arXiv:1711.02147v1 [math.GR] 6 Nov 2017
SUBGROUPS
A. S. DETINKO, D. L. FLANNERY, AND A. HULPKE
ABSTRACT. We give a method to describe all congruence images of a finitely generated Zariski dense groupH ≤SL(n,Z). The method is applied to obtain efficient algorithms for solving this problem in odd prime degreen; ifn= 2then we compute all congruence images only modulo primes. We propose a separate method that works for allnas long asH contains a known transvection. The algorithms have been implemented inGAP, enabling computer experiments with important classes of linear groups that have recently emerged.
1. INTRODUCTION
This paper further develops methods and algorithms for computing with linear groups over infinite domains. It is a sequel to [6].
Let H be a finitely generated subgroup ofSL(n,Z), n ≥ 2, that is Zariski dense in SL(n,R). By the strong approximation theorem,His congruent toSL(n, p)modulopfor all but a finite number of primesp[22, p. 391]. Ifn > 2andH is arithmetic, i.e.,Hhas finite index inSL(n,Z), then the congruence subgroup property guarantees thatHcontains aprincipal congruence subgroup of levelmfor somem, i.e., the kernelΓn,mof the reduc-tion modulomhomomorphismϕm : SL(n,Z) → SL(n,Zm). In that eventH contains a unique maximal principal congruence subgroup Γn,M, and we callM = M(H)thelevel ofH. The dense groupHis contained in a uniquely defined minimal arithmetic overgroup cl(H), namely the intersection of all arithmetic subgroups of SL(n,Z) containing H (its ‘arithmetic closure’) [6, Section 3.3]. The level ofH is defined to be the level ofcl(H), and is again denotedM(H). Sarnak [27] calls dense non-arithmetic H ≤SL(n,Z)athin
matrix group.
In [6] we developed practical algorithms to compute the levelMof a dense groupH≤ SL(n,Z)forn > 2. This was motivated by the fact thatM is the key component of our algorithms to compute with arithmetic subgroups of SL(n,Z) [5]. Once we haveM, we can findcl(H)and obtain further information about denseHvia computation withcl(H). Algorithms to computeMwere implemented and used to carry out extensive computer experiments, as detailed in [6, Section 4]. Our method requires the set Π = Π(H) of primespsuch thatϕp(H)6= SL(n, p); essentially, a computational realization of the strong approximation theorem.
2010 Mathematics Subject Classification: 20-04, 20G15, 20H25, 68W30.
The aim of the present work is twofold. First, in Section 2 we establish a general method to compute Π(H) for dense H ≤ SL(n,Z), based on the classification of maximal sub-groups of SL(n, p) as in [2] (see also [22, p. 397]). This is then applied in Section 3 to obtain efficient algorithms to computeΠ(H)for prime degreen(in which case the types of maximal subgroups of SL(n, p) are quite restricted). Moreover, for odd prime n, we build on this knowledge to describe the congruence images ofHmodulo all positive inte-gers. Arbitrary degreesnare treated in [8] (albeit with algorithms that are less efficient for primenthan those herein).
We also give an algorithm to computeΠ(H)for subgroupsHofSL(2n,Z)that contain a known transvection (a unipotent element t such that t−1n has matrix rank 1). This completes the task begun in [6, Section 3.2].
Another goal is to perform computer experiments successfully with low-dimensional dense representations of finitely presented groups that have recently been the focus of much attention. We compute Π and M for each group, thus enabling us to describe all of its congruence quotients. Experimental results are presented in Section 4.
We adhere to the following conventions and notation. Congruence images are sometimes indicated by overlining. Pre-images in H = hg1, . . . , gri ≤ SL(n,Z) of elements ofH¯ written as words in the g¯i are found by ‘lifting’: g¯kmii· · ·g¯mkss has pre-image gmkii· · ·gkmss. The set of prime divisors ofa∈Nis denotedπ(a). Throughout,Fis a field.
2. STRONG APPROXIMATION AND RECOGNITION OF CONGRUENCE IMAGES The core idea of our approach to computingΠ(H)is to find all primespsuch thatϕp(H) lies in a maximal subgroup ofSL(n, p). Here we provide some general methods for this purpose.
2.1. Large congruence images. LetHbe infinite. Given a positive integerk, we find all primespsuch thatϕp(H)has elements of order greater thank(cf. [30, Chapter 4] and [9, Section 3.5]).
Since a periodic linear group is locally finite, the finitely generated group H has an elementhof infinite order. We can findhquickly by random selection (see [9, Section 4.2, p. 107], and the discussion in Subsection 3.3 on randomly selecting elements with specified properties). For1 ≤i≤k, letmi be the greatest common divisor of the non-zero entries ofhi−1n, and letl= lcm(m1, . . . , mk). Ifp /∈π(l)then|ϕp(H)|> k. For eachp∈π(l) we check whether|ϕp(H)|< k. The preceding steps define a procedurePrimesForOrder that acceptskand infiniteH ≤SL(n,Z), and returns the (finite) set of primespsuch that |ϕp(H)|< k.
