Practical computation with linear groups over infinite domains
aa r X i v : . [ m a t h . G R ] M a y PRACTICAL COMPUTATION WITH LINEAR GROUPSOVER INFINITE DOMAINS
A. S. DETINKO AND D. L. FLANNERY
Abstract.
We survey recent progress in computing with finitely gener-ated linear groups over infinite fields, describing the mathematical back-ground of a methodology applied to design practical algorithms for thesegroups. Implementations of the algorithms have been used to performextensive computer experiments. Introduction
Motivation.
Linear groups (synonymously, matrix groups) have beenstudied from the beginning of group theory. Matrices afford a convenientrepresentation of groups that frequently arise in algebra, geometry, numbertheory, topology, and theoretical physics. Enhancements of technology andcomputer algebra systems have initiated a new phase in this classical subject,concerned with the design and implementation of algorithms for practicalcomputation.Computing with matrix groups over finite fields is well-established [29].The situation for linear groups over infinite domains is less advanced. Con-sequently we are motivated to obtain efficient methods, algorithms, andsoftware for computing in this class of groups.1.2.
Representing linear groups in a computer.
Input to any algo-rithm should be a finite set. Thus, in the first instance, we consider finitelygenerated linear groups. Certain linear groups that are not finitely gener-ated can still be designated by a finite set—say, of polynomials, in the caseof linear algebraic groups. Whereas an arbitrary linear group need not befinitely generated or algebraic, these are two major classes covering manyapplications.Finitely generated linear groups are amenable to symbolic computation.Let F be a field of characteristic p ≥
0, and suppose that G = h S i where S = { g , . . . , g r } ⊆ GL( n, F ). Then G is defined over a finitely generatedextension of the prime subfield of F . The classification of such field exten-sions implies that G is a subgroup of GL( n, L ), where L is a finite degreeextension of P (x , . . . , x m ), P is a number field or finite field F q of size q forsome p -power q , and the x i are algebraically independent indeterminates.This means that essentially we only have to deal with the aforementionedcategories of fields. All of these are supported by the computer algebrasystem Magma [5].
We could restrict the ground domain to the subring R ⊆ F generated byall entries of the g i and g − i . After replacing the original field by such a ring,we apply congruence homomorphism techniques to transfer computing over R to computing over a quotient ring R/ρ . If ρ is a maximal ideal then R/ρ isa finite field, and in that event the computational complexity is amelioratedby avoiding work over an infinite ring. We also gain access to the machineryfor matrix groups over finite fields. See [20, Section 2] for details.1.3.
Properties of linear groups.
We rely on classical theory of lineargroups [21, 34, 36]. Two basic properties are crucial in our endeavours.One of these provides background for the computational methods; the othersteers our overall strategy.First, we recall that each finitely generated linear group G is residuallyfinite. Moreover, G is ‘approximated’ by matrix groups of the same degreeover finite fields. This approximation is effected by congruence homomor-phisms ϕ ρ : GL( n, R ) → GL( n, R/ρ ). Since each non-zero element of R isabsent from at least one ideal, and R/ρ is a finite field if we choose ρ to bemaximal, the congruence images ϕ ρ ( G ) realize the finite approximation.A famous result of J. Tits [35] asserts that each finitely generated lineargroup over a field either is solvable-by-finite (virtually solvable), or containsa free non-abelian subgroup. The Tits alternative thereby divides finitelygenerated linear groups into two very different classes which require separatetreatment. 2. Computing with virtually solvable groups
Method of finite approximation.
Our techniques for computingwith solvable-by-finite groups are broad-based and uniform, enabling us tosolve a range of problems by similar algorithms. Underlying these featuresare deep results about the congruence subgroup G ρ := G ∩ ker ϕ ρ . Theorem 2.1.
