈ U g ′ . We can alsomake an existing word longer, since we can preserve the required properties nomatter how we continue it as long as it stays reduced. Consider the case when k w k ≤ R . Denote by D ⊂ W ρ the solid ball of radius R . To each reduced word ω we associate a set E ω ⊂ { v ∈ V ( K ) : k v k = 1 } × W ρ defined by E ω = { ( v, x ) : x ∈ ϕ v ( ω − )( D ) } . We need to show that there is a number k such that \ ω ∈ B k : ω does not contain g E ω = ∅ . Since the E ω are compact, it is enough to show that \ ω : ω does not contain g E ω = ∅ . We show that for each v ∈ V ( K ) with k v k = 1,( { v } × W ρ ) ∩ \ ω : ω does not contain g E ω = ∅ Assume to the contrary that there are at least two points( v , w ) , ( v , w ) ∈ { v } × W ρ ∩ \ ω : ω does not contain g E ω . Then w − w ∈ U h for some g = h ∈ S ′ . Property 5 in Proposition 21implies that there is some c > k ρ ( h ) w k > c k w k for all w ∈ U h .Then 2 R > k ϕ v ( h l ) w − ϕ v ( h l ) w k > c l k w k , a contradiction. Assume to thecontrary that { v } × W ρ ∩ \ ω : ω does not contain g E ω = { ( v , w ) } for a point w ∈ W ρ . Then w is fixed by all elements of S ′ except maybe for g ,which contradicts to property 6 in Proposition 21. So far we showed all requiredproperties of ω except that ϕ v ( ω ) w belong to the right set K g ′ . However, byproperties 2 and 5 in Proposition 21 there are at least two letters that we canappend to ω to fulfill these last requirement as well. For at least one of thesetwo, the word stays reduced.Now consider a reduced word g l · · · g for which ϕ v ( g l · · · g ) w = w . Thenwe also have for all 1 ≤ j < l/kϕ v ( g jk · · · g g l · · · g jk +1 )( ϕ v ( g jk · · · g ) w ) = ϕ v ( g jk · · · g ) w. (6)The above argument shows that out of the ( | S ′ |− k possibilities for g ( j +1) k · · · g jk +1 ,there is at least one for which (6) does not hold since the vector on the left29and side is longer than the one on the right. Although g jk · · · g g l · · · g jk +1 may not be reduced, if j < l/k −
1, we still get a reduced word ending with g ( j +1) k · · · g jk +1 after all possible reductions. This shows that |{ g ∈ B l | ϕ v ( g ) w = w }| < (cid:18) ( | S ′ | − k − | S ′ | − k (cid:19) l/k − | B l | giving the second half of the proposition. We show that the proposition holds if S ′ is the set of generators constructedin Proposition 20. Let q be a square-free integer and H < π q (Γ) a subgroup.Denote by q the product of those prime factors of q which are large enough sothat Propositions 10 and 20 hold. Let H = π q ( H ) < π q (Γ). Then clearly[ π q (Γ) : H ] ≥ q /q [ π q (Γ) : H ] ≫ [ π q (Γ) : H ] , hence if we show the claim for q and H , it will follow for q and H as well witha worse implied constant. Form now on we assume that q = q .Let H ♯ be the subgroup corresponding to H in Proposition 10. Let l ≤ c log[ π q (Γ) : H ] be an integer, where c is a sufficiently small constant. Let h ∈ Γ be such that π q ( h ) ∈ H ♯ and h ∈ B l , where B l is the set of reduced wordsof length l over the alphabet S ′ . If c is small enough, then k h k < [ π q (Γ) : H ♯ ] δ with the same δ for which Proposition 10 holds. Then by definition, h ∈ L δ ( H ♯ ).If we combine Propositions 10 and 20, then we get | B l ∩ L δ ( H ♯ ) | < | B l | − c for some c > | S ′ | = 2 M Set P k ( l ) = χ (2 k ) S ′ ( ω ), where ω ∈ B l . Since | B l | = 2 M (2 M − l − for l ≥
1, 1 = P k (0) + X l ≥ | B l | P k ( l ) . (7)By a result of Kesten [41, Theorem 3.], we havelim sup k →∞ ( P k (0)) /k = (2 M − /M . From general properties of Markov chains (see [62, Lemma 1.9]) it follows that P k (0) ≤ (cid:18) M − M (cid:19) k . Since χ (2 k ) S ′ is symmetric, we have P k (0) = P g [ χ ( k ) S ′ ( g )] , hence P k ( l ) ≤ P k (0)30or all l by the Cauchy-Schwartz inequality. Now we can write for k ≤ c log q/ χ (2 k ) S ′ ( L δ ( H ♯ )) = X l | B l ∩ L δ ( H ♯ ) | P k ( l ) ≤ X l | B l | − c P k ( l ) ≤ X l ≤ k/ (2 M ) l (cid:18) M − M (cid:19) k +(2 M − − c k/ X l ≥ k/ | B l | P k ( l ) < (2 M ) k/ M k + (2 M − − c k/ . The inequality between the third and fourth lines follows form (7).Note that π q [ χ (2 k ) S ′ ]( H ♯ ) is non-increasing with k . Let g ∈ π q ( H ), since S ′ issymmetric, we have π q [ χ (2 k ) S ′ ]( gH ♯ ) ≤ π q [ χ (4 k ) S ′ ]( H ♯ ) . Thus π q [ χ (2 k ) S ′ ]( H ) ≤ C n (cid:16) π q [ χ (4 k ) S ′ ]( H ♯ ) (cid:17) / , and this finishes the proof, since any positive power of q dominates C n if q islarge enough. In this section we aim to prove Proposition 8, but first we recall some results wewill need later on. We fix a square-free integer q = p · · · p n , and assume thateach prime divisor p i of q is bigger than some large but fixed constant. Then aswe saw at the beginning of Section 3, we have G := π q (Γ) = G p × . . . × G p n , where G p i = π p i (Γ) = L p i ⋉ U p i and L p i is a product of quasi-simple groupsgenerated by their elements of order p i and U p i is a p i -group. Furthermore,we have that U p i / [ U p i : U p i ] is isomorphic to ( F d i p i , +) for some integers d i which are bounded independently of q . We write L = L p × . . . × L p n and U = U p × . . . × U p n . The following result is a statement about abstract groupssatisfying certain assumptions that we will recall later in Section 4.2. We willalso show there that the group L satisfy these assumptions with parametersthat are independent of q . 31 roposition B ([61, Proposition 14]) . Let G be a group satisfying (A0)–(A5).For any ε > there is a δ > depending only on ε and the constants in (A0)–(A5) such that the following holds. If A ⊆ G is symmetric such that | A | < | G | − ε and χ A ( gH ) < [ G : H ] − ε | G | δ for any g ∈ G and any proper H < G , then | Q A | ≫ | A | δ . Assumptions (A0)–(A5) are more or less straightforward to check exceptfor (A4) which basically boils down to showing Proposition B for quasi-simplegroups that are the direct factors of L p i . This statement was first proved byHelfgott for SL ( Z /p Z ) [35] and for SL ( Z /p Z ) [36], and later it was extendedby Dinai [22] for SL ( F q ) for an arbitrary finite field F q . Now the statementis known for all finite simple groups of Lie type due to a recent breakthroughby Breuillard, Green, Tao [15], and Pyber, Szab´o [52]. We can either use [52,Theorem 4] or [15, Corollary 2.4]. The statement in the formulation of [52,Theorem 4] is the following. Theorem C.
Let L be a simple group of Lie type of rank r and A a generatingset of L . Then either Π A = L or | Π A | ≫ | A | ε , where ε and the impliedconstant depend only on r . The following useful Lemma is based on the Balog-Szemer´edi-Gowers Theo-rem and it is implicitly contained in [7].
Lemma D ([61, Lemma 15]) . Let µ and ν be two probability measures on anarbitrary group G and let K > be a number. If k µ ∗ ν k > k µ k / k ν k / K then there is a symmetric set A ⊂ G with K R k µ k ≪ | A | ≪ K R k µ k , | Q A | ≪ K R | A | andmin g ∈ A ( e µ ∗ µ ) ( g ) ≫ K R | A | , where R and the implied constants are absolute. Using the Lemma it is very easy to deduce Proposition 8 from the following
Proposition 26.
Let G be a Zariski-connected perfect algebraic group definedover Q . Let Γ < G ( Q ) be a finitely generated Zariski-dense subgroup. Thenfor any ε > , there is some δ > depending only on ε and G such that thefollowing holds. Let q be a square-free integer without small prime factors.If A ⊆ G q is symmetric such that | A | < | G q | − ε and χ A ( gH ) < [ G q : H ] − ε | G q | δ for any g ∈ G q and any proper H < G q , then | Q A | ≫ | A | δ .
32e defer the proof to the following sections, and now we show how it impliesthe
Proof of Proposition 8.
Assume that the conclusion of the proposition fails, i.e.that there is an ε such that for any δ , there is a q and there are probabilitymeasures µ and ν with k µ k > | G q | − / ε and µ ( gH ) < [ G q : H ] − ε for any g ∈ G q and for any proper H < G q , and yet k µ ∗ ν k ≥ k µ k / δ k ν k / . Take K = k µ k − δ in Lemma D. Note that by the third property of the set A , wehave χ A ( gH ) ≪ K R e µ ∗ µ ( gH ) ≤ K R max h ∈ G q µ ( hH ) ≪ | G q | Rδ [ G q : H ] − ε . Now | Q A | ≪ K R | A | contradicts Proposition 26, if δ is small enough. In fact,when q contains small prime factors, Proposition 26 does not apply, but we stillget a contradiction for π q ′ ( A ) and the group G q ′ , where q ′ is the product of theprime factors of q which are not too small for Proposition 26. Also note thatwhen q is smaller than a fixed constant we can get the contradiction by thetrivial inequality | Q A | ≥ | A | + 1. As mentioned before, Proposition B and Theorem C together imply Proposition26, when G is semisimple. When the unipotent radical U is nontrivial, we need todo some work which is carried out in this section. Recall the definition of G and L from the beginning of Section 4. Denote by pr the projection homomorphism G → L .The purpose of this section is to prove the following Proposition 27.
