Linear groups and computation - Dr Alla Detinko

Linear groups and computation Alla Detinko

RWTH Aachen University 25 May 2016

Alla Detinko

Linear groups and computation

1. Why linear groups? Fundamental mathematical model of transformations in science. Commonly used representation of groups in group theory and its applications. Convenient and efficient way to represent groups in computer.

Alla Detinko

Linear groups and computation

2. Computing with linear groups How to represent a group in computer? Permutations. Matrices over finite fields. Generators and relations. Project. Computing with groups given by a finite set of matrices over an infinite field. Develop general methods and techniques for computing with linear groups over an arbitrary infinite field. Obtain practical algorithms for solution of fundamental computational problems. Design software for computing in this class of groups aimed at solution of mathematical problems by straightforward computing. Alla Detinko

Linear groups and computation

History. L. Babai, R. Beals et. al. G. Nebe, W. Plesken et. al. B. Eick, G. Ostheimer et. al. (polycyclic groups). A. Cohen, W. de Graaf et. al. (algebraic groups).

Obstacles. Undecidability of basic computational problems. Complexity issues (e.g. such as growth of size of matrix entries). Lack of established methods for computing in this class of groups.

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Linear groups and computation

3. Finitely generated linear groups and computing. 3.1 Methods of computing. 3.1.1 Representation in computer: main domains. Given G := hSi, S = {g1 , . . . , gr }, gi ∈ GL(n, F), 1 ≤ i ≤ r, F is a field. Since G is finitely generated, it is defined over a finitely generated extension of the prime subfield of F. Moreover Lemma. For a subfield P ⊆ F there exist x1 , . . . , xm ∈ F (m ≥ 0) algebraically independent over P such that F is a finite extension of P(x1 , . . . , xm ). Examples: main domains 1. Q and algebraic number fields. 2. L = P(x1 , . . . , xm ), P is a number field or Fq . 3. A finite extension of L. Alla Detinko

Linear groups and computation

3.1.2 Method of finite approximation. Let G = hSi. Then G ≤ GL(n, R) for a finitely generated integral domain R ⊆ F determined by the entries of matrices in S ∪ S −1 . Lemma. R is ‘approximated’ by finite fields, i.e. for each maximal ideal ρ of R, R/ρ is a finite field; the intersection of maximal ideals of R is zero. Theorem (Mal’cev). The group G is residually finite. Moreover, G is approximated by matrix groups of degree n over finite fields.

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Linear groups and computation

3.1.3 Congruence homomorphism techniques. Given an ideal ρ ⊆ R, define the congruence homomorphism ϕρ : GL(n, R) → GL(n, R/ρ). Notation ker ϕρ := Γρ (principal congruence subgroup). G ∩ Γρ := Gρ (congruence subgroup). Method: Reduction to, e.g, finite fields via construction of a congruence homomorphism ϕρ such that Gρ satisfy some special properties.

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Linear groups and computation

3.2 What are finitely generated linear groups: classes and structure. Which algorithms do we need? 1 2 3

Recognition algorithms, i.e. testing the type of an input group. Investigation of the structure and properties of the input group. Library of basic functions.

Theorem (J. Tits, 1972). A finitely generated linear group over a field is either solvable-by-finite or it contains a non-cyclic free subgroup. Theorem (Selberg, Wehrfritz). A finitely generated linear group contains a normal subgroup N of finite index such that the torsion elements of N are all unipotent. In particular, if char F = 0 then N is torsion-free. Theorem (Lie-Kolchin-Mal’cev). A linear group G over a field F is solvable-by-finite iff G contains a unipotent-by-abelian subgroup of finite index. Alla Detinko

Linear groups and computation

4. Computing in solvable-by-finite groups. 4.1 Computational analogue of the method of finite approximation. Theorem. Given a finitely generated subgroup G of GL(n, R) there exist maximal ideals ρ of R such that (i) torsion elements of Gρ are unipotent, i.e. Gρ is torsion free if char R = 0. (ii) Gρ is unipotent-by-abelian if G is solvable-by-finite. We call ϕρ for ρ as above an W-homomorphism. Method. 1

Select ρ such that ϕρ is an W-homomorphism.

2

Construct ϕρ (G) (which is a matrix group over the finite field R/ρ).

3

Find a (finite) normal generating set N of the kernel Gρ , i.e. Gρ = hN iG . Alla Detinko

Linear groups and computation

Examples: W-homomorphisms.

