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?”
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