For every ε > there is an integer C such that the followingholds. Let A ⊆ G be a symmetric set such that pr( A ) = L , and χ A ( gH ) < [ G : H ] − ε | G | /C for any g ∈ G and any proper H < G . Then π q [ Q C A ] = G q for some q > q − ε . We need a couple of Lemmata. Let b G = b G × . . . × b G n be a direct productof groups. For each i , let β i : b G i → b L be a given homomorphism into a group b L .Denote by β : b G → b L the homomorphism induced by the β i in the obvious way.For each i write pr i : b G → b G i for the projection homomorphisms. We introducethe following distance for two elements g , g ∈ b G : d ( g , g ) = X i :pr i ( g ) =pr i ( g ) log | Ker( β i ) | . (8)33 emma 28. Let A ⊆ b G be a symmetric set with β ( A ) = b L and ∈ A . Assumethat for every g ∈ A.A.A with β ( g ) = 1 we have d (1 , g ) ≤ ε log | Ker( β ) | for some ε > . Then A can be covered with at most n | Ker( β ) | ε cosets of a subgroupof b G of order at most | b L | . We will apply this Lemma in the following setting: We will have normal sub-groups N p i E U p i which are normal in G p i as well, and we will set b G i = G p i /N p i and b L = L . The homomorphism β i will be the projection L p i ⋉ ( U p i /N p i ) → L p i .The purpose of the lemma is to find an element g in a product-set of A which isin the kernel of pr, but have a large conjugacy class. In a subsequent lemma wewill recover the normal subgroup generated by g in the product-set of a boundednumber of copies of A . This will allow us to increase N p i , and proceed to thenext step of the iteration. Proof of Lemma 28.
Let ψ : b L → b G be a map such that β ◦ ψ = Id , and ψ ( b L ) ⊆ A , this is possible due to the assumption β ( A ) = b L . By assumption, wehave d ( g − ψ ( β ( g )) , < ε log | Ker( β ) | for any g ∈ A , hence d ( ψ ( β ( g )) , g ) < ε log | Ker( β ) | . (9)Moreover, for any g, h ∈ b L , we have d ( ψ ( g ) ψ ( h ) , ψ ( gh )) < ε log | Ker( β ) | and d ( ψ ( g ) − , ψ ( g − )) < ε log | Ker( β ) | . These two inequalities mean that ψ is an ε | Ker( β ) | -homomorphism of type IIwith respect to d in the sense of Farah, see [25, Section 1]. Then by [25, Theorem2.1], there is a homomorphism ϕ : b L → b G such that d ( ψ ( g ) , ϕ ( g )) < ε log | Ker( β ) | for every g ∈ b L . Combining this with (9), we get that for every element g ∈ A ,there is h ∈ ϕ ( b L ) such that d ( g, h ) < ε log | Ker( β ) | . By definition, this meansthat there is an index set I ⊂ { , . . . , n } such that pr i ( g ) = pr i ( h ) if i / ∈ I and Q i ∈ I | Ker( β i ) | < | Ker( β ) | ε . For a fixed I , the elements g which satisfy thiscondition can be covered by at most | Ker( β ) | ε cosets of the group ϕ ( b L ). Thisproves the claim, because we have 2 n possibilities for I .As already promised, we show that we can recover the normal subgroup gen-erated by the element constructed in the previous lemma. We need to introducemore notation. Let p , . . . , p n be primes and with the notation as above, assumethat b G i is generated by its p i -elements. Assume that b G i = b L i ⋉ b U i is a semidi-rect product and b L = b L × . . . × b L n and that the kernel of β i is b U i . We assumefurther that b U i is isomorphic to ( F d i p i , +). Then writing b U i additively, we can as-sociate to it an F p i -vector space M i which is an b L i -module such that the action v g · v of g ∈ b L i on M i descends from the conjugation action u gug − of g ∈ b G i on b U i . We note that since b U i is commutative, its action by conjugationis trivial on itself, so the above is well-defined. Write b U = b U × . . . × b U n .34 emma 29. Let p , . . . , p n , b G i , b L i , b U i and M i satisfy the above assumptions.Furthermore, assume that no M i contain a one dimensional composition factor.Let A ⊆ b G be a symmetric set with ∈ A and β ( A ) = b L and let g ∈ A be anyelement with β ( g ) = 1 . Denote by N the smallest normal subgroup of b G thatcontains g .Then there is a constant c depending only on max d i such that Q c A ⊇ N. The proof of Lemma 29 requires
Lemma 30.
Let p be a prime and let H ⊆ GL( M ) be a group generated by its p -elements, where M is a vector space of dimension d over F p and p ≫ d . Assumethat no non-zero vector is fixed by H , i.e. the trivial representation is not a sub-representation of M . Then there is a constant c = c ( d ) , only depending on d ,such that P c H · v contains a non-zero H -subspace, for any = v ∈ M . Here and everywhere below H · v denotes the orbit of v under the action of H on M . Proof.
Let g ∈ H be an element of order p . Then x = g − ∈ End( M ) is anilpotent element. Let k be the largest integer such that x k is not zero. It isclear that k is at most d . So X d H ∋ ( g j − k = ((1 + x ) j − k ∀ ≤ j ≤ p − x j − X i =0 (cid:18) ji + 1 (cid:19) x i ! k = j k x k since x k +1 = 0.Since any element of F p can be written as the sum of at most k k -th powers, wehave that P d d H ⊇ F p x k . Thus P d d H ⊇ F p Hx k H , and therefore X d d H ⊇ h x k i , where h x k i is the ideal generated by x k in A = F p [ H ], the F p -span of H inEnd( M ).Now, we prove the lemma by induction on d . If A · v is a proper H -subspace,we get the claim by the induction hypothesis. If not, then h x k i · v is a non-zero H -subspace of P d d H · v , as we wished. Note that, if h x k i· v = 0 and A· v = M ,then h x k i · M = 0, which is a contradiction as M is a faithful A -module. Corollary 31.
Let p be a prime and let H ⊆ GL( M ) be a group generated byits p -elements, where M is a vector space of dimension d over F p and p ≫ d .Assume that none of the composition factors of M is one-dimensional. Thenthere is a constant c = c ( d ) , depending only on d , such that P c H · v is equal tothe H -subspace generated by v .Proof. Using Lemma 30, one can easily prove this, by induction on d .35 roof of Lemma 29. Since b U = b U × · · · × b U n is a normal subgroup of b G , we geta homomorphism θ from b G to Aut( b U ), b G acting on b U by conjugation. As b U iscommutative, θ factors through b L and we get back the action of b L i on b U i , forany i . Moreover, θ commutes with the projection homomorphisms pr i .Let g = ( u , . . . , u n ), where u i ∈ b U i , for any i . Denote by N i the normalsubgroup generated by u i in b G i . Translating Corollary 31 to the language ofmultiplicative groups, we get a constant c , depending only on max d i , such that Q c { h ( u , . . . , u n ) h − : h ∈ A } ⊇ N × · · · × N n . It is clear that N × · · · × N n is a normal subgroup of b G which contains g . Thus Q c A = Q c A.A. e A ⊇ N, as we wished.The above lemmata allows us to deal with the case when U is commutative.In the general case we will work with U/ [ U, U ], and recover a subset of U whichprojects onto U/ [ U, U ]. Then we will use the following lemma to recover U .This lemma is very similar to the main idea behind the papers [28] and [21]. Lemma 32.
Let b U be a finite k -step nilpotent group generated by m elements,and let A ⊆ b U be a subset such that A. [ b U , b U ] = b U . Then Q C ( k,m ) A = b U .Proof. Consider the lower central series b U = Γ ⊲ Γ ⊲ . . . ⊲ Γ k +1 = { } definedby Γ i +1 = [ b U , Γ i ]. Then for 1 ≤ i ≤ k , K i = Γ i / Γ i +1 is a commutative group. Itis well known (see [63, Corollary 1.12]) that for any i, j we have [Γ i , Γ j ] ⊆ Γ i + j and for any x, y, z ∈ U we have the identities (see [63, equations 1.4 and 1.5]):[ x, yz ] = [ x, z ][ x, y ] z [ xy, z ] = [ x, z ] y [ y, z ] , where x y = y − xy . Therefore the maps ϕ i : K × K i → K i +1 defined by ϕ i ( g Γ , h Γ i +1 ) = [ g, h ]Γ i +2 are well-defined, and they are homomorphisms in both variables.We show that for any i Q m ϕ i ( K , K i ) = K i +1 . (10)Let x , . . . , x m be generators for K . Then any element of K i +1 is of the form ϕ i ( x a , · · · x a ,m m , y ) · · · ϕ i ( x a l, · · · x a l,m m , y l )for some a · , · ∈ Z and y j ∈ K i . Using that ϕ i is a homomorphism in the firstvariable, we can expand this, then we can collect the factors containing x k using36he commutativity of K i +1 and finally we use that ϕ i is a homomorphism in thesecond variable and get that the above is equal to ϕ i ( x , y a , · · · y a l, l ) · · · ϕ i ( x m , y a ,m · · · y a l,m l ) . This proves the claim.To prove the lemma, we note that the above claim implies that if ( Q C A )Γ i +1 ⊇ Γ i , then ( Q m (2 C +2) A )Γ i +2 ⊇ Γ i +1 . This proves the lemma by induction, and approximating an element successivelyin b U / Γ i for larger and larger values of i . Proof of Proposition 27.
