Algorithms for arithmetic groups with the congruence subgroup property

Abstract

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.

Practical Computation with Linear Groups Over Infinite Domains

“…An associated algorithm computes |Γ n : H|. Although 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 membership testing and computing the index are both decidable, an arithmetic group H ≤ Γ n is 'explicitly given' as per [24].…”

“…The procedure IsSubnormal(H) tests whether H is subnormal in Γ n ; Normalizer(H) computes 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 generated by B ⊂ Γ n . Other procedures are given in [12,Section 3.2]. Many more algorithms could be developed along these lines.…”

Linear groups and computation

“…We first solve the orbit-stabilizer problem for subgroups of GL(n, Z m ) acting on Z n m , then solve it for Γ n,m acting on Z n , and patch together the two solutions. The second stage of the method uses the next theorem (see [37,Proposition 4.10 and Theorem 4.13]). A similar result is true over Z m , for use in the first stage.…”

Constructive Membership Tests in Some Infinite Matrix Groups

“…Let ϕ : G → SL n (p) the congruence homomorphism; Let H = ϕ(g) ; Let w be a word expression for ϕ(e) as a word in ϕ(g); if w evaluated in g equals e then return w; end Increment p to the next prime; end return failure; Algorithm 2: Word by congruence image Despite its simplicity, this method often works well to find short word expressions (as done for example in [5] to find candidates for generic word expression that then are explicitly proven) and sometimes works faster (and for longer word lengths) than the previous one. However there is no practical way to determine a priori a small modulus that would guarantee success.…”