Zariski density and computing in arithmetic groups

We develop practical techniques to compute with arithmetic groups H ≤ SL(n, Q) for n > 2. Our approach relies on constructing a principal congruence subgroup in H. Problems solved include testing membership in H, analyzing the subnormal structure of H, and the orbit-stabilizer problem for H. Effective computation with subgroups of GL(n, Z m ) is vital to this work. All algorithms have been implemented in GAP.In [8-10] we established methods for computing with finitely generated linear groups over an infinite field, based on the use of congruence homomorphisms. These have been applied to test virtual solvability and answer questions about solvable-by-finite (SF) linear groups.Computing with finitely generated linear groups that are not SF is a largely unexplored topic. Significant challenges exist: these groups comprise a wide class in which Proposition 1.2. In each of the following situations, Γ n = E n : (i) n ≥ 2 and R is Euclidean or semi-local; (ii) n ≥ 3 and R is a Hasse domain of a global field.Proof. See [16, 4.3.9, pp. 172-173]. P Remark 1.3. O P is a Hasse domain of a global field, F q [x] is Euclidean, and Z m is semi-local.Proposition 1.2 implies that ϕ m maps SL(n, Z) onto SL(n, Z m ). However, ϕ I : GL(n, R) → GL(n, R/I) may not be surjective.. If n > 2 or R = O P then E n , Γ n , and G n are finitely generated. None of the groups E 2 , Γ 2 , or G 2 is finitely generated when R = F q [x].Proof. If n ≥ 3 then Γ n = E n is finitely generated by [16, 4.3.11, p. 174]; hence so too is G n , by [16, 1.2.17, p. 29] and Dirichlet's unit theorem. See [16, 4.3.16, p. 175] and subsequent comments for the remaining claims. P The notation A ≤ f B means that A is of finite index in the group B. For n ≥ 3, Γ n = SL(n, Z) has the congruence subgroup property: H ≤ f Γ n is equivalent to H containing some Γ n,m [3,23]. On the other hand, Γ 2 does not have the CSP [31, §1.1]. Lemma 1.5. For any (1, . . . , 1, −1, 1, . . . , 1) with −1 in position k, this concludes the proof. P Proposition 1.6. If n ≥ 3 and i = j then Γ n,m = t ij (m) Γ n = E Γ n n,m (hence Γ n,m = E G n n,m ). Proof. See [3], [4], or [23]. P Remark 1.7. For n, m > 1, E n,m is not normal in Γ n . Remark 1.8. E n,m 1 ≤ E n,m 2 ⇔ Γ n,m 1 ≤ Γ n,m 2 ⇔ m 2 | m 1 .A PCS in Γ n for n ≥ 3 is the image under ϕ m of a PCS in Γ n .Corollary 1.9. Let I be an ideal of Z m , so Z m /I ∼ = Z a for some divisor a of m. If n ≥ 3 then the kernel Γ n,a of ϕ I on Γ n = SL(n, Z m ) is 1 n + ax ∈ Γ n | x ∈ Mat(n, Z m ) = ϕ m (Γ n,a ) = E Γ n n,a .Furthermore, Γ n,a = t ij (a) Γ n = t ij (a) Gn for any i and j = i.

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“…Postscript. For developments in the area since the publication of [11] (including further experiments with groups from [22]), see, e.g., [12,13].…”

Section: Methodsmentioning

confidence: 99%

Algorithms for arithmetic groups with the congruence subgroup property

Detinko¹,

Flannery²,

Hulpke³

2015

Journal of Algebra

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“…Since H should be dense in G, density testing is a preliminary task. A deterministic algorithm to test density of H is given in [30], together with a Monte-Carlo algorithm that tests density of H ≤ G(Z) for G = SL(n, C) or Sp(n, C); see also [13,Section 3.2]. These algorithms have been implemented in GAP [23] (see [14]).…”

Section: Density and Computing With Linear Groups Most Linear Groupsmentioning

confidence: 99%

“…Computing the level. The set π(M ) of prime divisors of M (H) coincides with Π(H), besides minor exceptions for n = 3, 4 and p = 2 (which are dealt with separately); see [13,Section 2.4]. Thus, the strong approximation algorithms cited in Section 3.2 yield π(M ).…”

Section: 41mentioning

confidence: 99%

Practical Computation with Linear Groups Over Infinite Domains

Detinko

Flannery

2019

Groups St Andrews 2017 in Birmingham

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“…This produced matrices in the respective group for which an upper bound for the length of a word expression was known. The dimensions considered were chosen for be ≤ 8, as the motivating examples from [6] do not exceed this bound.…”

Section: Examplesmentioning

confidence: 99%

“…The groups we are interested in here will be particular infinite matrix groups over rings of integers, namely G = SL n (Z) or G = Sp 2n (Z) with particular generating sets. This is motivated by recent work [6] on finitely generated subgroups of these groups: Given a subgroup S ≤ G given by generating matrices, one often would like to determine whether S has finite index in G, in which case S is called arithmetic.…”

Section: Introductionmentioning

confidence: 99%