Let F = Q. Then R = 1c Z, c ∈ Z. Define ρ = pZ, p 6= 2, p - c. Then ϕρ is a W -homomorphism; in particular, Gρ is torsion-free (Minkowski). Let F = Q(α) be a number field, f (t) be the minimal polynomial of α. Then R = 1c O, O is the ring of integers of Q(α), c ∈ Z, c 6= 0. If p > 2 is a prime dividing neither c nor the discriminant of f (t) then reduction modulo p is a W-homomorphism. Let F := Fp (x). Then R = 1c Fp [x], c = c(x) ∈ Fp [x]. If α ∈ F¯p , c(α) 6= 0 then reduction modulo ρ := (x − α) is a W-homomorphism. N.B. We can construct W-homomorphisms for all finitely generated integral domains R. Alla Detinko

Linear groups and computation

4.2 Algorithms: recognizing types of groups. Testing finiteness. Method (char F = 0): test whether the kernel Gρ of reduction modulo ρ for a W-homomorphism ϕρ is trivial. Testing virtual solvability (computational analogue of the Tits alternative). Method: for a W-homomorphism ϕρ test whether Gρ = hN iG is unipotent-by-abelian (via computing in enveloping algebras). Testing solvability, (virtual) nilpotency, testing whether the group is abelian-by-finite, central-by-finite etc.

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Linear groups and computation

4.3 Algorithms: investigating structure. Finite groups: construct an isomorphic copy over a finite field via W-homomorphisms, and then apply algorithms for matrix groups over finite fields. Solvable-by-finite groups. Motivation: theory of infinite soluble groups. Given a finitely generated solvable-by-finite group over a field F we can: I

Construction of a generating set of the completely reducible (i.e. block-diagonal) part of G. This includes testing whether G is completely reducible and whether G is unipotent (i.e. upper uni-triangular in some basis).

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Computing a ‘basis’ of the unipotent radical U (G), i.e. a finite set of matrices T such that torsion-free ranks of hT i and U (G) are the same. This includes computing Pr¨ufer rank and torsion free rank of G.

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Library of functions for computing with (virtually) nilpotent linear groups: this is a premium class of groups.

Software: Magma package http://magma.maths.usyd.edu.au/ magma/handbook/matrix_groups_over_infinite_fields. Alla Detinko

Linear groups and computation

5. Next step: computing with non-solvable-by-finite groups. 5.1 What are non- solvable-by-finite groups?

Ubiquity of non solvable-by-finite groups: a finitely generated linear group ‘most likely’ is not solvable-by-finite (see e.g. D. Epstein, 1971; R. Aoun, 2011). Computational obstacles: I

Undecidable basic algorithmic problems: - membership test is decidable in finitely generated solvable-by-finite subgroups of GL(n, Q) (Kopytov, 1968); - membership test is undecidable in SL(4, Z) (Michailova, 1958).

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Too wide class of groups.

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Lack of computational methods. Alla Detinko

Linear groups and computation

5.2 Why arithmetic groups ? A subgroup H ≤ GL(n, Q) of an algebraic Q-group T G ≤ GL(n, C) is T arithmetic if H is commensurable with GZ := G GL(n, Z), i.e., H GZ has finite index in both H and GZ . N.B. Arithmetic subgroups are finitely generated. Example. If G = SL(n, C) or Sp(n, C) then finite index subgroups of SL(n, Z), (resp. Sp(n, Z)) are arithmetic. Motivation. One of the central classes of linear groups. Fundamental algorithmic problems are known to be decidable (under some conditions!): Grunewald & Segal, 1980. Computing with arithmetic subgroups is currently in high demand (in particular, due to impact on number theory, topology, physics: see, e.g., P. Sarnak, Notes on thin matrix groups, preprint, 2012). Alla Detinko

Linear groups and computation

5.3 Arithmetic groups with the congruence subgroup property: computing 5.3.1 Set up: congruence subgroup property. Examples: SL(n, Q) vs SL(n, Z). 1

The only proper normal subgroups of SL(n, Q) lie in the finite center consisting of scalar matrices (i.e. SL(n, Q) is ‘close’ to simple).

2

Γn := SL(n, Z) contains infinitely many normal subgroups of finite index. For example, principal congruence subgroups (PCS) Γn,m of level m, i.e. kernels of ϕm : SL(n, Z) → SL(n, Zm ); here Zm := Z/mZ.

