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My survey of computing with matrix groups at the IPAM workshop on Zariski-dense subgroups in February 2015.

- 1. How dense is dense, or: Computing in linear groups Igor Rivin (Temple University)
- 2. Computing in matrix groups One goal is to understand examples. Another is to think of the computational aspects as another facet of the mathematics.
- 3. Computing in matrix group So, we want to develop quick practical methods on the one hand. And show the EXISTENCE of efficient algorithms on the other. If we are lucky, the two things are the same.
- 4. Basic question: given a collection of matrices A, B, C, D, …, what sort of group do they generate?
- 5. Easy version: the matrices live in a finite group (Say, SL(n, Z/pZ))
- 6. Hard version: the matrices live in, say, SL(n, Z)
- 7. The Easy version: Given a collection of elements in, say SL(n, Z/pZ), we can tell exactly what subgroup they generate (Neumann-Praeger, and others) in (probabilistic) polynomial time.
- 8. The hard version: Given a collection of matrices in SL(n, Z), what can we say?
- 9. Some history
- 10. Once upon a time People understood a lot about lattices. And they could use sophisticated analysis to answer questions about them.
- 11. Then, came the Apollonian packing (well, really, the work of Graham, Lagarias, Mallows, Wilks, Yan)
- 12. Where the group is thin
- 13. And nothing was ever the same
- 14. Super-strong approximation machine The Sarnak School (Sarnak, Helfgott, Bourgain, Gamburd, Kontorovich, Fuchs, Salehi-Golsefidy, Varju) and the non-Princeton school (Breuillard, Green, Guralnick, Tao, Pyber, Szabo…) developed a machine which allowed us to extend the results on lattices to thin groups.
- 15. But there were questions Such as: other than the Apollonian packings, do thin groups actually arise in nature?
- 16. What is nature? I had shown that a random subgroup of a linear group was Zariski dense (’11), and R. Aoun (slightly later) showed that a random subgroup was free, so together these results showed that a random subgroup was thin.
- 17. But… That is not the same as arising in nature (for example, most numbers are transcendental, but most numbers we run into are not…)
- 18. So, what is nature? Monodromy groups of families?
- 19. Monodromies Algebraic geometers mostly cared about Zariski closure, but still, there was one case (due to A’Campo) where something was shown to be arithmetic, and a couple of cases (Deligne- Mostow, M. Nori) which were NOT arithmetic (but in a product).
- 20. Calabi-Yau Then, we (= Elena Fuchs, Inna Capdeboscq, Sarnak, IR) saw the explicit examples of monodromy associated to Calabi-Yau three- folds (14 in all), and the question was: are they thin or arithmetic?
- 21. Are Calabi-Yaus thin? We had computers, but it was not clear what to do with them - we realized that no algorithms existed, and the questions were probably undecidable.
- 22. But then they were decided! C. Brav and H. Thomas showed that seven of the groups were thin (by showing that they played ping-pong), and T. N. Venkataramana and Singh showed that the other 7 were arithmetic (by finding many unipotent).
- 23. Still, this actually required thought, not just CPU cycles (though Brav/Thomas used those too).
- 24. So, the questions Given a collection of matrices in (say) SL(n, Z): Is the group they generate FINITE? Is their span Zariski dense? If yes, is it maximal? If not, what’s the closure? Is their span Arithmetic? Is their span PROFINITELY dense? If arithmetic, what is the index?
- 25. Finiteness Very well understood: a number of different practical algorithms (Babai, Babai-Rockmore, Detinko-Flannery- O’Brien) Basic idea of (one half of) Babai’s algorithm: If the group is finite, then trace is bounded by N (the dimension). Look at long product: if trace is bounded, probably finite, otherwise not.
- 26. Finiteness Basic idea of Detinko-Flannery-O’Brien: look at the intersection with a principal congruence subgroup (this can be computed): that is torsion free, so should be trivial. If it is, the congruence homomorphism is an isomorphism.
- 27. Finiteness Both algorithms are both theoretically and practically good (Babai’s implemented in GAP, DFO in MAGMA), the DFO algorithm works for any characteristic 0 ground field.
- 28. Zariski density Three years ago, no one knew a good algorithm (or at least admitted to it). Now there are several.