We will also need the following.
Lemma 2.1. Suppose thatϕp(H) = SL(n, p)for some primep. (i) Ifn≥3thenHis infinite.
(ii) Ifn= 2andp≥3thenHis infinite.
2.2. Irreducibility. This subsection recaps an argument from [6, Section 3.2].
We test whetherH ≤SL(n,Z)is absolutely irreducible by computing a Q-basisA = {A1, . . . , Am}of the enveloping algebrahHiQ, where theAiare words over a generating set ofH. Ifm =n2
thenHis absolutely irreducible, andϕp(H)is absolutely irreducible for any prime p not dividing ∆ := det[tr(AiAj)]; heretr(x) is the trace of a matrix x. Hence we have the following.
Lemma 2.2. IfHis absolutely irreducible thenϕp(H)is absolutely irreducible for almost
all primesp.
Ifp|∆thenϕp(H)might be absolutely irreducible. Testing for this as before is the last step in PrimesForAbsIrreducible(H), which returns the set of all primespsuch that ϕp(H)is not absolutely irreducible.
Note that ifH¯ is absolutely irreducible (e.g.,H¯ = SL(n, p)), andA¯is a basis ofhHi¯ Zp, thenAis a basis ofhHiQ.
2.3. Primitivity. Next, we give conditions for the congruence image of an (irreducible) primitive subgroup ofSL(n,Z)to be imprimitive. The main concern is primen; in such degrees an irreducible linear group is either primitive or monomial.
Lemma 2.3. IfH ≤ SL(n,Z) is not solvable-by-finite then ϕp(H) is not monomial for
almost all primesp.
Proof. SinceHhas a free non-abelian subgroup by the Tits alternative, givenk≥1there existg,h∈Hsuch thatc:= [gk, hk]6= 1
n. Then[¯gk,h¯k] = ¯c6= 1nfor almost all primes p. The lemma follows by takingkto be the exponent ofSym(n).
Lemma 2.4. For primen, an infinite solvable-by-finite primitive (irreducible) subgroupH
of SL(n,Z)is solvable.
Proof. LetKEHbe solvable of finite index. Sincenis prime,Kis scalar or irreducible. IfKwere scalar thenHwould be finite. ThusK is irreducible. IfKwere monomial over Qthen it would be finite once more; soKis primitive. LetAbe a maximal abelian normal subgroup ofK. ThenAis irreducible, andK =hA, gi because the fieldhAiQ is a cyclic
extension ofQ1nof degreen. Leth∈H anda∈A; thenhah−1 =bgkfor someb∈A. Ifn|kfor allhthenHnormalizes Aand is therefore solvable. Suppose thatnandkare coprime for someh. Sincea7→(bgk)a(bgk)−1is aQ-automorphism of the fieldhAi
Q, and
(bgk)nis fixed by this automorphism,(bgk)nis scalar. Hencebgkhas finite order. Suppose that a ∈ A has infinite order. Of course hah−1
6
= bgk, implying thathnormalizes hai. Sinceais irreducible (i.e.,haiis irreducible),Hnormalizes hAiQ, and again we conclude
thatHis solvable.
Corollary 2.5. Letnbe prime. IfH ≤ SL(n,Z) is infinite, non-solvable, and primitive, thenϕp(H)is primitive for almost all primesp.
to compute this finite set. First we find g, h ∈ H such that [gk, hk] 6= 1
n, where k is the exponent ofSym(n). (In our experimentsg, hare found by random selection; cf. [1] and see Subsection 3.3.) Letdbe the gcd of the non-zero entries of[gk, hk]−1
n. Then ϕp(H)is non-monomial ifp6∈π(d). Finally, we test whetherϕp(H)is monomial for each p∈π(d), using, e.g., [25].
2.4. Solvability. Zassenhaus’s theorem [29, p. 136] implies existence of a boundδ=δ(n) on the derived length of solvable subgroups ofSL(n,F)that depends only onn, not onF. See, e.g., [29, p. 136] for an estimate ofδdue to Dixon.
LetH ≤SL(n,Z)be non-solvable. We sketch a procedurePrimesForSolvable(H, δ) that returns the set of primespsuch thatϕp(H)is solvable andϕp(H)6= SL(n, p). Take a non-trivial iterated commutator inH. As usual, we do this by random selection inH, or by lifting toHfrom a (non-solvable) congruence image: pick[¯h1, . . . ,¯hδ+1]6= 1ninH; then¯ g= [h1, . . . , hδ+1]∈His as required. Letdbe the gcd of the non-zero entries ofg−1n. Thenϕp(H)is non-solvable ifp6∈π(d). Solvability ofϕp(H)forp∈π(d)can be tested using [25]. We have proved the following.