Let
A, G, L, U and pr be the same as in the proposition.First we prove the proposition in the case, when U is commutative. Then each U p i is isomorphic to ( F d i p i , +) for some integers d i . Denote d = max d i . For each p i , we give a sequence of normal subgroups { } = N (0) p i E N (1) p i E . . . E N ( l ) p i E U p i such that each of them is a normal subgroup in G p i as well. Write N ( m ) = N ( m ) p × . . . × N ( m ) p n They will satisfy the following properties:( Q C A ) N ( k − ⊇ N ( k ) (11)[ N ( k ) : N ( k − ] ≥ [ U : N ( k − ] ε/ d (12)[ U : N ( l ) ] < q ε , (13)where C is a constant depending only on d .Assume that m >
0, and N ( k ) p i is defined for k < m and they satisfy (11)and (12). If [ U : N ( m − ] < q ε , then we can set l = m −
1, and we are done.Assume the contrary. To apply Lemma 28 we take b G i = G p i /N ( m − p i , L = b L and we let β i : b G i → b L be the homomorphism induced by pr. Consider thegroup b G = G/N ( m − and the set A = AN ( m − ⊂ b G . By assumption, A ⊂ G cannot be covered with less than | Ker( β ) | ε | G | − /C cosets of a subgroup of G of index | Ker( β ) | . We can assume that C is so large that | G | /C < | Ker( β ) | ε/ and q is so large that 2 n < | Ker( β ) | ε/ . Then A cannot be covered by lessthan 2 n | Ker ( β ) | ε/ cosets of a subgroup of b G of order | b L | . Using Lemma 28 wefind an element g ∈ A.A.A such that β ( g ) = 1 and d ( g, > ε log | Ker( β ) | / d ( · , · ) is defined by (8). Let N ( m ) p i be the smallest normal subgroup of G p i that contains N ( m − p i and π p i ( g ). Then (11) follows from Lemma 29 appliedfor b G = G/N ( m − and A = A.A.A . However, we need to check the conditionthat the L p i -modules M i defined in the paragraph preceding Lemma 29 donot contain one dimensional composition factors. Suppose the contrary. Wecan assume that one of the M i contains the trivial representation as a sub-representation, actually for this purpose we might need to enlarge N ( m − . By[34, Theorem A], it follows that M i is completely reducible, hence we can write37 i = M ′ i ⊕ M ′′ i as the sum of L p i -modules such that the action on M ′ i is trivial.Then there is a proper normal subgroup N ⊳ U p i of G p i corresponding to M ′′ i such that G p i acts trivially on U p i /N , hence G p i /N is isomorphic to the directproduct L p i × ( U p i /N ). This contradicts to the assumption that G and hence G p is perfect.Let q ′ be the product of primes p i for which N ( m − p i = N ( m ) p i . Then q ′ ≥ e d ( g, /d > | Ker( β ) | ε/ d , since | Ker( β i ) | ≤ p di . The groups U p i are p i -groups, hence [ N ( m ) : N ( m − ] ≥ q ′ .This implies (12), since [ U : N ( k − ] = | Ker( β ) | . Therefore we proved equations(11)–(13).Equations (11) and (13) together imply that there is an integer q > q − ε such that π q ( Q lC A ) ⊇ U q . Since pr( A ) = L and G q = L q U q , this proves the proposition when U iscommutative.In the general case, running the above argument for the group G/ [ U, U ], weget π q ( Q lC A ) . [ U q , U q ] ⊇ U q . Then Lemma 32 applied for the group b U = U q and for the set π q ( Q lC A ) ∩ U q finishes the proof.It is worth mentioning that the result from [34] depends on the classificationof the finite simple groups. But we do not really need this result as the involvedrepresentations are coming from a representation over Q and therefore, for largeenough p , the picture modulo p is similar to the picture over Q . L ( Z /q Z ) We list the assumptions mentioned in Proposition B. When we say that some-thing depends on the constants appearing in the assumptions (A1)–(A5) wemean C and the function δ ( ε ) for which (A4) holds.(A0) L = L × · · · × L n is a direct product, and the collection of the factorssatisfy (A1)–(A5) for some sufficiently large constant C .(A1) There are at most C isomorphic copies of the same group in the collection.(A2) Each L i is quasi-simple and we have | Z ( L i ) | < C .(A3) Any non-trivial representation of L i is of dimension at least | L i | /C .(A4) For any ε >
0, there is a δ > µ and ν are probability measures on L i satisfying k µ k > | L i | − / ε and µ ( gH ) < | L i | − ε g ∈ L i and for any proper H < L i , then k µ ∗ ν k ≪ k µ k / δ k ν k / . (14)(A5) For some m < C , there are classes H , H , . . . , H m of subgroups of L i having the following properties.( i ) H = { Z ( L i ) } .( ii ) Each H j is closed under conjugation by elements of L i .( iii ) For each proper H < L i , there is an H ♯ ∈ H j , for some j , with H . C H ♯ .( iv ) For every pair of subgroups H , H ∈ H j , H = H , there is some j ′ < j and H ♯ ∈ H j ′ for which H ∩ H . C H ♯ .One may think about (A5) that there is a notion for dimension of the sub-groups of L i .In this section, we will check these assumptions. In the beginning of Sec-tion 3, we have already checked (A1) and (A2). By a result of V. Landazuriand G. Seitz [44], we also know that (A3) holds.Assume that (A4) does not hold for L i ; then there is an ε such that for any δ one can find probability measures on L i with the following properties: k µ k > | L i | − / ε , (15) µ ( gH ) < | L i | − ε , ∀ g ∈ G, ∀ H (cid:12) G, (16) k µ ∗ ν k ≥ k µ k / δ k ν k / . (17)One can easily see that ε is less than 1 /
2. Choose δ such that Rδ is less than ε/ (1 / − ε ) and 2 ε / (1 + ε ), where ε is the constant from Theorem C and R is the constant from Lemma D.By Lemma D and (17), there is a symmetric subset A of L i such that, k µ k − δ ′ ≪ | A | ≪ k µ k − − δ ′ , (18) | Π A | ≪ k µ k − δ ′ | A | , (19)min s ∈ A (˜ µ ∗ µ )( s ) ≫ k µ k − δ ′ | A | , (20)where δ ′ = Rδ and the implied constants are absolute.First we claim that A is a generating set of L i . If not, it generates a propersubgroup H . Hence, on one hand, we have that(˜ µ ∗ µ )( A ) ≤ (˜ µ ∗ µ )( H ) = X h ∈ H (˜ µ ∗ µ )( h )= X h ∈ H X g ∈ L i µ ( g ) µ ( gh )= X g ∈ L i µ ( g ) µ ( gH ) < | L i | − ε by (16) . (21)39n the there hand(˜ µ ∗ µ )( A ) ≫ k µ k δ ′ by (20), > | L i | δ ′ ( − / ε ) by (15) . (22)We get a contradiction, by (21), (22) and δ ′ < ε/ (1 / − ε ).Assume Π A = L i ; then since A is a generating set of L i , by Theorem C, | Π A | ≫ | A | ε . Hence, by (19) and (18), we have that k µ k ( − δ ′ ) ε ≪ | A | ε ≪ k µ k − δ ′ . (23)We get a contradiction by (23) and δ ′ < ε / (1 + ε ).So we have Π A = L i . By (19), (18) and (15), we have | L i | = | Π A | ≪ k µ k − δ ′ | A | ≪ k µ k − − δ ′ < | L i | (1 / − ε )(2+2 δ ′ ) , which is a contradiction by δ ′ < ε/ (1 / − ε ). Overall we showed that (A4) holdsfor L i .As L i is a quasi-simple finite group over a finite field which is of a boundeddegree extension of its prime field, property (A5) is a direct consequence of [61,Proposition 24]. Let N be a constant such that | G p | < p N for all p . We consider two cases. Thefirst case is when | pr( A ) | < q − ε/ NC | L | , where C is a constant such that anynontrivial representation of L p is of dimension at least p /C (cf. assumption (A3)in section 4.2). As we have seen in section 4.2, the group L satisfies the assump-tions (A0)–(A5), hence Proposition B is applicable. Then we get | Q pr( A ) | ≫| pr( A ) | δ . By the pigeonhole principle A contains at least | A | / | pr( A ) | elementsof a coset of Ker(pr). Then it follows that | Q A | ≫ | pr( A ) | δ | A | . Note that | pr( A ) | > | L | ε | G q | − δ by the assumption we made in the proposition on the set A . This proves the proposition in the first case (see (24) below).Now we consider the second case, i.e. when | pr( A ) | ≥ q − ε/ NC | L | . Then by[11, Lemma 5.2], there is a set A ′ ⊂ pr( A ) and integers K j such that for every g ∈ A ′ we have |{ x ∈ L p j : ∃ h ∈ A ′ s . t . π p ··· p j − ( h ) = π p ··· p j − ( g ) and π p j ( h ) = x }| = K j and the integers K j satisfy | A ′ | = Y K j ≥ [ Y (2 log p j ) − ] | A | . Denote by q the product of primes p j for which K j ≥ p − / Cj | L j | . Then q/q < q ε/ N , if all the primes p j are sufficiently large. By a theorem of Gowers4030] (see also [50, Corollary 1]) it follows that if B , B , B ⊆ L p j are sets with | B i | ≥ p − / Cj | L j | , i = 1 , ,