3

Γ2 := SL(2, Z) contains normal subgroups of finite index which do not contain a principal congruence subgroup Γ2,m for any m (XIXth century; Klein, Fricke). Moreover, SL(2, Z) contains non-central normal subgroups of infinite index (e.g. ht21 (m)iΓn is of infinite index for m > 5). Alla Detinko

Linear groups and computation

Theorem (Bass, Mennicke et al, 1965-67). Every normal non-central subgroup of SL(n, Z), n > 2, contains a principal congruence subgroup Γn,m for some m. Corollary. Each arithmetic subgroup of SL(n, Z), n ≥ 3, contains a principal congruence subgroup Γn,m for some m. In other terms SL(n, Z), n ≥ 3, satisfies the congruence subgroup property (CSP).

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Linear groups and computation

Examples. SL(2, Z) does not have CSP, while SL(n, OF ), n ≥ 3, Sp(2n, OF ), n ≥ 2, satisfy CSP, OF is ring of integers of a number field F that is not totally imaginary. For details see current surveys on the congruence subgroup problem. 5.3.2 Approach to computing with arithmetic subgroups which satisfy CSP.

(i) Find a principal congruence subgroup Γn,m of H; (ii) Reduce computing to matrix groups over Zm . N.B. A principal congruence subgroup Γn,m of H is not unique, but the maximal PCS Γn,M is unique.

Alla Detinko

Linear groups and computation

5.4 Structure of arithmetic subgroups and computing 5.4.1 Principal congruence subgroup and computing Notation. Let tij = 1n + eij (m) be an elementary matrix of level m, i 6= j; here eij (m) is n × n matrix with m in position (i, j) and zeros everywhere else. Denote En,m = htij (m)|1 ≤ i, j ≤ n, i 6= ji elementary group of level m. Properties of Γn,m . Fact 1. Γn,m = hEn,m iΓn . Fact 2. Γn,m2 ≤ En,m . Fact 3. If |Γn : H| ≤ c then H ≥ Γn,m2 , m = lcm{1, . . . , c}.

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Linear groups and computation

Proposition. Construction of a PCS in an arithmetic subgroup of SL(n, Z), n > 2, is decidable. Proof. Follows from Todd-Coxeter procedure and Fact 3.  Proposition. Let H be an arithmetic subgroup of SL(n, Z). Then the following problems are decidable. Membership test of g ∈ SL(n, Z) in H. Computing an upper bound on the index |SL(n, Z) : H|. Proof. Computing a PCS of H is decidable.  Conclusion. Arithmetic subgroups of SL(n, Z), n > 2, are explicitly given in terms of Grunewald & Segal. Hence solution of all decision problems obtained by Grunewald & Segal valid for arithmetic subgroups of SL(n, Z), n > 2. N.B.: most of algorithms of Grunewald & Segal are not practical. Alla Detinko

Linear groups and computation

5.4.1 Computing in congruence images: subgroups of GL(n, Zm ). Let m = pk11 . . . pkt t where pi are distinct primes and ki ≥ 1. Then GL(n, Zm ) ∼ = GL(n, Zpk1 ) × . . . × GL(n, Zpkt ). t

1

Denote K := {h ∈ GL(n, Zpk )| h ≡ 1n mod pk−1 } (congruence subgroup of level pk−1 ). Then K is a (finite) p-group and GL(n, Zpk )/K ∼ = GL(n, p). Conclusion: computing with matrix groups over Zm is reduced via congruence homomorphism method to computing over finite fields and computing in finite p-groups.

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Linear groups and computation

6. Algorithms for computing with arithmetic subgroups 6.1 Computing principal congruence subgroups: from finite approximation to strong approximation. Aim. Find the level M of the maximal principal congruence subgroup of an arithmetic subgroup H of Γn := SL(n, Z), n > 2. Feature. The method requires computing congruence images of H modulo all primes. N.B.: no need for a generating set of the principal congruence subgroup Γn,M , although we can compute it too.

Alla Detinko

Linear groups and computation

What is the strong approximation? Example:GL(n, Z) vs SL(n, Z). GL(n, Z) does not surject onto GL(n, p) modulo all primes p ≥ 5. SL(n, Z) surjects onto SL(n, p) modulo all primes. Let H be an arithmetic subgroup of SL(n, Z), n > 2. Fact 1. H surjects onto SL(n, p) modulo all but a finite number of primes p. Reason: H is Zariski dense in SLn (strong approximation theorem). Moreover Fact 2. H surjects onto SL(n, p) iff p does not divide the level M of the maximal principal congruence subgroup of H (besides small exceptions for n = 3, 4, p = 2). These provide fast algorithms for computing the level M of the maximal principal congruence subgroup of H. Alla Detinko

Linear groups and computation

6.2 Algorithms: library of functions 6.2.1 Computing the level and related procedures Given: arithmetic subgroup H of SL(n, Z). LevelMaxPCS(H): computes the level M of the maximal principal congruence subgroup Γn,M of H. IsIn(H, g): membership test of g ∈ Γn in H. Index(Γn : H): computing the index.

