Algorithms for arithmetic groups with the congruence subgroup - PowerPoint PPT Presentation

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Algorithms for arithmetic groups with the congruence subgroup property Dane Flannery; joint work with Alla Detinko National University of Ireland, Galway Groups St Andrews 2013

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Tits alternative: a finitely generated linear group over a field F either is SF (solvable-by-finite), or contains a noncyclic free subgroup. We established uniform methodology for computing in the first class of the Alternative, essentially any F : deciding virtual properties, further computing, e.g., calculating ranks of an SF group. (See also work of Assmann and Eick, Beals.) Computing with finitely generated linear groups that are not SF is relatively unexplored. Some fundamental algorithmic problems undecidable. As a starting point, we restrict to arithmetic (sub)groups in the second class of the Alternative. Grunewald and Segal proved decidability of algorithmic problems for ‘explicitly given’ groups.

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL A subgroup H ≤ GL( n, Q ) of an algebraic group G ≤ GL( n, C ) defined over Q is arithmetic if it is commensurable with G Z := G ∩ GL( n, Z ) , i.e., H ∩ G Z has finite index in both H and G Z . Fact (Bass-Lazard-Serre, Mennicke): for n ≥ 3 , Γ n = SL( n, Z ) has the congruence subgroup property (CSP): H ≤ f Γ n ⇔ H contains some principal congruence subgroup (PCS) Γ n,m = kernel of reduction mod m surjection ϕ m : Γ n → SL( n, Z m ) . Note: Γ 2 does not have the CSP . (Let R be a commutative ring with 1 . The kernel of the congruence homomorphism ϕ I : GL( n, R ) → GL( n, R/I ) induced by the natural map R → R/I is called a principal congruence subgroup.) Key idea to compute with arithmetic groups in SL( n, Z ) , n ≥ 3 , is to use congruence homomorphism techniques and computing with matrix groups over finite rings.

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Generation of congruence subgroups Let t ij ( m ) for i � = j denote the transvection with m in position ( i, j ) , 1 s down the main diagonal, and zeros elsewhere. Γ n is generated by all transvections t ij = t ij (1) . In fact Γ n , thus SL( n, Z m ) , is 2 -generated. Lemma . For n ≥ 3 , and any i � = j , Γ n,m = � t ij ( m ) � Γ n . Lemma . A PCS of SL( n, Z m ) for n ≥ 3 is ϕ m ( a PCS of Γ n ) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Sury and Venkataramana proved that if n ≥ 3 then Γ n,m has generating set { t ij ( m ) g | 1 ≤ i < j ≤ n, g ∈ Σ } , where Σ = { 1 n , ( k, l ) , 1 n − 2 e ii − 2 e i +1 ,i +1 + e i +1 ,i | 1 ≤ k < l ≤ n, 1 ≤ i ≤ n − 1 } ; ( k, l ) denoting the permutation matrix obtained from 1 n by swapping rows k and l , and e rs = t rs − 1 n . Note that the number of generators is independent of m . It is not known whether the above is a minimal-sized generating set for Γ n,m ; although we know that Γ ′ n,m = Γ n,m 2 and Γ n,m / Γ n,m 2 has rank n 2 − 1 , so a generating set for Γ n,m has size ≥ n 2 − 1 .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Maximal congruence subgroups Let n ≥ 3 . Lemma . H ≤ f GL( n, Z ) contains a unique maximal PCS (of Γ n ); i.e., ∃ unique m > 0 such that Γ n,m ≤ H , and Γ n,k ≤ H ⇒ Γ n,k ≤ Γ n,m . Note that Γ n,m 1 ≤ Γ n,m 2 ⇔ m 2 divides m 1 . Corollary . Each subgroup of GL( n, Z m ) contains a (perhaps trivial) unique maximal PCS of SL( n, Z m ) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Subnormality For R = Z or Z m , let Z n,k denote the inverse image of the scalars of GL( n, R/kR ) in GL( n, R ) under ϕ k . The level ℓ ( h ) of h = [ h ij ] ij ∈ GL( n, R ) is the ideal of R generated by { h ij | i � = j, 1 ≤ i, j ≤ n } ∪ { h ii − h jj | 1 ≤ i, j ≤ n } . Then ℓ ( A ) := � a ∈ A ℓ ( a ) for A ⊆ GL( n, R ) . Theorem (J. S. Wilson). For n ≥ 3 , H ≤ GL( n, R ) is subnormal if and only if Γ n,k e ≤ H ≤ Z n,k ( † ) for some k , e > 0 . If ( † ) holds then e ≥ d − 1 where d is the depth of H ; and the least possible e is bounded above by a function of n and d only.