3, then B .B .B = L p j . This implies thatpr( π q ( A.A.A )) = L q . For more details see the argument on [61, pp. 26]. Now using Proposition 27for the set π q ( A.A.A ) we get an integer q | q with q > q − ε/ N such that π q [ Q C A ] = G q for some constant C independent of q . Thus | Q C A | >q − ε/ | G q | . It is a general fact (see the proof of [35, Lemma 2.2]) that | Q C A | < (cid:16) | A.A.A || A | (cid:17) C − | A | (24)whenever A is a symmetric set in a group. This finishes the proof. Let us first show the necessary part. Let G be the Zariski-closure of Γ. Denoteby G ◦ , the connected component of G , and let Γ ◦ = G ◦ ∩ Γ. It is clear that Γ ◦ is a normal finite-index subgroup of Γ, and so Γ ◦ is also generated by a finiteset S ◦ . We start by showing that G ( π q (Γ ◦ ) , π q ( S ◦ )) form expanders as q runsthrough square free integers with large prime factors assuming G ( π q (Γ) , π q ( S ))form expanders. To this end, first we show that Γ ◦ is a “congruence” subgroup,i.e. it contains a congruence kernel Γ( q ) = ker(Γ π q −→ G q ) if the prime factorsof q are large enough. To prove this claim, we notice that G ◦ and the quotientmap ι : G → G / G ◦ are defined over Q . Hence ι (Γ( q )) = ( ι (Γ))( q ) for any q with large prime factors. On the other hand, since G / G ◦ is a finite Q -group,( ι (Γ))( q ) = 1 for any q with large prime factors, which completes the argumentof the our claim. Now it is pretty easy to show that G ( π q (Γ ◦ ) , π q ( S ◦ )) formexpanders as q runs through square free integers with large prime factors. Forthe sake of completeness we present one argument: it is well-known that ourdesired condition holds if and only if the Haar measure is the only finitelyadditive Γ ◦ -invariant measure on b Γ ◦ , where b Γ ◦ is the profinite completion of Γ ◦ with respect to { Γ ◦ ∩ Γ( q ) } . By the above discussion b Γ ◦ is a finite-index opensubgroup of b Γ the profinite closure of Γ with respect to { Γ( q ) } ; thus one caneasily deduce our claim. As a consequence we get a uniform upper bound on | π q (Γ ◦ ) / [ π q (Γ ◦ ) , π q (Γ ◦ )] | . On the other hand, [ G ◦ , G ◦ ] and the quotient map ι ′ : G ◦ → G ◦ / [ G ◦ , G ◦ ] are defined over Q . Hence again we have that ι ′ and π q commute with each other for any q with large prime factors. Thus one cancomplete the proof of the necessary part using the facts that Γ ◦ is Zariski-densein G ◦ and G ◦ does not have any proper open subgroup. Next we show that the condition that the connected component of the Zariskiclosure of Γ is perfect is sufficient for the Cayley graphs to form a family of41xpanders. The argument which shows this using Propositions 7 and 8 is basedon the ideas of Sarnak and Xue [55] and Bourgain and Gamburd [7] and it iscommon to all of the papers [7]–[11] and [61]. In the previous section we havealready remarked that Γ ◦ = Γ ∩ G ◦ is finitely generated. Using Proposition 7for Γ ◦ , we get a symmetric set S ′ ⊂ Γ ◦ such that if q is square-free and coprimeto the denominators of the entries in the elements of S , H ≤ π q (Γ) and l is aninteger with l > log q , then π q [ χ ( l ) S ′ ]( H ) ≪ [ π q (Γ ◦ ) : H ] − δ . We show that the Cayley graphs G ( π q (Γ ◦ ) , π q ( S ′ )) are expanders and laterwe will see that this implies the statement of the theorem. Denote by T = T q the convolution operator by χ π q ( S ′ ) in the regular representation of π q (Γ ◦ ). I.e.we write T ( µ ) = χ π q ( S ′ ) ∗ µ for µ ∈ l ( π q (Γ ◦ )). We will show that there is aconstant c < q such that the second largest eigenvalue of T is less than c . By a result of Dodziuk [23]; Alon [2]; and Alon and Milman [4](see also [37, Theorem 2.4]) this then implies that G ( π q (Γ ◦ ) , π q ( S ′ )) is a familyof expanders.Consider an eigenvalue λ of T , and let µ be a corresponding eigenfunc-tion. Consider the irreducible representations of π q (Γ ◦ ); these are subspaces of l ( π q (Γ ◦ )) invariant under T . Denote by ρ the irreducible representation thatcontains µ . Recall form Section 3 that π q (Γ ◦ ) = Y p | q prime π p (Γ ◦ ) . We only consider the case when the kernel of ρ does not contain π p (Γ ◦ ) for any p | q , otherwise we can consider the quotient of π q (Γ ◦ ) by π p (Γ ◦ ), and we canreplace q by a smaller integer. Then ρ is the tensor-product of nontrivial repre-sentations of the groups π p (Γ ◦ ), hence the dimension of ρ is at least | π q (Γ ◦ ) | ε forsome ε > ρ )). This in turn implies thatthe multiplicity of λ in T is at least | π q (Γ ◦ ) | ε , since the regular representation l (Γ / Γ q ) contains dim( ρ ) irreducible components isomorphic to ρ .Using this bound for the multiplicity, we can bound λ l by computing thetrace of T l in the standard basis: λ l ≤ | π q (Γ ◦ ) | − ε Tr( T l ) = | π q (Γ ◦ ) | − ε | π q (Γ ◦ ) |k π q [ χ ( l ) S ′ ] k , where k · k denotes the l norm over the finite set π q (Γ ◦ ). This proves thetheorem, if we can show that k π q [ χ ( l ) S ′ ] k ≪ | π q (Γ ◦ ) | − / ε/ (25)for some l ≪ log q .First apply Proposition 7 with H = { } . It gives π q [ χ ( l ) S ′ ](1) ≪ | π q (Γ ◦ ) | − ε for l > log q and for some ε >
0. If l is even then π q [ χ ( l ) S ′ ](1) > π q [ χ ( l ) S ′ ]( g ) for any42 ∈ π q (Γ) by the Cauchy-Schwartz inequality and the definition of convolution(recall that S is symmetric). Then we get the estimate k π q [ χ ( l ) S ′ ] k ≪ | π q (Γ ◦ ) | − ε/ . Observe that if we repeatedly apply Proposition 8 for the measures µ = ν = π q [ χ (2 k l ) S ′ ], then we get (25) in finitely many steps. To justify the use of Propo-sition 8, we remark that since S ′ is symmetric, we have (cid:16) π q [ χ (2 k l ) S ′ ]( gH ) (cid:17) ≤ π q [ χ (2 k +1 l ) S ′ ]( H )that can be bounded using Proposition 7. This shows that G ( π q (Γ ◦ ) , π q ( S ′ ))are expanders indeed.To finish the proof we show the same for the family G ( π q (Γ) , π q ( S )). Write c = c ( G ( π q (Γ ◦ ) , π q ( S ′ ))), recall the definition from the introduction. Assumethat the elements of S ′ are the product of at most m elements of S . Considera set A = V ( G ) = π q (Γ) of vertices with | A | ≤ | V ( G ) | /
2, and denote by N k ( A )the set of vertices that can be joined to an element of A by a path of length atmost k in G ( π q (Γ) , π q ( S )). I.e. by definition N k ( A ) = ( Q k S ) .A. Also, it is easy to see that | N k ( A ) | ≤ | S | k − | ∂A | + | A | , so it is enough to give alower bound on | N k ( A ) | . We clearly have | N | Γ / Γ ◦ | ( A ) | ≥ | Γ / Γ ◦ | max g ∈ π q (Γ) | A ∩ gπ q [Γ ◦ ] | . This finishes the proof if say | A ∩ π q (Γ ◦ ) | < | A | / (2 | Γ / Γ ◦ | ) or if | A ∩ π q (Γ ◦ ) | > | π q (Γ ◦ ) | /
4. If both of these inequalities fail, then by the expander property of G ( π q (Γ ◦ ) , π q ( S ′ )) already proved, we conclude that N m ( A ) > | A | + c/ | S ′ | min {| A | / (2 | Γ / Γ ◦ | ) , | π q (Γ ◦ ) | / } which proves the theorem. Remark 33.
