Zariski Density and Computing in Arithmetic Groups (original) (raw)
Algorithms for Experimenting with Zariski Dense Subgroups
Experimental Mathematics
We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL(n, Z). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n if H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.
Experimenting with Zariski Dense Subgroups
2017
We give a method to describe all congruence images of a finitely generated Zariski dense group H ≤ SL(n,Z). The method is applied to obtain efficient algorithms for solving this problem in odd prime degree n; if n = 2 then we compute all congruence images only modulo primes. We propose a separate method that works for all n as long as H contains a known transvection. The algorithms have been implemented in GAP, enabling computer experiments with important classes of linear groups that have recently emerged.
Algorithms for arithmetic groups with the congruence subgroup property
Journal of Algebra, 2015
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, Zm) is vital to this work. All algorithms have been implemented in GAP.
Large Galois groups with applications to Zariski density
2013
The complexity of doing this in$\mathop{SL}(n, \mathbb{Z})$ is of order O(n4lognlog∣mathcalG∣)logepsilonO(n^4 \log n \log \|\mathcal{G}\|)\log \epsilonO(n4lognlog∣mathcalG∣)logepsilon and in mathopSp(2n,mathbbZ)\mathop{Sp}(2n, \mathbb{Z})mathopSp(2n,mathbbZ) the complexity is of order O(n8lognlog∣mathcalG∣)logepsilonO(n^8 \log n\log \|\mathcal{G}\|)\log \epsilonO(n8lognlog∣mathcalG∣)logepsilon In general semisimple groups we show that Zariski density can be confirmed or denied in time of order O(n14log∣mathcalG∣logepsilon),O(n^14 \log \|\mathcal{G}\|\log \epsilon),O(n14log∣mathcalG∣logepsilon), where epsilon\epsilonepsilon is the probability of a wrong "NO" answer, while ∣mathcalG∣\|\mathcal{G}\|∣mathcalG∣ is the measure of complexity of the input (the maximum of the Frobenius norms of the generating matrices). The algorithms work essentially without change over algebraic number fields, and in other semi-simple groups. However, we restrict to the case of the special linear and symplectic groups and rational coefficients in the interest of clarity.
Computing in arithmetic groups with Voronoï's algorithm
Journal of Algebra, 2015
We describe an algorithm, meant to be very general, to compute a presentation of the group of units of an order in a (semi)simple algebra over Q. Our method is based on a generalisation of Voronoï's algorithm for computing perfect forms, combined with Bass-Serre theory. It differs essentially from previously known methods to deal with such questions, e.g. for units in quaternion algebras. We illustrate this new algorithm by a series of examples where the computations are carried out completely.
The strong approximation theorem and computing with linear groups
Journal of Algebra
We obtain a computational realization of the strong approximation theorem. That is, we develop algorithms to compute all congruence quotients modulo rational primes of a finitely generated Zariski dense group H ≤ SL(n, Z) for n ≥ 2. More generally, we are able to compute all congruence quotients of a finitely generated Zariski dense subgroup of SL(n, Q) for n > 2.
International Journal of Group Theory, 2013
Let FFF be a finite extension of BbbQBbb QBbbQ, ${Bbb Q}_p$ or a global field of positive characteristic, and let E/FE/FE/F be a Galois extension. We study the realization fields of finite subgroups GGG of GLn(E)GL_n(E)GLn(E) stable under the natural operation of the Galois group of E/FE/FE/F. Though for sufficiently large nnn and a fixed algebraic number field FFF every its finite extension EEE is realizable via adjoining to FFF the entries of all matrices ginGgin GginG for some finite Galois stable subgroup GGG of GLn(BbbC)GL_n(Bbb C)GLn(BbbC), there is only a finite number of possible realization field extensions of FFF if GsubsetGLn(OE)Gsubset GL_n(O_E)GsubsetGLn(OE) over the ring OEO_EOE of integers of EEE. After an exposition of earlier results we give their refinements for the realization fields E/FE/FE/F. We consider some applications to quadratic lattices, arithmetic algebraic geometry and Galois cohomology of related arithmetic groups.