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL As special cases we obtain Proposition . Suppose that H ≤ � Γ n = GL( n, R ) has level l . Then Γ n,l ≤ H � Γ n = � H, Γ n,l � ≤ Z n,l . Corollary . H � � Γ n if and only if ℓ ( H ) is the level of the maximal PCS in H . Lemma . H ≤ Γ n = SL( n, R ) is normal in Γ n precisely when it is Γ n : H Γ n = H � normal in � Γ n . Note: if H = � S � then ℓ ( H ) = ℓ ( S ) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Let m = p k 1 1 · · · p k t t where the p i are distinct primes and k i ≥ 1 . Define a ring isomorphism χ : Z m → Z p k 1 1 ⊕ · · · ⊕ Z p kt t by a i ≡ a mod p k i χ ( a ) = ( a 1 , . . . , a t ) , i . Proposition . � � (i) χ extends to isomorphisms GL( n, Z m ) → × t n, Z p ki i =1 GL and � � i SL( n, Z m ) → × t n, Z p ki i =1 SL . i (ii) Let I = � a � be an ideal of Z m , and let I i be the ideal of Z p ki i generated by a i ≡ a mod p k i i . Denote by K I , K I i the kernels of � � ϕ I , ϕ I i on GL( n, Z m ) , GL n, Z p ki respectively. Then i • χ ( K I ) = × t i =1 K I i ; • χ ( K I ∩ SL( n, Z m )) = × t i =1 ( K I i ∩ SL( n, Z m )) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL To answer computational questions about H ≤ GL( n, Z p k ) , consider ϕ p : GL( n, Z p k ) → GL( n, p ) . Approach is then twofold: computing with ϕ p ( H ) in GL( n, p ) , and computing in the finite nilpotent group ( p -group) ker ϕ p ∩ H . We take advantage of efficient algorithms available for both cases. This yields algorithms to, e.g., test membership construct presentations test subnormality and bound depth test solvability, nilpotency etc. for subgroups of GL( n, Z m ) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Let H be a finitely generated subgroup of Γ n = SL( n, Z ) , n ≥ 3 . Vital assumption: H contains some Γ n,m for known m . We apply the menu of algorithms for computing with subgroups of ϕ m (Γ n ) = SL( n, Z m ) , and established knowledge of PCS in Γ n . Some procedures straightforward, e.g.; IsSubgroup ( L, H ) : for finitely generated L ≤ Γ n , returns true if and only if ϕ m ( L ) ≤ ϕ m ( H ) . Normalizer ( H ) returns N Γ n ( H ) , which is the full preimage in Γ n of N SL( n, Z m ) ( ϕ m ( H )) .

A RITHMETIC SUBGROUPS OF SL( n, Q ) C OMPUTING WITH MATRIX GROUPS OVER Z m A LGORITHMS FOR ARITHMETIC GROUPS OF GIVEN LEVEL Theorem . If Γ n,r is the maximal PCS in H , then ϕ m (Γ n,r ) is the maximal PCS in ϕ m ( H ) . IsSubnormal ( H ) Output: true and an upper bound d on its depth if H is subnormal in Γ n ; false otherwise. l 1 := Level ( H ) , l 2 := Level ( MaxPCS ( H )) . 1 � � l e If ∄ e such that l 2 1 then return false , else return true and 2 � d := e ′ + 1 where e ′ := least e such that l 2 � l e 1 . IsNormal ( H ) returns true iff l 2 = l 1 .