The above proof implies a variant of Proposition 8 that is useful insome applications. Compare the statement below with [12, Lemma 2 in Section7]. Let q be a square-free integer and G be a Zariski-connected, perfect algebraicgroup defined over Q , and write G = G ( Z /q Z ). For every ε >
0, there is a δ > µ be a probability measure which satisfies thefollowing version of the assumptions in Proposition 8 for some ε >
0. I.e. k µ k > | G | − / ε and µ ( gH ) < [ G : H ] − ε | G | δ for any g ∈ G and for any proper subgroup H < G . Let f ∈ l ( G ) be a complexvalued function on the group G such that X g ∈ a G ( Z /q ′ Z ) f ( g ) = 043or all a ∈ G and q ′ | q with q ′ = 1. This condition is equivalent to saying that f isorthogonal to those irreducible subrepresentations in the regular representationof G that factor through G/ G ( Z /q ′ Z ) for some q ′ = 1. Then using the argumentin the proof above, we can write k µ ∗ f k < q − δ k f k (26)for some δ > ε and G . Indeed, repeated application of Propo-sition 8 shows the analogue of (25) for µ ( L ) in place of π q [ χ ( l ) S ′ ] for some integer L which depend on ε and G . Combining this with the lower bounds for mul-tiplicities of the eigenvalues we get the inequality (26). We also note that thestatement in this remark also holds if we consider a group G which satisfies theassumptions (A0)–(A5) listed in Section 4.2 instead of taking G = G ( Z /q Z ). A Appendix: Effectivization of Nori’s paper
In this section, we address the non-effective parts of Nori’s argument in [51] andpresent alternative effective arguments. Most of the arguments in the mentionedarticle are effective. We only need to present effective proofs of [51, Proposi-tion 2.7, Lemma 2.8 and Theorem 5.1]. It should be said that in this articleby effective we mean that there is an algorithm to find the implied constants.Alternatively one can say the mentioned functions are recursively defined. Weshould say that these results are far from the best possible. In fact using theclassification of finite simple groups, Guralnick [34, Theorem D] showed that if p > max { n + 2 , } , then Nori’s statement hold for any subgroup of GL n ( F p )which is generated by its p -elements and has no normal p -subgroup . Unfortu-nately this last condition does not allow us to apply this sharp result.Before stating the main results of this section, we recall very few termsfrom [51] and refer the reader to the mentioned article for the undefined terms.Here R always denotes a finitely presented integral domain unless otherwisementioned. Definition 34 (Definition 2.2 in [51]) . An R -submodule L of M n ( R ) is calleda k -strict Lie subalgebra of M n ( R ) if1. L is a Lie ring.2. There is a submodule L ′ of M n ( R ) such that M n ( R ) = L ⊕ L ′ , and L ′ is locally free of rank n − k . Definition 35 (Definition 2.5 in [51]) . Let U n , N n and Y n,k be the schemeswhich represent the following functors from Z [ n − ]-algebras A to sets:1. N n ( A ) := { x ∈ M n ( A ) | x n = 0 } , 44. U n ( A ) := { x ∈ M n ( A ) | ( x − n = 0 } ,3. Y n,k ( A ) := { x = ( x , . . . , x k ) ∈ N n ( A ) | L x is a k -strict Lie subalgebra } ,where L x = Ax + · · · + Ax k . Definition 36.
Let L be a k -strict Lie subalgebra of M n ( A ) and H be a closedsubgroup-scheme of ( GL n ) A := GL n × Spec A . Let L be the A -scheme whichrepresents the functor S S ⊗ A defined for all commutative A -algebras. Letus define two closed subschemes of ( GL n ) A : e ( L ( n ) ) := exp( L ∩ ( N n ) A ) , H ( u ) := H ∩ ( U n ) A , for any Z [1 / (2 n − A . Definition 37 (Definition 2.3 and Remark 2.18 in [51]) . Let L and H be as inDefinition 36. Then ( L, H ) is called an acceptable pair if the following hold:1. The projection H → Spec( A ) is a smooth morphism with all the fibersconnected.2. Lie( H /A ) = L .3. ( e ( L ( n ) )) red = ( H ( u ) ) red .In this case, L or H are called acceptable.In this section, let X = { X , . . . , X m } and R [ X ] be the ring of polynomialsin the variables X , . . . , X m with coefficients in the ring R . If F is a subset ofa ring R , then h F i denotes the ideal generated by F in R .Here are the main results of this section: Lemma 38 (Effective version of Lemma 2.8 in [51]) . Let R be a computablenoetherian integral domain with quotient field K and z ∈ Y n,k ( R ) . If L z ⊗ R K is acceptable, then we can algorithmically find a non-zero element g ∈ R suchthat L z ⊗ R R g ⊆ M n ( R g ) is also acceptable. (For the definition of a computable ring, see [5, Chapter 4.6].) Lemma 39 (Effective version of Proposition 2.7 in [51]) . For a given k, n , wecan give presentations of finitely many integral domains R i and algorithmicallyfind elements z i ∈ Y n,k ( R i ) such that1. L z i is acceptable if char( R i ) = 0 .2. z i : Spec( R i ) → Y n,k is a locally closed immersion and Y n,k = G i z i (Spec( R i )) . In order to prove Lemma 38, we need to show the following effective versionof certain results from EGA [31]. 45 heorem 40 (Effective version of Theorem 9.7.7 (i) and Theorem 12.2.4 (iii) in[31]) . Let R be a computable integral domain. Let F = { f , . . . , f l } ⊆ R [ X ] and A = R [ X ] / h F i . If the generic fiber of the projection map
Spec( A ) → Spec( R ) issmooth and geometrically irreducible, then one can compute a non-zero element g ∈ R such that the projection map Spec( A g ) → Spec( R g ) is smooth and all ofits fibers are geometrically irreducible. Finally we shall use Lemma 38 and Lemma 39 to get the following:
Theorem 41 (Effective version of a special case of Theorem 5.1 in [51]) . Let G be a perfect, Zariski-connected, simply-connected Q -group. Let Γ ⊆ G ( Q ) bea Zariski-dense subgroup generated by a finite symmetric set S . Then one caneffectively find p = p ( S ) such that for any p > p one has π p (Γ) = G ( F p ) . A.1 Proof of Theorem 40.
In this section, first we show the “generic flatness” in Lemma 42 and the “genericsmoothness” in Lemma 45. Then we reduce the general case of Theorem 40 tothe hyperplane case and finish it as in [31].
Lemma 42.
Let R be a computable noetherian integral domain. Let K be thequotient field of R . Let F = { f , . . . , f l } ⊆ R [ X ] and A = R [ X ] / h F i . Then wecan algorithmically find a non-zero element g ∈ R such that1. A g is a free R g -module.2. h F i g = h F i K ∩ R g [ X ] , where h F i g (resp. h F i K ) is the ideal generated by F in R g [ X ] (resp. K [ X ] ).3. All the fibers of the projection map Spec( A g ) → Spec( R g ) are equidimen-sional.Proof. For any ordering of X i , we compute the Gr¨obner basis of h F i K andmultiply all the head coefficients which appear in the process. Let g ∈ R be theproduct of these head coefficients. Now the basic information on the Gr¨obnerbasis gives us the first and the second parts.Now without loss of generality we can and will assume that { X , . . . , X d } is a maximal independent set modulo h F i K . Since the head coefficients of theGr¨obner basis are units in R g , for any p ∈ Spec( R g ), { X , . . . , X d } is a maximalindependent set modulo h F i k ( p ) ⊆ k ( p )[ X ], where k ( p ) = R p / p R p . Hence, by[5, Theorem 9.27], we have that the Gelfand-Kirillov dimension of A ⊗ k ( p ) is d , which proves the third part. Definition 43.
For F = { f , . . . , f l } ⊆ R [ X ], Let J e ( F ) denote the idealgenerated by the e × e minors of [ ∂f i /∂X j ] in R [ X ] / h F i . Lemma 44.
Let k be an algebraically closed field and F = { f , . . . , f l } ⊆ k [ X ] .Let A = k [ X ] / h F i . Then the projection map Spec( A ) → Spec( k ) is smooth ifand only if J m − d ( F ) = A , where m = X and d = dim A . roof. This is essentially Jacobi’s criteria.
Lemma 45.
Let R be a computable noetherian integral domain, F = { f , . . . , f l } be a subset of R [ X ] , and A = R [ X ] / h F i . If the generic fiber of the projectionmap
Spec( A ) → Spec( R ) is smooth, then one can compute a non-zero element g ∈ R such that the projection map Spec( A g ) → Spec( R g ) is smooth.Proof. Since clearly Spec( A ) is locally finitely presented, it is enough to compute g ∈ R such that1. The projection map Spec( A g ) → Spec( R g ) is flat.2. For any p ∈ Spec( R g ), the projection map Spec( A g ⊗ k ( p )) → Spec( k ( p ))is smooth, where k ( p ) is the algebraic closure of k ( p ) = R p / p R p .By Lemma 42, we can compute a non-zero element g ∈ R such that the pro-jection map Spec( A g ) → Spec( R g ) is flat and all of its fibers are of a fixeddimension d . On the other hand, since the generic fiber is smooth, by Lemma 44, J m − d ( F ) ∩ R is non-zero. By computing a Gr¨obner basis of J m − d ( F ), we cancompute a non-zero element g ∈ J m − d ( F ) ∩ R . It is easy to check that g = g g gives us the desired property. Lemma 46.
Let R be an infinite computable noetherian integral domain, F afinite subset of R [ X ] and A = R [ X ] / h F i . Then we can compute a matrix [ a ij ] ∈ GL m ( R ) , a non-zero element g ∈ R , elements x d +1 ∈ A g , f ∈ R g [ X ′ , . . . , X ′ d ] and p ∈ R g [ X ′ , . . . , X ′ d , T ] , where X ′ i = P a ij X j , such that the following hold1. R g [ X ′ , . . . , X ′ d ] ∩ h F i g = 0 .2. A g is an integral extension of R g [ X ′ , . . . , X ′ d ] .3. A gf = ( R g [ X ′ , . . . , X ′ d ]) f [ x d +1 ] ≃ ( R g [ X ′ , . . . , X ′ d , T ] / h p i ) f .Proof. Let K be the quotient field of R . By [45], we can compute a matrix[ a ij ] ∈ GL m ( R ) and elements r d +2 , . . . , r m ∈ R such that the following hold1. X ′ , . . . , X ′ d are algebraically independent in A ⊗ K .2. X ′ j are integral over K [ X ′ , . . . , X ′ d ].3. S − A = K ( X ′ , . . . , X ′ d )[ x d +1 ], where S = K [ X ′ , . . . , X ′ d ] \{ } and x d +1 = X ′ d +1 + P mi = d +2 r i X ′ i .where X ′ i = P a ij X j .Again computing Gr¨obner basis of the ideal generated by F in K ( X ′ , . . . , X ′ d )[ X ′ d +1 , . . . , X ′ m ]with respect to various orderings, we can compute the minimal polynomialsof X ′ i over K ( X ′ , . . . , X ′ d ). Since X ′ i are integral over K [ X ′ , . . . , X ′ d ] and the47ing of polynomials over a field is integrally closed, all the minimal polyno-mials are monic polynomials with coefficients in K [ X ′ , . . . , X ′ d ]. Hence wecan compute a non-zero element g ∈ R such that A g is an integral exten-sion of R g [ X ′ , . . . , X ′ d ]. Moreover writing X ′ i as a polynomial in terms of x d +1 with coefficients in K ( X ′ , . . . , X ′ d ), we can find f ∈ K [ X ′ , . . . , X ′ d ] such that A f ⊗ K = K [ X ′ , . . . , X ′ d ] f [ x d +1 ]. We can also compute the minimal polyno-mial p of x d +1 over K ( X ′ , . . . , X ′ d ). Now let f be the product of f by theproduct of all the denominators of the coefficients of the minimal polynomial.It is clear that these choices satisfy the desired properties. Proof of Theorem 40.
