computing with matrix groups, or "how dense is dense"

How dense is dense, or:

Computing in linear groupsIgor Rivin

(Temple University)

Computing in matrix

groups

One goal is to understand examples.

Another is to think of the computational aspects

as another facet of the mathematics.

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.

Basic question: given a collection of matrices

A, B, C, D, …, what sort of group do they

generate?

Easy version: the matrices live in a

finite group (Say, SL(n, Z/pZ))

Hard version: the matrices live in,

say, SL(n, Z)

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.

The hard version:

Given a collection of matrices in SL(n, Z), what

can we say?

Some history

Once upon a time

People understood a lot about lattices.

And they could use sophisticated analysis to

answer questions about them.

Then, came the Apollonian packing

(well, really, the work of Graham, Lagarias, Mallows, Wilks, Yan)

Where the group is thin

And nothing was ever the

same

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.

But there were questions

Such as: other than the Apollonian packings, do

thin groups actually arise in nature?

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.

But…

That is not the same as arising in nature (for

example, most numbers are transcendental, but

most numbers we run into are not…)

So, what is nature?

Monodromy groups of families?

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).

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?

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.

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).

Still, this actually required thought, not just

CPU cycles (though Brav/Thomas used those

too).

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?

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.

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.

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.

Zariski density

Three years ago, no one knew a good algorithm

(or at least admitted to it).

Now there are several.

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)

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”?

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.

Zariski density: Algorithm

2

Group is Zariski-dense if and only if the adjoint

representation is irreducible and does NOT have

finite image.

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).

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.

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.

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?

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.

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)

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.

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?

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.

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!

Arithmeticity (slides borrowed from

Alla Detinko)

As a finale, we show some results:

Calabi-Yau (from preprint of

Hoffman-van Straten)

Index of arithmetic

subgroups

Note: the algorithm is (obviously) practical, but

no complexity bounds are known.

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?”