There exist maximal ideals ρ of R such that (i) All torsion elements of G ρ are unipotent. In particular, G ρ is torsion-free if char R = 0 . (ii) If G is solvable-by-finite then G ρ is unipotent-by-abelian as long asone of the following holds: char R > n ; char R = 0 and char( R/ρ ) >n ; R is a Dedekind domain of characteristic zero and p ∈ ρ \ ρ p − for some odd prime p . See [36, Chapter 4] or [20, Section 2] for a proof of Theorem 2.1 (i).Proofs, and extra conditions on R and ρ guaranteeing the outcome in The-orem 2.1 (ii), are given in [37].Our method begins by selecting ρ according to the strictures of Theo-rem 2.1, and computing the congruence image ϕ ρ ( G ) ≤ GL( n, R/ρ ). Thenwe examine the structure of G ρ . We call ϕ ρ for ρ as in Theorem 2.1a W-homomorphism . Algorithms to compute W-homomorphisms ϕ ρ andtheir corresponding congruence images ϕ ρ ( G ) were developed in [18, 20]. RACTICAL COMPUTATION WITH LINEAR GROUPS OVER INFINITE DOMAINS 3
These compute normal generators of G ρ , i.e., a finite set N ⊆ G such that G ρ = h N i G . The set N is found by means of a presentation of ϕ ρ ( G ), com-puted using algorithms for matrix groups over finite fields [4, 29]. For ourpurposes, any relevant information about G ρ can be deduced from N ; thefull normal closure h N i G is not needed.2.2. Recognizing the type of a matrix group.
Armed with practicalmethods, we proceed to the development of algorithms. Given S ⊆ GL( n, F )we must first recognize the ‘type’ of G = h S i . Once this is done, G canbe investigated using tools that are most appropriate for the group type.Below we note algorithms to recognize the type of G (each of which requiresselection of a single W-homomorphism ϕ ρ ). These algorithms additionallyjustify that the relevant problems are decidable for finitely generated lineargroups over infinite fields.2.2.1. Finiteness.
In characteristic zero, G is finite if and only if G ρ = h N i G = 1. If char F = p > G ρ is a p -group, i.e., unipotent. See [20, Section 4].2.2.2. Virtual solvability and other properties.
We can recognize whether G is solvable-by-finite: a computational realization of the Tits alternative. Forthis it is enough to test whether G ρ is unipotent-by-abelian, i.e., conjugateto a block-triangular group with all main diagonal blocks abelian. This testis carried out using manipulations with the enveloping algebra of G ρ over F , as explained in [18, Section 3]. Although it decides whether a finitelygenerated linear group contains a free non-abelian subgroup, our algorithmdoes not construct one.Algorithms to test whether G is solvable, (virtually) nilpotent, abelian-by-finite, or central-by-finite. use a mix of ideas similar to the above [18,Section 5].2.3. Investigating the structure of linear groups.
Finite groups. If G is found to be finite then we can obtain an iso-morphic copy over a finite field F q . In characteristic zero, G ∼ = ϕ ρ ( G ) forany W-homomorphism ϕ ρ ; in positive characteristic, repeated selection of ρ may be needed to get an isomorphism ϕ ρ [20, Section 4.3]. Algorithms formatrix groups over finite fields may then be applied to ϕ ρ ( G ) ≤ GL( n, q ) toanswer questions about the original group G .2.3.2. Solvable groups.
Linear groups play a central role in the theory ofinfinite solvable groups. However, in designing algorithms for solvable lineargroups we encounter serious obstacles, such as lack of decidability of vari-ous problems [25, Chapter 9]. To further illustrate this point, we make acomparison with polycyclic groups. Virtually polycyclic groups are finitelygenerated and Z -linear. On the other hand, finitely generated (virtually)solvable linear groups need not be finitely presentable, they might have sub-groups that are not finitely generated, and they do not satisfy the maximal A. S. DETINKO AND D. L. FLANNERY condition on subgroups. Computing becomes feasible with groups of finitePr¨ufer rank, which are solvable-by-finite and Q -linear. Hence, we can testwhether a finitely generated linear group G over a number field F has finiterank. Furthermore, if G is (virtually) solvable then we can: compute thetorsion-free rank (Hirsch number) of G , and bounds on its Pr¨ufer rank; testwhether | G : H | is finite, for a finitely generated subgroup H of G ; constructa generating set of the completely reducible part of G (this includes test-ing whether G is completely reducible or unipotent). More generally, thesealgorithms work for solvable-by-finite groups G over any field, albeit withqualifications on G in positive characteristic. The papers [18, 19] containlengthier discussion of the above.Nilpotent-by-finite linear groups are more tractable. Algorithms for theseare given in [10] and [18, Section 5]. Computing with polycyclic linear groupsis a separate topic (see, e.g., [2, 3]) beyond the remit of our survey.2.3.3. Implementation.
Many of our algorithms for virtually solvable groupswere developed jointly with Eamonn O’Brien. Implementations are availablein
Magma ; see [17]. Experimental results are reported in [18, Section 6],[19, Section 4.5], and [20, Section 5].3.
Dense and arithmetic groups
The methods of Section 2 could be developed further. However, to movebeyond virtually solvable groups, new ideas are required.Each linear group H is contained in an algebraic group, with the Zariskiclosure of H being the ‘smallest’ such overgroup. We will suppose that H isa dense (in the Zariski topology) subgroup of an algebraic group. Note thatan algorithm to compute the Zariski closure of a finitely generated lineargroup is given in [9].The most interesting case is Q -groups G ≤
GL( n, C ), i.e., G is defined bya set of polynomials with coefficients in Q . For a subring R ⊆ C , denote G ∩
GL( n, R ) by G ( R ). Recall that H ≤ G ( Q ) is arithmetic if H ∩ G ( Z ) hasfinite index in H and in G ( Z ). In particular, finite index subgroups of G ( Z )are arithmetic. Arithmetic groups are finitely generated and dense. If H ≤G ( Z ) is dense but not arithmetic, then we call H a thin matrix group (after[31]). A major open problem is testing whether finitely generated subgroupsof G ( Z ) are arithmetic. In [11] we provide an algorithm (implemented in Magma ) to test arithmeticity when G is solvable; showing that the problemis decidable with this proviso. The algorithm computes a generating set ofan arithmetic subgroup in G ( Z ), compares its Hirsch number with that ofthe input H ≤ G ( Q ), and tests integrality of H .3.1. Density and computing with linear groups.
Most linear groupsare not virtually solvable [1, 22]. So we cannot expect to handle every finitelygenerated linear group H that is not virtually solvable by a single uniformmethod. Selecting one ideal at a time might not suffice for all problems. RACTICAL COMPUTATION WITH LINEAR GROUPS OVER INFINITE DOMAINS 5
We are viewing H as a subgroup of some algebraic Q -group G , whichmay be assumed semisimple by a standard reduction [8, Chapters 3 and4]. Since H should be dense in G , density testing is a preliminary task. Adeterministic algorithm to test density of H is given in [30], together with aMonte-Carlo algorithm that tests density of H ≤ G ( Z ) for G = SL( n, C ) orSp( n, C ); see also [13, Section 3.2]. These algorithms have been implementedin GAP [23] (see [14]).3.2.
From finite to strong approximation.
We have expanded the con-gruence homomorphism methodology to cover dense subgroups. For certain G , and H ≤ G ( Z ) dense in G , a celebrated result known as the strong ap-proximation theorem [27, Window 9] enables us to compute all congruencequotients of H modulo primes. Both SL( n, C ) and Sp( n, C ) are suitable ex-amples of such G ; from now on G stands for either of these two groups. Strongapproximation implies that if H ≤ G ( Z ) is dense then ϕ p ( H ) = ϕ p ( G ( Z )) forall but finitely many primes p . Denote the set of these exceptional primesby Π( H ). We have developed practical algorithms to compute Π( H ), thusrealizing strong approximation computationally; see [13, Section 3.2], [15],[16]. Our methods for computing Π( H ) draw on classifications of maximalsubgroups in SL( n, p ) and Sp( n, p ), and subgroups of GL( n, p ) with a knowntransvection. Actually, once we have Π( H ) we can find all congruence quo-tients of H [15, 16].3.3. From density to arithmeticity.
Let n > H ≤ G ( Z ) be dense.Then H lies in a unique ‘minimal’ arithmetic group cl( H ), namely the in-tersection of all arithmetic groups in G ( Z ) containing H . Algorithms forarithmetic subgroups of G ( Z ) can therefore be used to study dense sub-groups as well.We gain much mileage from the fact that Γ n := G ( Z ) has the congru-ence subgroup property : each arithmetic group H in Γ n contains a principalcongruence subgroup (PCS), which is the kernel of a congruence homomor-phism ϕ m : Γ n → GL( n, Z m ) for some m . Here Z m = Z /m Z , and m iscalled the level of the PCS. The maximal PCS of H is unique, and its level M = M ( H ) is defined to be the level of H . Similarly, for dense H ≤ Γ n , weassign M ( H ) as the level of cl( H ).3.4. Computing via the congruence subgroup property.
The bedrockof our method for computing with dense groups is the congruence subgroupproperty. It splits our method into two overlapping parts: finding M ( H ),and computing in GL( n, Z m ).3.4.1. Computing the level.
The set π ( M ) of prime divisors of M ( H ) coin-cides with Π( H ), besides minor exceptions for n = 3, 4 and p = 2 (which aredealt with separately); see [13, Section 2.4]. Thus, the strong approximationalgorithms cited in Section 3.2 yield π ( M ). We can also compute the largestpower of p dividing M ( H ) for each p ∈ π ( M ). These two steps constitute A. S. DETINKO AND D. L. FLANNERY the procedure
LevelMaxPCS , which accepts Π( H ) and a generating set S ofa dense group H ≤ Γ n , and returns its level.3.4.2. Computing with matrix groups over Z m . Algorithms for subgroupsof GL( n, Z m ) have intrinsic value. We reduce computing to the situationsof matrix groups over finite fields, and groups of prime-power order. Twomajor steps in the reduction are as follows. Say m = p k . . . p k t t where the p i are distinct primes and all k i are non-zero. Then (essentially by the ChineseRemainder Theorem)(i) GL( n, Z m ) ∼ = GL( n, Z p k ) × · · · × GL( n, Z p ktt )(ii) GL( n, Z p k ) /K ∼ = GL( n, p ), where K = { h ∈ GL( n, Z p k ) | h ≡ n mod p k − } is a p -group.We also use the fact K ∩ G almost always does not have a proper supplementin G , for G = SL( n, Z p k ) or Sp( n, Z p k ) [13, Theorem 2.5].3.5. Algorithms for arithmetic subgroups.
Let H ≤ Γ n be arithmetic.In the application of Section 3.4 to designing algorithms for H , the mainsteps are LevelMaxPCS , and computing with matrix groups over finite rings Z m . One example is the membership test IsIn ( g, H ) which determineswhether g ∈ Γ n is in H ; it merely checks whether ϕ M ( g ) ∈ ϕ M ( H ). Weemphasize that our results imply decidability of membership testing in arith-metic groups in Γ n . An associated algorithm computes | Γ n : H | . Al-though the index could be calculated in the congruence image, i.e., as | ϕ M (Γ n ) : ϕ M ( H ) | , in practice | Γ n : H | is found as a byproduct of computing M [13, Section 2.4.2] (see [12, Section 6] and [13, Section 4]). Since mem-bership testing and computing the index are both decidable, an arithmeticgroup H ≤ Γ n is ‘explicitly given’ as per [24]. Other notable algorithmicproblems for arithmetic subgroups are therefore decidable too.3.6. Further computation with arithmetic subgroups.
Structural analysis.
Arithmetic groups are matrix groups defined overrings, and so their (sub)normal structure is of interest. The procedure
IsSubnormal ( H ) tests whether H is subnormal in Γ n ; Normalizer ( H ) com-putes a generating set of the normalizer of H in Γ n ; NormalClosure ( B )computes a generating set of the normal closure in Γ n of the group gener-ated by B ⊂ Γ n . Other procedures are given in [12, Section 3.2]. Manymore algorithms could be developed along these lines.3.6.2. The orbit-stabilizer problem.
Let n > H ≤ SL( n, Z ) be arith-metic. Given u, v ∈ Q n , the procedure Orbit ( u, v ) tests whether there is g ∈ SL( n, Z ) such that gu = v , and computes g if it exists. Stabilizer ( H, u ) re-turns the (finitely generated) stabilizer of u in H . Both procedures solve therelated orbit and stabilizer problems for the congruence image over Z M andfor the maximal PCS in H acting on Q n . The outputs are then combined.See [12, Section 4]. RACTICAL COMPUTATION WITH LINEAR GROUPS OVER INFINITE DOMAINS 7
Experiments.
The algorithms of this section are joint work with Alex-ander Hulpke. Below we review some experiments illustrating our
GAP implementation of the algorithms and their practicality; see [13, 15, 16] formore.3.7.1. Integral representations of the fundamental group h x, y, z | zxz − = xy, zyz − = yxy i of the figure-eight knot complement are constructed in[26]. For non-zero T ∈ Z , let β T ( x ) = X T and β T ( y ) = Y T where X T = " − T − T T − T − T , Y T = " − − T − TT − . Then β T is a homomorphism and β T ( h x, y i ) ≤ SL(3 , Z ) is arithmetic. Con-struction of these representations was motivated by long-standing prob-lems; such as the conjecture that each arithmetic group in SL( n, Z ) hasa 2-generator finite index subgroup. The conjecture has been settled affir-matively [28]. Still, the subgroups h X T , Y T i merit closer scrutiny. Earlierattempts to compute | SL(3 , Z ) : h X T , Y T i| were stymied by the fact thatthis index may be arbitrarily large. We were able to compute indices us-ing our algorithms (see [13, Section 4.1]). For example, let T = 100; theindex 2 · · · ·
193 were found in 892 . G ( d, k ) = h U, T i where U = d d − k − , T = . For fourteen pairs d, k of integers, G ( d, k ) ≤ Sp(4 , Z ) is the monodromygroup of a generalized hypergeometric ordinary differential equation associ-ated to Calabi-Yau threefolds. Seven of these groups are arithmetic, whilethe rest are thin [32, 33]. To investigate the latter, one could attempt toconstruct arithmetic groups in Sp(4 , Z ) containing them [6]. We successfullycomputed cl( G ( d, k )) for the seven thin groups [13, Table 3]; e.g., it took 25seconds to find the level 2 and the index 2 of G (12 , Where to next?
We outline avenues for future research.New methods and algorithms for algebraic groups and Lie algebras wouldhave an impact on computing with virtually solvable groups. Despite signif-icant progress (cf. Section 2), key algorithmic questions are still unresolved.One of these is membership testing. This problem is known to be decid-able for groups of finite rank. The main challenge is handling the unipotentradical, which is a torsion-free nilpotent group that may not be finitely gener-ated. Lie algebra methods due to P. Hall, and computing in ambient solvable
A. S. DETINKO AND D. L. FLANNERY algebraic groups, are possible approaches. These are similarly promising inthe design of algorithms for structural analysis of virtually solvable lineargroups. We also expect a number of new algorithms for computing with(virtually) nilpotent and (virtually) polycyclic linear groups.Methods based on algebraic group techniques will be productive in ap-plications to non-virtually solvable groups (cf. Section 3). Arithmeticitytesting is open in general, even for subgroups of SL( n, Z ). Indeed, it is notknown whether the problem is decidable. Computing generating sets andpresentations of arithmetic subgroups are supplementary problems (cf. [8,Chapter 6], [7]). Construction of free subgroups would aid in the study ofmatrix groups that are not virtually solvable; ‘large’ free subgroups, i.e.,those that are dense in the Zariski closure, are especially useful. Testingfreeness of finitely generated linear groups is yet another priority.We await breakthroughs that apply computational methods to the so-lution of hard problems in group theory, other areas of mathematics, andfarther afield (cf. Section 3.7). Here we point to computing linear repre-sentations of finitely presented groups: in contrast to the same problem forfinite groups, much remains to be done. Acknowledgments.
We are indebted to our collaborators Willem de Graaf,Alexander Hulpke, and Eamonn O’Brien. We also thank MathematischesForschungsinstitut Oberwolfach, and the International Centre for Mathe-matical Sciences, UK, for hosting our visits under their ‘Research in Pairs’and ‘Research in Groups’ programmes. A. S. Detinko is supported by aMarie Sk lodowska-Curie Individual Fellowship grant (Horizon 2020, EUFramework Programme for Research and Innovation).
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