By Lemma 45 and Lemma 42, we can compute a non-zeroelement g ∈ R such that the projection map Spec( A g ) → Spec( R g ) is smoothand A g is a free R g -module. Let g ∈ R , f and p be the parameters which aregiven by Lemma 46. Changing R to R g g and using the above results, we canand will assume that1. A is a free R -module.2. The projection map Spec( A ) → Spec( R ) is smooth,3. A is an integral extension of R [ X , . . . , X d ] and the latter is the ring ofpolynomials,4. A f ≃ ( R [ X , . . . , X d , T ] / h p i ) f .Let B = R [ X , . . . , X d , T ] / h p i . Since the generic fiber of Spec( A ) over R is geo-metrically irreducible, so is the generic fiber of Spec( B ) over R . Hence by virtueof [31, Lemma 9.7.5], we can compute a non-zero element g ∈ R such that allthe fibers of Spec( B g ) → Spec( R g ) are geometrically irreducible. In particu-lar, all the fibers of Spec( A g f ) ≃ Spec( B g f ) → Spec( R g ) are geometricallyirreducible. This means for any p ∈ Spec( R g ) A g f ⊗ k ( p )is an integral domain. If it is a non-zero ring, then A g ⊗ k ( p ) is also an integraldomain. On the other hand, A g f ⊗ k ( p ) = 0 if and only if f is either zero or azero-divisor in A g ⊗ k ( p ).By a similar argument as in Lemma 42, we can compute a non-zero element g ∈ R such that ( A/ h f i ) g is a free R g -module. Let g = g g . We claim thatall the fibers of Spec( A g ) → Spec( R g ) are geometrically irreducible. By theabove discussion, it is enough to show that for any p ∈ Spec( R g ), f is not eitherzero nor a zero-divisor in A g ⊗ k ( p ).Let λ f ( x ) = f x be the map of multiplication by f in A g . Since A g is anintegral domain and f is not zero, we have the following short exact sequenceof R g -modules: 0 → A g λ f −−→ A g → ( A/ h f i ) g → . p ∈ Spec( R g ) we have the following exact sequenceTor(( A/ h f i ) g , k ( p )) → A g ⊗ k ( p ) → A g ⊗ k ( p ) . Since ( A/ h f i ) g is a free R g -module, Tor(( A/ h f i ) g , k ( p )) = 0. Therefore f isneither a zero nor a zero-divisor in A g ⊗ k ( p ). Thus by the above discussion, weare done. A.2 Proof of Lemma 38.
By Definition 37, a pair of a Lie ring and an algebraic group scheme is acceptableif and only if it satisfies three properties. In this section, we show how one canuse Theorem 40 to guarantee the first property. The second property is achievedusing smoothness and the definition of the Lie algebra of a smooth group scheme.The third property is dealt with in Lemma 52.
Lemma 47.
Let G be an algebraic group and X an irreducible subvariety. If ∈ X = X − , then Y dim G X = X · · · · · X, is the group generated by X .Proof. This is clear.
Lemma 48.
Let G be an algebraic group and X an irreducible subvariety. Then Y dim G X · X − = ( X · X − ) · · · · · ( X · X − ) is the group generated by X.Proof. It is a consequence of Lemma 47.
Lemma 49.
Let G be an algebraic group and X i irreducible subvarieties whichcontain . Let e X = ( X · X − ) · · · · · ( X k · X − k ) . Then Q dim G ( e X · e X − ) is thegroup generated by X i .Proof. By Lemma 48, it is enough to observe that X i ⊆ e X . Lemma 50.
Let R be a computable integral domain whose characteristic is atleast n and z = ( z , . . . , z k ) ∈ Y n,k ( R ) . Let K be the quotient field of R , H bethe K -subgroup scheme of ( GL n ) K which is generated by exp( tz i ) and H be theclosure of H in ( GL n ) R . Then we can algorithmically find an element g ∈ R and a finite subset F = { f , . . . , f l } ⊆ R g [ GL n ] such that H ×
Spec( R ) Spec( R g ) ≃ R g [ GL n ] / h F i as closed subschemes of ( GL n ) R g . roof. It is clear that, for any i , the image of a z i : A K → ( GL n ) K a z i ( t ) := exp( tz i ) , is a 1-dimensional irreducible K -algebraic subgroup of ( GL n ) K . Hence byLemma 49 we can find an algebraic morphism Φ : A (2 k − n → ( GL n ) K whoseimage is exactly H . Hence by means of the elimination method we can computea presentation for H , i.e. F = { f , . . . , f l } ∈ R [ GL n ] such that H ≃ Spec( K [ GL n ] / h F i K ) , as K -varieties. Now by the second part of Lemma 42, we can compute a non-zeroelement g ∈ R such that H g := H ×
Spec( R ) Spec( R g ) ≃ Spec( R g [ GL n ] / h F i g ) . Lemma 51.
Let R be a computable integral domain, K be the quotient field of R , z ∈ Y n,k ( R ) , L = L z and H be a closed subgroup of ( GL n ) K . If ( H , L ⊗ K ) is anacceptable pair, then we can algorithmically find a non-zero element g ∈ R suchthat Lie( H )( R g ) = L g , where H is the closure of H in ( GL n ) R and L g = L ⊗ R g .Proof. By [51, Lemma 2.12], we know that H is generated by exp( tz i ). Henceby Lemma 49 and Theorem 40, we can compute a non-zero element g ∈ R anda finite subset F = { f , . . . , f l } ⊆ R [ GL n ] such that1. The projection map H g := H ×
Spec( R ) Spec( R g ) → Spec( R g ) is smooth.2. As R g -schemes, H g ≃ Spec(( R [ GL n ] / h F i ) g ) ≃ Spec( R g [ X ] / h ˜ F i ) , where X = { X , . . . , X n +1 } , ˜ F = F ∪ { X n +1 D ( X , . . . , X n ) − } and D is the determinant of the first n variables.Since H g is a smooth R g -scheme, Lie( H g /R g ) = Ker(Jac( ˜ F )), where Jac( ˜ F ) =[ ∂ ˜ f i /∂X j ] is the Jacobian of( X , . . . , X n +1 ) ( ˜ f ( X , . . . , X n +1 )) ˜ f ∈ ˜ F . By Gauss-Jordan process, we can compute a non-zero element g ∈ R such thatKer(Jac( ˜ F )) g is a free R g -module. We can also compute an R g -basis. Sincewe know that L ⊗ K = Ker(Jac( ˜ F )) g ⊗ R g K and we have R g -basis for bothof them, we can compute a non-zero element g such that L g = Ker(Jac( ˜ F )) g ,which finishes the proof. Lemma 52.
Let R , K , z , L , H and H be as in Lemma 51. If ( H , L ⊗ K ) isan acceptable pair, then we can algorithmically find a non-zero element g ∈ R such that ( e ( L ( n ) ) × Spec( R ) Spec( R g )) red = ( H ( u ) × Spec( R ) Spec( R g )) red . roof. Since L is given through an R -basis, we can compute a non-zero element g ∈ R and an R g -basis for the dual of L . Hence we can compute a presenta-tion for L . Thus using elimination method we can compute a presentation of e ( L ( n ) ) g := e ( L ( n ) ) × Spec( R ) Spec( R g ). We can also compute a presentation of H ( u ) .Since ( H , L ⊗ K ) is an acceptable pair, we have( e ( L ( n ) ) × Spec( R ) Spec( K )) red = ( H ( u ) × Spec( R ) Spec( K )) red . So having a presentation of both sides over R g , one can easily compute g ∈ R such that( e ( L ( n ) ) × Spec( R ) Spec( R g )) red = ( H ( u ) × Spec( R ) Spec( R g )) red holds for g = g g . Proof of Lemma 38.
One can repeat Nori’s argument [51, Lemma 2.8] and getthe effective version using Theorem 40, Lemma 51 and Lemma 52.
A.3 Proof of Lemma 39.
In this section, first we give a precise presentation of Y n,k . Then using Lemma 38by an inductive argument we get the desired result. Definition 53.
Let F = { f , . . . , f l } and F ′ = { f ′ , . . . , f ′ l ′ } be two subsets of R [ X ], where X = { X , . . . , X m } . Then let V ( F ) denote the closed subschemeof A mR defined by the relations F , and W ( A mR ; F, F ′ ) := V ( F ) \ V ( F ′ ) . If a and b are two ideals of R [ X ], then V ( a ) denotes the closed subscheme of A mR defined by a and W ( A mR ; a , b ) := V ( a ) \ V ( b ) . Definition 54.