Lemma 2.6. IfHis non-solvable thenϕp(H)is non-solvable for almost all primesp. We get better bounds on derived length for irreducible groups in prime degree.
Lemma 2.7. Letnbe prime. An irreducible solvable subgroupGofGL(n,F)has derived lengthd≤6.
Proof. A monomial group G is an extension of its subgroup of diagonal matrices by a solvable transitive permutation group of prime degree. Such permutation groups are meta-cyclic, so d ≤ 3. Suppose thatG is primitive.By [30, Theorem 3.3, p. 42], there exists EEGof derived length at most2, such thatG/Eis isomorphic to a subgroup ofSL(2, n). Sinceδ(SL(2, n))≤4(see, e.g., [29,§21.3]), we getd≤6as required.
Remark2.8. Ifn= 2,3andG≤SL(n,F)thend≤4,d≤5, respectively.
2.5. Isometry. We say that G ≤ GL(n,F) is an isometry group if it preserves a non-degenerate bilinear (symmetric or alternating) form. On the other hand, sinceSL(2,F) = Sp(2,F), we say thatGis not an isometry group ifGdoes not preserve a non-degenerate bilinear form forn >2.
Lemma 2.9. LetG≤GL(n,F)be absolutely irreducible. ThenGis an isometry group if and only iftr(g) = tr(g−1
)for allg∈G.
Proof. Suppose thattr(g) = tr(g−1)for allg∈G. As their characters are equal, the
iden-tity and contragredient representations ofGare therefore equivalent; i.e.,g= Φ(g⊤
)−1
Φ−1
for someΦ∈GL(n,F). Rearranging this equality, we see thatGpreserves the form with
matrixΦ.
The procedure PrimesForIsometryaccepts an absolutely irreducible subgroupH of SL(n,Z)that is not an isometry group. It selectsh∈Hsuch thata:= tr(h)−tr(h−1
We will need to check not only whether a congruence image ofHpreserves a form, but whether it lies in the similarity group generated by a full isometry group and all scalars. This is achieved withPrimesForSimilarity(H), which selectsh = [h1, h2]∈H such
thata := tr(h)−tr(h−1
) 6= 0. Clearlyϕp(H) is in a similarity group only ifp ∈ π(a). Hence
Lemma 2.10. Suppose thatH ≤ SL(n,Z)is absolutely irreducible and not an isometry group. Then for almost all primesp,ϕp(H)does not lie in a similarity group overZp.
3. ALGORITHMS FOR STRONG APPROXIMATION
We proceed to formulate an algorithm that realizes strong approximation in prime degree n. That is, the algorithm computes Π(H) for any dense inputH ≤ SL(n,Z). We also computeΠfor dense subgroups ofSL(2n,Z)containing a transvection.
3.1. Density in prime degree. For the entirety of this subsection,nis prime.
By [2] (cf [22, p. 397]), the setCof maximal subgroups ofSL(n, p)is a union of certain subsetsC1, . . . ,C9. For eachi, we determine all primespsuch that ϕp(H) could be in a group inCi. Hence, we provide criteria forHto surject ontoSL(n, p)for almost all primes p. These conditions turn out to be equivalent to density. They constitute the background of our main algorithm and obviate any need to test density of the input (as in, say, [6, Section 5]).
We start with an auxiliary statement forC9(called classSin [3, Chapter 8]).
Lemma 3.1. There is a bound in terms ofnon the order of subgroups ofSL(n, p)that are contained solely in groups inC9.
Proof. Suppose that U ≤ SL(n, p) lies only inC9 and not in Ci fori 6= 9. The perfect residuumU∞
(i.e., the last term of the derived series ofU) is therefore a simple absolutely irreducible subgroup ofSL(n, p). If we show that the order ofU∞
is bounded, thenU ≤ Aut(U∞
)also has bounded order. Thus, without loss of generality,U = U∞
from now on.
Prime degree faithful representations of quasisimple groups are classified in [24, Theo-rem 1.1]. The orders of the groups in classes (10)–(27) of this classification are bounded absolutely (i.e., by a bound not depending on n or p). The orders of groups in classes (2)–(9) are bounded by a function ofn.
Class (1) groups are of Lie type in characteristic pin the Steinberg representation [15], whose degreenis thep-part of the group order. For each class of groups of Lie typeGm(p), this p-part ispa witha ≤ 1 for only finitely many values ofm. So class (1) is finite for primen.
Finally we come to the case excluded by [24, Theorem 1.1], namelyU/Z(U)∼= Alt(m) form >18. As a consequence of [16, 18], there are only finitely many degreeslsuch that Sym(l)and thusAlt(l)has a faithful (projective) representation of degreem.