- 29. Zariski Density: Algorithm 1 Based on strong approximation: The main observation is a theorem of T. Weigel: If some modular projection is surjective (for p> 3), then the group is Zariski-dense (and converse is strong approximation)
- 30. Zariski Density: Algorithm 1 So, in practice, pick a moderately large prime p, reduce mod p, use Neumann-Praeger to see if onto. What if the answer is “NOT ONTO”?
- 31. Zariski density: Algorithm 1 In practice: try another random prime, then quit. In theory: Use E. Breuillard’s bound, reduce mod a prime bigger than his bound, Zariski-dense if and only if the projection is onto. Problem: we know that this is a polynomial algorithm, but we don’t know the constants.
- 32. Zariski density: Algorithm 2 Group is Zariski-dense if and only if the adjoint representation is irreducible and does NOT have finite image.
- 33. Zariski Density: Algorithm 2 We know how to check finiteness, for irreducibility use Burnside (the group should span the matrix algebra): polynomial time! Bad degree (as function of dimension)! (best bound I know is 14, so not so great in any reasonable dimension).
- 34. Zariski density: Algorithm 3 (IR) Fact 1: (Prasad-Rapinchuk) If you have two non-commuting elements in G, one of which has Galois group (of char. poly) equal to the Weyl group of the ambient group, then G is Zariski dense. Fact 2: (IR, Jouve-Kowalski-Zywina) a long word in the generators of a Z. Dense subgroup has Galois group the Weyl group with probability exponentially close (in length) to 1.
- 35. Zariski Density: Algorithm 3(IR) So, algorithm is: compute two long words. If they commute, NO. If they don’t commute, compute Galois group (of one of them). If same as Weyl group. YES, otherwise, NO.
- 36. Zariski density: Algorithm 3 Problem: exponent in exponential convergence is NOT effective (since based on super-strong approximation). Lesser problem: how to compute Galois group?
- 37. Zariski Density: Algorithm 3 Weyl groups are usually the symmetric group, or the signed permutation group - turns out that there are (with some major caveats) good algorithms.
- 38. Zariski Density: Algorithm 3 For example, a polynomial time (but not practical) algorithm to check that the Galois group is the symmetric group is to check that the characteristic polynomial of the first five exterior powers of the companion matrix are irreducible (and the discriminant of the original polynomial is square free). Running time: polynomial of degree around 40(!) We use a different algorithm, to get the running time of Zariski-density checker to a fourth degree polynomial in the dimension for SL(n, Z), and an eighth degree polynomial for Sp(2n, Z) (the running time is linear in the log of the height of the generateng set, in both cases)
- 39. What if not Zariski dense? An algorithm to compute Zariski closure (using Groebner bases) is given by Derksen-Jeandel- Koiran (’07). Not obviously practical (worst case at least doubly exponential), but there may be ways to make it so over Q.
- 40. Profinite Density Fact: a random group is profinitely dense with probability bounded away from zero (Capdeboscq- IR, ’15), but how do you tell? Only algorithm I know: check every prime (primes and 4 and 9 are enough) until Breuillard’s bound, so simply exponential. Can one do better?
- 41. Arithmeticity Three years ago, we had no clue: it was easy to see that seeing if you have the whole group was semi- decidable (keep multiplying until you get the generators). Since then: Detinko-Flannery-Hulpke gave an algorithm to compute the index of an arithmetic subgroup.
- 42. Arithmeticity Their algorithm should be a semi-decision procedure: if you let it run, it will give you either the index or a lower bound on index (together with “don’t know”). But it is not (yet). In particular, the arithmetic Calabi-Yau monodromies (as described in Venkataramana’s talk) can not yet be detected WITHOUT thinking!
- 43. Arithmeticity (slides borrowed from Alla Detinko) As a finale, we show some results:
- 46. Calabi-Yau (from preprint of Hoffman-van Straten)
- 47. Index of arithmetic subgroups Note: the algorithm is (obviously) practical, but no complexity bounds are known.
- 48. Index of arithmetic subgroups The work of Detinko/Flannery/Hulpke reduces the question of “Is a subgroup finite index in a given arithmetic group?” to “Does a given set of matrices generate a given arithmetic group?”