For any z = ( z , . . . , z k ) ∈ M n ( R ) k , fix the standard R -basis of M n ( R ) and view z i as column vectors in this basis. Let F z be the set of all themaximum dimension minors of the matrix (cid:2) z · · · z k (cid:3) and a z be the idealgenerated by F z .We also consider the case R = Z [ X ], where X = { X i ′ ij | ≤ i, j ≤ n, ≤ i ′ ≤ k } and set x = ( x , . . . , x k ), where the ij -th entry of x i ′ is X i ′ ij . In this case, F n,k := F x and a n,k := a x . Remark 55.
We sometimes identify the functor M n with A n Z . This way, any z ∈ M n ( R ) gives rise to a ring homomorphism φ z from Z [ A kn ] to R and it isclear that φ z ( a n,k ) = a z . Lemma 56.
Let ( A, m ) be a pair of a local ring and its maximal ideal. Let φ : A n → A n be an A -linear map. Then the following statements are equivalent: . φ is surjective.2. φ : ( A/ m ) n → ( A/ m ) n is invertible.3. φ is invertible.Proof. This is clear.
Lemma 57.
Let R be any commutative ring. Then z = ( z , . . . , z k ) ∈ W ( A kn Z ; 0 , a n,k )( R ) if and only if M n ( R ) /L is locally of dimension n − k , where L = Rz + · · · + Rz k .Proof. By the definition of W ( A kn Z ; 0 , a n,k )( R ), it is straightforward to checkthat z ∈ W ( A kn Z ; 0 , a n,k )( R ) if and only if a z = R .On the other hand, M n ( R ) /L is locally of dimension n − k if and only iffor any p ∈ Spec( R ) there are z ( p ) k +1 , . . . , z ( p ) n ∈ M n ( R ) such that M n ( R p ) = ( k X i =1 R p z i ) + ⊕ n i = k +1 R p z ( p ) i . (27)Let φ p : R n p → M n ( R p ) be the following R p -linear map φ p ( r , . . . , r n ) := X i r i z ( p ) i , where R n p is the direct sum of n copies of R p and z ( p ) i = z i for any i ≤ k .By Lemma 56, it is clear that (27) holds if and only if φ p is invertible. It iseasy to see that the latter is equivalent to a z = k ( p ), where k ( p ) = R p / p R p and z = ( z , . . . , z k ) ∈ M n ( k ( p )) k . Let S p = R \ p . By the definition, it is clear that a z = k ( p ) if and only if S − p a z = R p . The latter holds for any p ∈ Spec( R ) ifand only if a z = R , which completes the proof. Definition 58.
Let z = ( z , . . . , z k ) ∈ ( R n ) k . We sometimes view such avector in two other ways: as an n × k matrix whose i -th column is z i ; or a k -tuple of n × n matrices whose i -th entry is z i written in matrix form withrespect to the standard basis.Let J ⊆ { , . . . , n } be of order k . Then z J denotes the k × k submatrixof z whose rows are determined by J . For a vector v ∈ R n , v J denotes thesubvector of size k determined by J .For a given a ∈ Mor( A kn Z , A n Z ) and any subsets J, J ′ ⊆ { , . . . , n } of order k , we define f ( a ) J,J ′ ∈ Mor( A kn Z , A k Z ) as follows: f ( a ) J,J ′ ( z ) = z J ′ adj( z J ) a ( z ) J − det( z J ) a ( z ) J ′ . Also let F ( a ) n,k be the set consisting of all the entries of f ( a ) J,J ′ for all the possible J and J ′ . 52 emma 59. Let R be any commutative ring and a ∈ Mor( A kn Z , A n Z ) . Then z = ( z , . . . , z k ) ∈ W ( A kn Z ; F ( a ) n,k , F n,k )( R ) if and only if1. M n ( R ) /L z is locally of dimension n − k , where L z = Rz + · · · + Rz k ,2. a ( z ) ∈ L z .Proof. Let Y n,k = W ( A kn Z ; ∅ , F n,k ) and let Y ( a ) n,k be the functor from commu-tative rings to sets such that Y ( a ) n,k ( R ) = { z ∈ Y n,k ( R ) | a ( z ) ∈ L z } . By Lemma 57, it is enough to show that Y ( a ) n,k ( R ) = W ( A kn Z ; F ( a ) n,k , F n,k )( R ) . Let us view z as an n × k matrix. Then if z ∈ Y n,k , then it belongs to Y ( a ) n,k ( R )if and only if there is ~r = ( r , . . . , r k ) such that z ~r = a ( z ). The latter holds ifand only if for any J ⊆ { , . . . , n } of order k we have z J ~r = a ( z ) J .We claim that if z ∈ Y ( a ) n,k ( R ) then there is a unique ~r which satisfies theequations z J ~r = a ( z ) J for all the subsets J of order k in { , . . . , n } . To showthis claim it is enough to notice that det( z J ) ~r = adj( z J ) a ( z ) J and the idealgenerated by det( z J ) as J runs through all the subsets of order k contains 1 as z ∈ Y n,k ( R ).We also observe that if z ∈ Y ( a ) n,k ( R ), then for any J and J ′ we have z J ′ adj( z J ) a ( z ) J = det( z J ) z J ′ ~r = det( z J ) a ( z ) J ′ . Hence z ∈ W ( A kn Z ; F ( a ) n,k , F n,k )( R ), i.e. Y ( a ) n,k ( R ) ⊆ W ( A kn Z ; F ( a ) n,k , F n,k )( R ).Let z ∈ W ( A kn Z ; F ( a ) n,k , F n,k )( R ). We claim that if det( z J ) is a unit in R forsome J , then z ∈ Y ( a ) n,k ( R ). To see this it is enough to check that ~r = det( z J ) − adj( z J ) a ( z ) J satisfies all the equations z J ′ ~r = a ( z ) J ′ . In particular, for any local ring R , wehave Y ( a ) n,k ( R ) = W ( A kn Z ; F ( a ) n,k , F n,k )( R ) . For an arbitrary commutative ring R , let again z ∈ W ( A kn Z ; F ( a ) n,k , F n,k )( R ). Bythe above discussion, for any p ∈ Spec( R ), we have that z ∈ Y ( a ) n,k ( R p ), i.e. thereis a unique ~r p ∈ R k p such that z J ~r p = a ( z ) J for any J . On the other hand, bythe uniqueness argument, since the ideal generated by det( z J ) is equal to R ,there is ~r ∈ R k such that for any p and any J we have z J ~r = a ( z ) J in R k p . Nowone can easily deduce that z J ~r = a ( z ) J in R k , which means z ∈ Y ( a ) n,k ( R ) andwe are done. 53 efinition 60. Let a ij ∈ Mor( A kn Z , A n Z ) be the following morphism a ij ( z ) := [ z i , z j ] = z i z j − z j z i , for any 1 ≤ i, j ≤ k . Let e F n,k := S ki,j =1 F ( a ij ) n,k . Corollary 61.
For any commutative ring R , we have Y n,k ( R ) = W ( A kn Z ; e F n,k , F n,k )( R ) . Proof.
This is a direct consequence of Lemma 59
Lemma 62.
Let F and F ′ = { f ′ , . . . , f ′ l ′ } be two subsets of Z [ X ] , where X = { X , . . . , X m } . Assume that h F i is a radical ideal. Then we can com-putationally determine if W ( A m Z ; F, F ′ ) is nonempty, and if it is, then we cangive a presentation of an integral domain R and z ∈ W ( A m Z ; F, F ′ )( R ) such that1. z : Spec( R ) → W ( A m Z ; F, F ′ ) is an open immersion.2. For any given d ∈ R , we can computationally describe the complement of z (Spec( R [ d ])) in W ( A m Z ; F, F ′ ) .Proof. It is clear that W ( A m Z ; F, F ′ ) is empty if and only if h F ′ i ⊆ p h F i , whichcan be computationally determined. To show the rest, first we claim that wecan assume that F ′ = ∅ . To show this claim, we start with the following openaffine covering: W ( A m Z ; F, F ′ ) = ∪ i W ( A m Z ; F, { f ′ i } ) . And, we notice that W ( A m Z ; F, { f ′ i } ) ≃ W ( A m +1 Z ; F ∪ { f ′ i X m +1 − } , ∅ ) . Nowif we find R and z for F ∪ { f ′ i X m +1 − } and F ′ = ∅ , then one can see that thefirst assertion still holds and the complement of z (Spec( R [ d ]) in W ( A m Z ; F, F ′ ) isequal to the union of its complement in W ( A m Z ; F, { f ′ i } ) and W ( A m Z ; F ∪{ f ′ i } , F ′ ).So without loss of generality, we can and will assume that F ′ = ∅ . By[5, Chapter 8.5], we can compute a primary decomposition ∩ i p i of h F i . Since h F i = p h F i , p i is a prime ideal for any i . If h F i is a prime ideal, let c = 1 and R = Z [ X ] / h F i ; otherwise, let c ∈ ∩ i ≤ p i \ p (we can computationally find such c ) and R = ( Z [ X ] / p )[ c ]. Clearly this choice of R satisfies the first assertionin the statement of Lemma. Now let d ∈ R be a given element. Then one caneasily check that the complement of the natural open immersion of Spec( R [ d ])in W ( A m Z ; F, ∅ ) is isomorphic to W ( A m Z ; F ∪ { c } , ∅ ) ∪ W ( A m Z ; F ∪ { d } , ∅ ) . Proof of Lemma 39.