Alla Detinko

Linear groups and computation

6.2.1 Subnormal structure IsSubnormal(H): tests whether H is subnormal in Γn . Normalizer(H): returns a generating set of NΓn (H). NormalClosure(H): returns a generating set of hHiΓn . IsSL(H): tests whether H = SL(n, Z).

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Linear groups and computation

6.2.3 Orbit-stabilizer problem Given an arithmetic subgroup H of SL(n, Z), and vectors u, v ∈ Qn . Orbit(H, u, v): tests whether ∃ g ∈ H such that g(u) = v and find a g if such exists. Stabilizer(H, u): returns a generating set of StabH (u). N.B.: StabH (u) is a finitely generated group. Method: solution of orbit-stabilizer problem for ϕm (H) acting on Zn and Γn,m ≤ H. Details: see in Section 4 in Journal of Algebra 421 (2015) 234-259.

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Linear groups and computation

6.3 Applications and experimental results 6.3.1 Arithmetic groups and low dimensional topology. Example(Long & Reid, 2011). Define βT (F ) = hXT , YT i ≤ SL(3, Z), where     −1 0 0 −1 + T 3 −T T 2 0 −1 2T , YT =  −T 2 1 −T , T ∈ Z. XT =  −T 0 1 T 0 −1 Theorem (Long & Reid, 2011) Fix an integer T 6= 0. Then the groups βT (F ) are arithmetic and ∩T >0 βT (F ) = 1. Problem (Long & Reid): what are the indices |SL(3, Z) : βT (F )|? N.B.: |SL(3, Z) : βT (F )| → ∞ as T → ∞.

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Linear groups and computation

T 1 2 3 4 5 11 15 19 50 100

M 5 25 3 3 73 27 23 53 367 5·113 797 33 53 67·151 193 67·307 5 2 56 23·1019 27 56 29·67·193

index 23 31 220 3·7 5 13 2 3 13·1801 233 3·72 11·79 24 33 510 13·31·61·3463 6 2 2 3·5 7·1110 19·31·157·199·4051 29 315 512 73 11·13·312 1093 4 10 2 2 3 5·7 11·17·1910 31·43·127·733 227 32 525 73 11·31·79·509·148483 246 36 525 75 11·13·312 67·1783

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Linear groups and computation

t(sec.) 0.3 0.7 5.8 3.5 62.8 616.4 56.4 108.1 1137.9 186.2

6.3.2 Computing with monodromy groups. Let 

1 1 1  0 U :=  d d 0 −k



0 0 0 0  , 1 0  −1 1



1  0 T :=  0 0

0 1 0 0

0 0 1 0



0 1  0  1

with d, k ∈ Z. Then G(d, k) ≤ Sp(4, Z) is the monodromy group of a generalized hypergeometric ordinary differential equation. For 14 pairs (d, k) the group G(d, k) is a monodromy group associated to Calabi-Yau threefolds; seven of these are arithmetic while the rest are ‘thin’. Problem (D. van Straten et. al.). ˆ k)| of Sp(4, Z) which contains G(d, k), and Find an arithmetic subgroup G(d, ˆ k)|. compute the index |Sp(4, Z) : G(d,

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Linear groups and computation

(d, k) (1, 3) (1, 2) (2, 3) (3, 4) (4, 4) (6, 5) (9, 6) (5, 5) (2, 4) (1, 4) (16, 8) (12, 7) (8, 6) (4, 5)

M 2 2 8 22 · 32 26 3 2 · 32 2 · 35 2 · 53 24 22 210 5 2 · 32 27 25

index 6 10 26 · 3 · 5 29 · 35 · 52 220 · 32 · 5 210 · 36 · 52 28 · 314 · 52 8 2 · 33 · 58 · 13 211 · 32 · 5 25 · 5 240 · 32 · 5 217 · 36 · 52 224 · 32 · 5 213 · 3 · 5

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t(sec) 3.910 3.306 4.797 7.155 8.064 9.988 10.671 10.312 5.106 3.515 16.841 21.446 10.771 7.605

Linear groups and computation