• PrimesForAbsIrreducible(H) • PrimesForMonomial(H)
• PrimesForSolvable(H, δ), whereδis a bound on the derived length of a solvable linear group of degreen
• PrimesForSimilarity(H)
• PrimesForOrder(H, k)wherekis a bound on element orders for groups of degree ninC6∪ C9.
Theorem 3.2. Assuming termination for inputH,PrimesForDense(H)returnsΠ(H). Proof. Each prime returned must lie in Π(H). Conversely, let p be a prime such that ϕp(H)6= SL(n, p). Thenϕp(H)is in a group in someCi,1≤i≤9. For eachi, we show that (at least) one of the subsidiary procedures returnsp.
C1: hereϕp(H)is reducible, sopis returned byPrimesForAbsIrreducible(H). C2:pis returned byPrimesForMonomial(H).
C3: for prime n, the stabilizers of extension fields are solvable, so p is returned by
PrimesForSolvable(H, δ).
C4,C7: since the degree of a tensor product is the product of the factor degrees, andnis
prime, these classes are empty.
C5is empty over fields of prime size.
C6consists of groups whose structure depends onnbut not onp[3, Section 2.2.6]. The
number of such groups (and thus the largest order of an element in any one of them) is bounded, and soPrimesForOrder(H, k)returnsp.
C8: the groups in this class preserve a form moduloZ(SL(n, p)). Hence the derived
group ofϕp(H)preserves a form andpis returned byPrimesForSimilarity(H). C9: by Proposition 3.1, the number of groups in this class is finite. Thus (as withC6)
PrimesForOrder(H, k)returnsp.
Remark3.3. UsingGAPand tables in [3, Chapter 8], we can calculate bounds on the order of groups inC6∪ C9(and hence bounds on their element orders) for smalln. Forn= 2,3,
5,7,11, these bounds are10,21,60,84,253, respectively.
Remark3.4. PrimesForDensesimplifies in small degrees. Ifn ≤ 3 then the groups in C2 are solvable, so PrimesForSolvableoverrides PrimesForMonomial. In degree 2,
PrimesForSimilarityis also redundant.
IfPrimesForDense(H)terminates thenΠ(H)is finite, i.e.,H is dense [26, p. 3650]. Next we prove the converse. This leads to a characterization of density inSL(n,Z).
Lemma 3.5. IfH is irreducible, not solvable-by-finite, and not an isometry group, then
Π(H)is finite.
Proof. Each constituent output set is finite by Lemmas 2.2, 2.3, 2.6, 2.10, and 3.1.
Lemma 3.6. IfHis infinite, non-solvable, primitive, and not an isometry group, thenΠ(H)
Proof. As the previous proof, but relying on Corollary 2.5 instead of Lemma 2.3.
Lemma 3.7. Suppose that ϕp(H) = SL(n, p)for some primep, wherep > 3ifn = 2.
ThenHis infinite, non-solvable, and primitive. Furthermore,H is not an isometry group.
Proof. SinceSL(n, p)is absolutely irreducible and non-solvable, the same is true ofH. A monomial subgroup ofSL(n,Z)cannot surject ontoSL(n, p)because it has an abelian nor-mal subgroup whose index is too snor-mall. The remaining assertion follows from Lemmas 2.1
and 2.9.
Lemmas 3.6 and 3.7 yield
Corollary 3.8(Cf. p. 396 of [22] and Proposition 1 of [23]). Ifϕq(H) = SL(n, q)for one
primeq >3, thenϕp(H) = SL(n, p)for almost all primesp.
Corollary 3.9. The following are equivalent.
(i) His dense.
(ii) Hsurjects ontoSL(n, p)modulo some primep, wherep >3ifn= 2.
(iii) His infinite, non-solvable, primitive, and not an isometry group.
(iv) His irreducible, not solvable-by-finite, and not an isometry group.
Remark3.10. Letn= 2. ThenHis dense if and only ifHis not solvable-by-finite; which is equivalent toHbeing infinite and non-solvable.
To round out the subsection, we give one more set of criteria for density in odd prime degree.
Lemma 3.11. Letn >2. IfHcontains an irreducible element and is not solvable-by-finite thenHis dense.
Proof. We appeal to Lemma 3.5. Leth ∈ H be irreducible. Suppose that H preserves a form with (symmetric or skew-symmetric) matrixΦ. Thenx 7→ Φx⊤
Φ−1
defines aQ -automorphism ofhhiQ of order2. ButhhiQis a field extension of odd degreen. HenceH
is not an isometry group.
Corollary 3.12. Forn > 2, a finitely generated subgroup ofSL(n,Z)is dense if and only if it contains an irreducible element and is not solvable-by-finite.
Remark3.13. Lemma 3.6 allows us to replace ‘not solvable-by-finite’ in Lemma 3.11 and Corollary 3.12 by ‘infinite non-solvable primitive’, or by ‘infinite non-solvable’ ifn = 3 (cf. [19, p. 415], [20, Theorem 2.2]).
3.2. Algorithms for groups with a transvection. In [6, Section 3.2] we gave a straight-forward procedurePrimesForDenseto computeΠ(H)ifHis dense inSL(2n+ 1,Z)or Sp(2n,Z) and contains a known transvection. The caseH ≤ SL(2n,Z) was left open. Now we close that gap.
Lemma 3.14. Suppose thatH ≤SL(2n,Z)contains a transvectiont. ThenHis dense if and only ifN :=htiH is absolutely irreducible andtr(h)6= tr(h−1
Proof. Suppose thatHis dense. ThenN is absolutely irreducible by [6, Corollary 3.5]. If tr(h) = tr(h−1)for allh∈N, then by Lemma 2.9 there is a form with matrixΦsuch that
hΦh⊤
= Φ. SinceNEHandN is absolutely irreducible,hΦh⊤
=αΦfor allh∈Hand someα∈Q(see, e.g., [3, Lemma 1.8.9, p. 41]). This contradicts density ofH.
Now suppose thatN is absolutely irreducible andtr(h) 6= tr(h−1
) for someh ∈ N. Thenϕp(N)is absolutely irreducible andϕp(tr(h))6=ϕp(tr(h−1))for almost all primes p. So there arep >3andg∈ϕp(N)such thatϕp(N)is absolutely irreducible andtr(g)6=
tr(g−1
). Sinceϕp(N)is generated by transvections, the theorem of [31, p. 1] implies that ϕp(N) = SL(2n, p)orSp(2n, p). Since the latter possibility is ruled out by Lemma 2.9, we must haveϕp(H) = SL(2n, p)and soHis dense (see [23, Proposition 1]).
The procedurePrimesForDense(H, t), based on
Lemma 3.14, accepts dense H ≤ SL(2n,Z) containing a transvection t, and returns Π(H). It combines PrimesForAbs-Irreducible(N) and PrimesForIsometry(N), checking whether ϕp(H) = SL(2n, p) for each p in the union of their outputs. See [6, Section 3] for an algorithm to compute a basis ofhNiQ without computing (a full
generat-ing set of) the normal closureN. Similarly, the application ofPrimesForIsometrydoes not require computingN, and just randomly selectsh∈N such thattr(h)6=tr(h−1
).
3.3. General considerations. We comment further on the operation of our algorithms. When selecting (pseudo-)random elements ofSL(n,Z)for some subprocedures, we seek just one element with a nominated property. These will be plentiful in dense subgroups. Hence we do not aim for any semblance of a random distribution, but randomly take words of length5in the given generators. If these repeatedly fail to have the desired property then we gradually increment the word length.
At the start of the calculation we also select (e.g., by computing the orders, or invoking composition tree on images of H modulo different primes [25]) a prime p0 > 3 such
that ϕp0(H) = SL(n, p0). The properties of elements that we are seeking may then be
maintained modulo p0. That is, instead of searching inH, we search for an element¯hin
ϕp0(H)that has the desired properties (overZp0) and lift to the pre-imageh∈H.
Each of the subsidiary procedures forPrimesForDense(H)returns a positive integerd divisible by every primepsuch thatϕp(H)is in the respective class of maximal subgroups ofSL(n, p). However,dcan have prime factors not inΠ(H). Furthermore, these factors might be so large as to make factorization ofdimpractical, or make the test of the congru-ence image overly expensive. Thus we do not factordfully, but only attempt a cheap partial factorization (e.g., by trial division and a Pollard-ρ algorithm). Ifddoes not factorize, or has large prime factors (magnitudes larger than the entries of the input matrices), then we compute another positive integerd′
using the same algorithm but with different choices of random elements, and replacedbygcd(d, d′
).
4. EXPERIMENTING WITH LOW-DIMENSIONAL DENSE SUBGROUPS
to obtain important information about groups. In particular we describe all congruence images ofH, as explained in the next subsection.
4.1. Computing all congruence images. Let H ≤ SL(n,Z) be dense. As in [6, Sec-tion 2.4.1], define Π(˜ H) = Π(H)∪ {2} ifϕ2(H) = SL(n,2)and ϕ4(H) 6= SL(n,4);
whereas Π(˜ H) = Π(H) otherwise. Note that the disparity between Π(˜ H) and Π(H) can arise only whenn ≤ 4, and M(H)is even but 2 6∈ Π(H). By [6, Theorem 2.18],
˜
Π(H) =π(M(H)). Ifn >2thenϕk(H) = ϕk(cl(H))for allk; soΠ(˜ H) = ˜Π(cl(H)). We may therefore assume thatHis arithmetic, of levelM. Leta= gcd(k, M), sok=abc, π(b)⊆π(a), andgcd(c, a) = 1. Thenϕk(H)∼=H/(H∩Γk)∼=HΓk/Γkis a subgroup of
Γab/Γk×Γc/Γk. It is not difficult to show thatϕk(H)splits as a direct product ofΓab/Γk withQ := ((HΓk)∩Γc)/Γk. SinceΓab/Γk ∼= SL(n,Z)/Γc, this expressesϕk(H)as a direct product ofQwith a subgroup isomorphic toSL(n,Zc). Hence the task in describ-ing all congruence images ofHboils down to computing with the quotientQofϕk(H)in
SL(n,Zk); in effect, ranging over all divisors ofM. Ifn= 2then the congruence subgroup property does not hold, and we can only handlek=pprime.
In some of the examples below we describe the congruence quotient modulo the level M, exhibiting which parts of its structure arise for various prime powers. We give this as an ATLAS-style composition structure [4] (separating composition factors by dots; cf. [3]), marked up to show the prime powers for which each factor first arises. We emphasize that these have been generated ‘semiautomatically’ usingsomecomposition series that refines the congruence structure, not necessarily the nicest or best possible. One example from Table 1 is a group of level34
5·19with quotient structure
34
34 .33
33 .33
32 .52
.2.2.2
5 .3.3
3,5
.L2(19)
19
In the standard notationLm(q) := PSL(m, q), this has congruence imageL2(19)modulo
19, which is a simple direct factor not interacting with the other primes. The quotient modulo 3has structure 3.3 (and is almost certainly the group 32). The quotient modulo
5 is 52
.2.2.2.3.3, forming a subdirect product with the quotient of order 3in which the full factor3.3is glued together. Modulo9the group possesses a factor33
(of the possible 33·3−1
= 38
), modulo 27another factor 33
, and modulo 81 a factor 34
. (Since34
is the prime power dividing the level, the quotient modulo 243 would contain a full 38.) The
structural analysis in [6, Section 2] proves that the exponent for pi+1
cannot be smaller than the exponent for pi. The name indicates all proper prime powers dividing the level. Thus ‘empty’ factors
pa
are possible if the group has no elements on that level.
Experimental results are displayed in Tables 1 and 2 (writing Am for Alt(m) and Sm forSym(m)). Our actual implementation computesΠ =˜ π(M)rather thanΠ(H). We do not stateΠ(H); as noted above, this set almost always coincides withπ(M).
Experiments were performed on a 2013 MacPro with a 3.7 GHz Intel Xeon E5 utilizing up to 8GB of memory. The software can be accessed at
http://www.math.colostate.edu/˜hulpke/arithmetic.g. Some
4.2. Low-dimensional dense subgroups. Our examples in this subsection come from a family of low-dimensional representations of finitely presented groups, as defined in [19, 20, 21]. For each test groupH we computeΠ(H), incidentally justifying density ofH. Thereafter we computeM(H), |SL(n,Z) :H|, and the congruence quotients ofH.
4.2.1. Adopting the notation of [19, p. 414], let
Γ :=hx, y, z|zxz−1=xy, zyz−1=yxyi;
this is the fundamental group of the figure-eight knot complement. LetF = hx, yi. In [19] two families of representationsβT,ρk ofΓinSL(3,Z)were constructed. Section 4 of [6] reports on
experiments withβT for a range ofT andρk fork = 0,2,3,4,5. The groupsρk(Γ),ρk(F)for
k 6= 0,2,3,4,5are of special interest (see [19, Section 5]). However, neither the methods of [19] nor those of [6] facilitate proper study ofρkfor suchk.
Recall that
ρk(x) =
1 −2 3 0 k −1−2k 0 1 −2
,ρk(y) =
−2−k −1 1 −2−k −2 3 −1 −1 2
,ρk(z) =
0 0 1 1 0 −k 0 1 −1−k
.
The results of experiments with ρk(Γ)andρk(F)are collected in Table 1; here M is the level
(which turns out to be the same for bothΓandF), IndexΓis|SL(3,Z) : cl(ρk(Γ))|, and IndexΓ,F is
|cl(ρk(Γ)) : cl(ρk(F)|). The last column is the congruence image ofρk(F)moduloM. Fork= 1,
6,10the groups surject modulo2but not modulo4.
Determination of the relevant primes was instantaneous. The time to calculate level and index increased roughly with the level, from a few seconds fork= 1to about15minutes fork= 20.
k M IndexΓ IndexΓ,F StructureF
1 22
34
210
315
13 22
34
34 .33
33 .
22 .33
32 .3.3
3 .L3(2)
2 6 22
31·43 210
33
7·432
331·631 2·3·5 22
.31.31.2 31
.L2(43)
43
.L2(31)
31
.L3(2)
2 7 34
5·19 26
317
5·13·192
31·127 22
32
34
34 .33
33 .33
32 .52
.2.2.2 5
.3.3 3,5
.L2(19)
19 10 22
34
11·37 214
316
72
13·19·372
67 22
32
5 34
34 .33
33 .
22 .33
32
.11.11.2 11
.3 3
.3 3,11 .L2(37)
37
.L2(11)
11
.L3(2)
2 15 229·241 26
33
5·97·181·2412
19441 2·3·19 229.229.2 229
.L2(241)
241
.L2(229)
229 20 409·421 24
33
5·7·4212
55897·59221 22
3·17 409.409.2 409
.L2(421)
421
.L2(409)
409 TABLE 1.
4.2.2. Next we look at triangle groups∆(p, q, r) =ha, b|ap=bq= (ab)r= 1i
. In [20] representations of∆(3,3,4)inSL(3,Z)are defined by
a7→a1=
0 0 1 1 0 0 0 1 0
, b7→b1(t) =
1 2−t+t2
3 +t2 0 −2 + 2t−t2
−1 +t−t2 0 3−3t+t2
(−1 +t)2
These representations are faithful for allt ∈R, and ift ∈ Zthen the images are dense and non-conjugate for differentt[20, Theorem 1.1]. Ift = 1then the group is conjugate to the one con-structed by Kac and Vinberg [19, p. 422]. PutH1(t) =ha1, b1(t)i.
In [20, p. 8], the following faithful dense representationsH2(t) =ha2(t), b2iof∆(3,4,4)were
constructed:
a7→a2(t) =
1 4 + 3t2
/4 3(6−t+t2 )/2 0 −(4 +t+t2
)/2 −3−t2 0 (4 + 2t+t2
)/4 (2 +t+t2 )/2
,
b7→b2=
0 0 1
1 0 −1
0 1 1
.
In [21, p. 13], faithful representations of∆(3,3,4)inSL(5,Z)are defined by
a7→a3(k) =
1 0 −3−2k−8k2
−1 + 10k+ 32k3
−5−16k2 0 4(−1 +k) −13−4k 3 + 16(1 +k)2
−4 + 16k 0 1−k+ 4k2
3−2k+ 8k2
−2(1 + 3k+ 16k3
) 3 + 16k2
0 k 2k 1−2k−8k2
1 + 4k
0 0 3k 3(−1 +k−4k2
) −2
,
b7→b3(k) =
0 0 −3−2k−8k2
−1 + 10k+ 32k3
−5−16k2 0 1 3 + 4k −13−8k−16k2
4−16k 0 0 −2(1 +k+ 4k2
) 6k+ 32k3
−3−16k2 1 0 −2(1 +k) −1 + 2k+ 8k2
−1−4k
2k 0 1−2k −4k 1
.
Askranges overZ, theH3(k) =ha3(k), b3(k)iare dense and pairwise non-conjugate.
It is known thatH1(t),H2(t),H3(k)are thin [20, 21]. For each of these groups we computed
its levelM and the index of its arithmetic closure inSL(n,Z)for several values of the parameters. See Table 2.
Fort≡1 (mod 4), theH1(t)as far as we tested surject ontoSL(3,2)but not ontoSL(3,Z4).
Runtimes for degree3groups were consistent with the previous example. In degree5, identi-fication of primes was again instantaneous, while the calculation of level and index took about6
minutes forH3(0)and20minutes forH3(3). So we did not try largerk.
4.2.3. Random generators. We constructed subgroups of SL(n,Z) for n = 3 and5 generated by a pair of pseudo-random matrices (via the GAPcommand RandomUnimodularMat). More
than half of the groups so generated surject onto SL(n, p)modulo all primesp(and modulo4). We attempted to verify whether each group is arithmetic by expressing its generators as words in standard generators ofSL(n,Z)and running a coset enumeration with the presentation from [28]. As the enumeration never terminated, we suspect that these groups are not arithmetic (and indeed a random finitely generated subgroup ofSL(n,Z)is likely to be thin [12, 26]).
4.2.4. Further experimentation. Comparing congruence images with finite quotients (obtained, e.g., by the low-index algorithm of [14, Section 5.4]) may help to decide whether a dense rep-resentation of a finitely presented group is faithful, or justify that a group is thin. For example, low-index calculations with the finitely presented groupΓas in Subsection 4.2.1 expose quotients (such asSym(23),Sym(29),Alt(11)≀C2, to name just a few) that cannot be congruence images of
anyρk(Γ), as they do not have representations of suitably small degree. Thusρk cannot be faithful
and normal inΓ; hence a representation ofΓinSL(3,Z)is arithmetic precisely when its restric-tion onF is arithmetic [19, p. 420]; but any virtually free group cannot have a faithful arithmetic representation inSL(n,Z)forn >2.
To illustrate another potential application of our algorithms, we show that faithful dense repre-sentations of the triangle groups∆(3,3,4),∆(3,4,4)inSL(3,Z)orSL(5,Z)are not arithmetic; this includes H1(t),H2(t), H3(k)as in Subsection 4.2.2 (cf. [20, 21]). Indeed, ∆(3,3,4) and
∆(3,4,4)each have a quotient isomorphic to Alt(20). This is not a congruence quotient of an arithmetic group inSL(3,Z)orSL(5,Z), becauseAlt(20)does not have a faithful representation inSL(3, p)orSL(5, p)for anyp.
We also use this example to compare the capability of our algorithm with that of the low-index algorithm. Congruence quotients ofρk(Γ)(modulo any integerm > 1, includingmnot dividing
the level) produced by our algorithms expose quotients ofΓ(such asSL(n, p)for largep) that are infeasible to find through a low-index computation, because these groups do not have a faithful permutation representation of sufficiently small degree. Using a homomorphism search [14, Sec-tion 9.1.1], we find thatΓhas34normal subgroupsN such thatΓ/N ∼= SL(3,5). Applying our algorithm, we identify80values ofkin the range1, . . . ,100, such that5 6∈ Π(ρk(Γ)). For these
k, the kernels of the induced surjectionsΓ → ρk(Γ) → SL(3,5)expose just4 of the34normal
subgroups. This prompts us to conjecture that theρkwill not expose allSL(n, p)quotients ofΓ.
A L G O R IT H M S F O R E X P E R IM E N T IN G W IT H Z A R IS K I D E N S E S U B G R O U P S 1 3
H1(1) 2 2
3·52
19 213
33 55
13·193 31·127
22 .56
52 .3
19 .2.2
5,19
.32
.2
3 .22
.3
3,5
.A6 19
.L3(2) 2
H1(2) 2
3
27 7 25
23 .25
22 .22
.3
2 H1(5) 2
3
7·19·31 223 36
52 72
192 313
127·331
23 .
22 .3
31 .192
.2.2.2
19 . 3
7,19,31
.A6 31
.L2(7) 7
.L3(2) 2 H1(9) 2267 29327211217·31·67
22 .672
.2.22
.3
67
.L3(2) 2 H1(10) 2
3
3·7 216
34 72
13·19 25 23
.24 22
.7.7
7 .3.3.2
3,7
.22
.3
2,3,7
H1(12) 237·31 21635527319·31·331 25 23
.24 22
.31.31.2
31 .22
2,31
. 3
2,7,31
.L2(7) 7 H1(50) 2
3
601 214
33 54
72
13·43·601·9277 25 23
.24 22
.601.601.2
601 .22
.3
2
,601
H1(100) 2 3
3·19·43 219 39
5·73
11·13·19·43·127·631 25 23
.24 22
.43.43
43
.19.19
19,43
.3.3.2
3,19,43
. 22
.3
2,3,19,43
H2(2) 2213 27327·61 22 22
.13.13.2.2
13 .22
.3
2,13
.L2(13) 13
.2
2,13
H2(10) 2 2
5·109 210
34 52
72
31·571 22 22
.109.109.2.2
109
.22 2,109
.3
2
.L2(109) 109
.A5 5
. 2
2,5,109
H2(12) 283417 2383157·13·307 26 28
.26 27
.25 26
.23 25
.23 24
.34 34
.22 23
.33 33 .
22 .33
32 .172
.2
17 .3.3.2
3,17
. 22 2,3,17
. 3
2,17
. 2
2,3,17
.L2(17) 17 H2(50) 2
2 52
13·193 216 34
57 72
31·61·1783 22 22
.53 52
.193.193.2
193
.13.13.2.2
13,193
.22 2,193
.3
2
.L2(193) 193
.L2(13) 13
.A5 5
. 2
2,5,13,193
H3(0) 2373192 25231455732192131·127·151·181·911·2801 28 23
.71 8
73 .26
22 .78
72 .1912
192 .192
.2.2.2
19 .26
.3
2,19
.L3(2) 2,7
H3(1) 26672 2873652731141723126720·449·761·26881 221 26
.211 25
.28 24
.25 23
.24 22
.6712 672
.67.67.2
67 .22
.3
2,67
H3(2) 2 3
13·2112 233 39 56 75 135 312 37·532
61·113·197
·21120
·1361·30941·292661
216 23
.214 22
.21112 2112
.2112
.2.2.2
211
.13.13.13.13
13,211
. 26
.3
2,13,211
.L2(169) 13
.L3(2) 2
H3(3) 267·112412 210638513781120192 · 29231·412061
·1723·2801·3221·579281
221 26
.211 25
.28 24
.25 23
.24 22
.1112 112
.4112 412
.41.41
41
.11.11.2
11,41
.
7.7.7.7.2
7,11,41
. 22
.3
2,7,11,41
.L2(7).L2(7) 7
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SCHOOL OFCOMPUTERSCIENCE, UNIVERSITY OF ST ANDREWS, NORTHHAUGH, ST ANDREWS
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IRELAND, GALWAY, UNIVERSITYROAD, GALWAYH91TK33, IRELAND
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