Following Nori’s proof of [51, Proposition 2.7] and usingLemma 38, Corollary 61, and Lemma 62 whenever needed, one can easily provethis lemma. 54 .4 Proof of Theorem 41.
Lemma 63.
Let S ⊆ GL n ( Q ) be a finite set of matrices. Let Γ be the groupgenerated by S . Assume that the Zariski-closure of Γ in ( GL n ) Q is Zariski-connected. Then we can compute a square-free integer q and a finite subset F = { f , . . . , f l } ⊆ Z [1 /q ][ GL n ] such that Γ ⊆ GL n ( Z [1 /q ]) and its Zariski-closure in ( GL n ) Z [1 /q ] is isomorphic to Z [1 /q ][ GL n ] / h F i .Proof. Since S is a finite set of matrices, we can find an odd prime p such thatΓ ⊆ GL n ( Z p ). Hence by changing Γ to Γ ∩ GL (1) n ( Z p ) (the first congruencesubgroup is denoted by GL (1) n ( Z p )), we can and will assume that Γ is torsionfree. It is worth mentioning that we are allowed to make such a change becauseof the following:1. We can compute representatives for the cosets of Γ ∩ GL (1) n ( Z p ) in Γ. Thuswe can compute a generating set for Γ ∩ GL (1) n ( Z p ).2. Since we have assumed that the Zariski-closure H of Γ in ( GL n ) Q is Zariski-connected, the Zariski-closure of Γ ∩ GL (1) n ( Z p ) in ( GL n ) Q is also H .We find a presentation for Q [ H ] and then similar to the proof of Lemma 50 wecan finish the argument.Since Γ is torsion-free, the Zariski-closure of the cyclic group generated byany element of Γ is of dimension at least one. Hence by the virtue of Lemma 49it is enough to find a presentation of the Zariski-closure G of the cyclic groupgenerated by γ ∈ S . We can compute the Jordan-Chevalley decomposition γ u · γ s of γ . Let G u (resp. G s ) be the Zariski-closure of the group generatedby γ u (resp. γ s ). Then G ≃ G u × G s as Q -groups [6, Theorem 4.7]. Using thelogarithmic and exponential maps, one can easily find a presentation of G u . Soit is enough to find a presentation of G s . We can compute all the eigenvalues λ , . . . , λ n of γ s . By [6, Proposition 8.2], in order to find a presentation of k [ G ],where k is the number field generated by λ i , it is enough to find a basis for thefollowing subgroup of Z n { ( m , . . . , m n ) ∈ Z n | Y i λ m i i = 1 } , i.e. all the character equations, which is essentially done in [53]. So far we founda finite subset F ′ of k [ GL n ] such that k [ G s ] ≃ k [ GL n ] / h F ′ i . Since G s is definedover Q , we have that Q [ G s ] ≃ Q [ GL n ] / ( h F ′ i ∩ Q [ GL n ]). On the other hand,using Gr¨obner basis we can find a generating set F s for h F ′ i ∩ Q [ GL n ], whichfinishes our proof. Lemma 64.
Let L be a smooth Z [1 /q ] -subgroup scheme of ( GL n ) Z [1 /q ] . Let F ⊆ Z [1 /q ][ GL n ] such that Z [1 /q ][ GL n ] / h F i . If the generic fiber L of L is asimply-connected semisimple Q -group, then1. we can algorithmically find a positive integer p such that for any prime p > p , the special fiber L p := L ×
Spec( Z [1 /q ]) Spec( F p ) of L over p is asemisimple F p -group. . we can algorithmically find a positive integer p such that, for any p > p , L ( Z p ) is a hyper-special parahoric in L ( Q p ) .Proof. The second part is a consequence of the first part as it is explained in [60,Section 3.9.1]. Here we only prove the first part.We can compute the Lie algebra l of L . Since L is not a nilpotent Lie algebra,not all the elements of a basis of l can be ad-nilpotent. Hence we can find anad-semisimple element x of l . Since l is a semisimple Lie algebra and x is asemisimple element, the centralizer c l ( x ) of x in l is a reductive algebra and c l ( x ) / z ( c l ( x )) is a semisimple Lie algebra (if not trivial), where z ( c l ( x )) is thecenter of c l ( x ). If c l ( x ) is not commutative, then repeating the above argumentwe can find x ′ ∈ c l ( x ) \ z ( c l ( x )). We can compute the eigen- values λ i (resp. λ ′ i )of ad( x ) (resp. ad( x ′ )) and find λ = λ i − λ j λ ′ i ′ − λ ′ j ′ , for any i, j, i ′ , j ′ , then c l ( x + λx ′ ) = c l ( x ) ∩ c l ( x ′ ). By repeating this process,we can compute a number field k over which l splits and we can also computea Cartan subalgebra. Hence we can compute a Chevalley basis x i for l ⊗ Q k .Looking at the commutator relations, we can compute an element a of k suchthat P i O k [1 /a ] x i form a Lie subring of l ⊗ Q k , where O k is the ring of integersin k . Thus for any p which does not divide N k/ Q ( a ) the special fiber L p is asemisimple F p -group, as we wished. Lemma 65.
Let H be a perfect Zariski-connected Q -subgroup of GL n . Let F bea finite subset of Q [ GL n ] such that Q [ H ] ≃ Q [ GL n ] / h F i . Then one can computea square-free integer q and a finite subset F ′ of Z [1 /q ][ GL n ] such that1. The Zarsiki-closure H of H in ( GL n ) Z [1 /q ] is defined by F ′ .2. The projection map H →
Spec( Z [1 /q ]) is smooth.3. We can compute a generating set for H ( Z [1 /q ]) .4. π p ( H ( Z [1 /q ])) is a perfect group if p ∤ q .Proof. By [32, Algorithm 3.5.3], we can compute the unipotent radical anda Levi subgroup of H . Therefore we can effectively write H as the semidirectproduct of a semisimple Q -group L and a unipotent Q -group U . We can computea square-free integer q and Z [1 /q ]-group schemes L and U such that:1. The projection maps to Spec( Z [1 /q ]) are smooth.2. All the fibers are geometrically irreducible.3. L acts on U .4. The generic fiber of L (resp. U , H := L ⋉ U ) is isomorphic to L (resp. U , H ). 56t is worth mentioning that the first and the second items are consequences ofTheorem 40 and the rest are easy. Using logarithmic and exponential maps, wecan effectively enlarge q , if necessary, and assume that for any p ∤ q we have[ U p ( F p ) , U p ( F p )] = [ U , U ] p ( F p ), where U p = U × Spec( Z [1 /q ]) Spec( F p ) ([ U , U ] p isdefined in a similar way). We can also get a generating set for U ( Z [1 /q ]). ByLemma 64, we can enlarge q and assume that L ( Z p ) is a hyper-special parahoricsubgroup of L ( Q p ) for any p ∤ q . In particular, by further enlarging q , we havethat L ( Z [1 /q ]) is an arithmetic lattice in (the non-compact semisimple group) L ( R ) · Q p | q L ( Q p ). Thus we have1. By the classical strong approximation theorem, we have that π p ( L ( Z [1 /q ])) = L p ( F p )is a product of quasi-simple groups.2. By [33], we can compute a generating set Ω for L ( Z [1 /q ]). Thus we geta generating set for H ( Z [1 /q ]).On the other hand, since H is perfect, the action of L on u / [ u , u ] has nonon-trivial fixed vector, where u = Lie( U ). This is equivalent to say that theelements of Ω do not have a common non-zero fixed vector. Fix a basis B of u / [ u , u ] and let X γ := [ γ ] B − I , where the [ γ ] B is the matrix associated withthe action of γ and I is the identity matrix. Let X be a column blocked-matrixwhose blocked-entries are X γ for γ ∈ Ω. By our assumption, X is of full rank,i.e. the product of its minors of maximum dimension is a non-zero element of Z [1 /q ]. Hence by enlarging q , if necessary, we can assume that the elementsof π p (Ω) do not fix any non-trivial element of U p ( F p ) / [ U p ( F p ) , U p ( F p )]. Henceby the above discussion, for any prime p ∤ q , we have π p ( H ( Z [1 /q ])) = L p ( F p ) ⋉ U p ( F p )is a perfect group, which finishes our proof. Proof of Theorem 41.
Let q be a square-free integer given by Lemma 65. Let G be the Zariski-closure of G in ( GL n ) Z [1 /q ] . Lemma 65 provides us with aneffective version of Theorem A for the group G ( Z [1 /q ]). On the other hand,we have already proved the effective versions of [51, Theorem B and C]. Hencefollowing the proof of Proposition 16, one can effectively compute a positivenumber δ such that: for any proper subgroup H = H + of G p ( F p ) + , one has that { γ ∈ G ( Z [1 /q ]) | k γ k S ≤ [ π p ( G ( Z [1 /q ])) : H ] δ } is in a proper algebraic subgroup. In particular, if Ω generates a Zariski-densesubgroup of G , then π p (Γ) + = G p ( F p ) + for any p > max γ ∈ Ω {k γ k /δS } , where S = { p | p is a prime divisor of q } . eferences [1] H. Abels, G. A. Margulis and G. A. Soifer, Semigroups containing proximallinear maps , Israel J. Math. (1995), 1–30.[2] N. Alon, Eigenvalues and expanders , Combinatorica No. 2. (1986), 83–96.[3] N. Alon, A. Lubotzky and A. Wigderson,
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Department of Mathematics, Princeton University, Princeton,NJ 08544, USA andAnalysis and Stochastics Research Group of the Hungarian Academyof Sciences, University of Szeged, Szeged, Hungary e-mail address:e-mail address: