Some generic properties of some occasionally

obscure groups Igor Rivin, Temple University

MSRI Hot Topics Thin Groups and Super Strong Approximation

In the beginning...

Thurston’s philosophy: everything is hyperbolic.

(Actually Poincare’s philosophy, adopted by Thurston).

Examples:Poincare (?): Almost all oriented surfaces are hyperbolic.

Thurston: Almost all Dehn surgeries on a knot are hyperbolic.

Thurston: Almost all surface bundles over the circle are hyperbolic ???

The last one seems

reasonable...

Thurston: A surface bundle (viewed as a mapping torus of a map F) is hyperbolic if and only if F is “pseudo-anosov”.

And

Surface automorphisms, are periodic, reducible, or pseudo-anosov (Thurston’s classification), so seems reasonable that a random element is pseudo-anosov.

What is “random”?

Two interpretations (for a discrete group):

Pick a (symmetric) generating set, look at long random products.

Look at elements in a Cayley ball.

Third interpretation for

matrix groups

Look at all elements of bounded “height” (for example, matrix norm).

But

It took a while to make it precise. More precisely:

It never rains but when it pours

(2006)

I. Kapovich

Suggested that it would be easier to translate the problem to the symplectic group (the Torelli map sends the mapping class group of genus g to Sp(2g, Z), and is surjective -- see Farb-Margalit).

Key ingredientCasson’s criterion: A map F is pseudo-anosov if P = characteristic polynomial of T(F) -- has the following propertie:

P is irreducible.

P is not cyclotomic.

P is not of the form

What interpretation of

randomness do we use?

Random walk or Cayley ball? Or “Archimedean height”?

Yes to all three, sort of.

Graph walk

Put generators of the group onto vertices of a (Perron-Frobenius, undirected) graph.

Allow multiplication if there is an edge.

“Stupid” random walk

Corresponds to the complete graph.

Cayley graph

Corresponds to walk on the defining automaton (group has to be bi-automatic).

Proof

In brief, a form of sieving, using ideas on equidistribution of walks on finite groups, and the already mentioned Borel-Chavdarov result (IR 2007, published DMJ ’08, FM ’09), and property tau

Results (SL(2, Z) Hua-Reiner)

Results (SL(3, Z) Hua-Reiner)

Results (SL(3, Z) transvections)

Results (SL(4, Z) Hua-Reiner)

Archimedean height

Completely different, we can do it anyway (using Nevo-Sarnak (Acta, 2010), see also Gorodnik-Nevo’s book and papers).

The method also works for Out(Fn)

There, the result is that a random outer automorphism is an IWIP (irreducible with irreducible powers).

Need the Galois group to be not intransitive, so just show it is the full symmetric group (generalized by Kowalski et al).

Argument is good

Because gives completely effective bounds (in terms of property tau constants, which are effective as per Kassabov, Shalom), exponential convergence.

Argument is bad

Because only seems to be for finite index subgroups of the mapping class group

In particular, no result at all for the Torelli subgroup (kernel of T).

Maher’s result is the opposite:

Not effective

Very general (works for any “nonelementary” subgroup of the mapping class group).

Argument geometric, so does not seem to work for Out(Fn)

More Maher

Maher’s argument also gives growth of the translation length in the curve complex.

Which shows that a random Dunfield-Thurston manifold is hyperbolic.

What is the state of the art for the

MCG?Bestvina-Fujiwara (2008) gave ineffective but general result via quasimorphisms.

Malyutin (2010), using the Bjorkman-Hartnick central limit theorem showed (by soft methods) that the convergence in BF is superpolynomial (but can’t show exponential).

State of the art continued.

Joseph Maher can get exponential convergence (still ineffective?) 2011

In his original paper also shows increasing distance in the curve complex (also follows from BF).

State of the art, continued more

(2011)Lubotzky-Meiri and Malestein-Souto can do Torelli, by a “Prym map” trick (going to double covers, and getting a linear representation of Torelli, an idea of Looijenga, and of Grunewald-Lubotzky).

Lubotzky-Meiri can do Torelli of Out(Fn)

Question 1:

Can you do sieving directly on the mapping class group (using CSP a la Ellenberg)?

Back to our scheduled programSuperstrong approximation

allows us to catch up a little to Maher, since we can prove all of our results for thin subgroups of linear groups.

But also, we can show that a random subgroup (of a Zariski dense subgroup) is Zariski dense (IR 2010, R. Aoun 2011)

And also

Generic subgroup of a linear group is free (R. Aoun 2011).

Generic subgroup of a word-hyperbolic group is free (Gilman-Miasnikov-Osin).

Combining the last couple of slides

Generic subgroup (in higher rank, anyhow) is thin.

Very thin

In fact, for a two-generator subgroup with random long-word generators, Hausdorff dimension goes to zero (Fuchs-IR, 2012)

And yetA generic (2-generator) subgroup of a linear group of rank > 1 is profinitely dense (modular reductions are ALL surjective) with positive probability. For SL(2, Z) need 3 random generators. (Capdebosq-IR, 2012).

Historical note

First example of a free profinitely dense subgroup was constructed by Stephen Humphreys (’87), second by Soifer-Venkataramana (’00), independently, various other people probably also.

So, thin groups are good

Because there are lots of them.

Because they are easy to verify: Lubotzky, Weigel: surjective for one prime implies p-adic density implies Zariski density (as pointed out in A. Rapinchuk’s talk).

Well, actually...

Not clear how easy...

Question 2:

Given matrix generators for a subgroup G of (say) SL(n, Z), can one give a bound for how far a certificate of Zariski-density or lack thereof lies? (in terms of the matrix coefficients)

Question 3

Same question as Question 2, but for profinite density.

Thin is bad

Since, as far as I know, the bounds are not very effective (theoretically effective?) unlike lattices.

Lattices are good

Because get good bounds

Lattices are bad

Because thin on the ground, and it seems very difficult to check if something is a lattice.

Model question:

Given a set of matrices A, B, C..., do they generate SL(n, Z)?

Easier questionDoes <A, B, C, ...> contain M?

Undecidable for SL(n>3, Z)! (Mihailova, 195?) -- uses F2xF2, shows equivalence to Post correspondence.

Open(!!!) for SL(3, Z)

Easy for SL(2, Z)

Back to Model question

The Mihailova result would seem to suggest that the model question is undecidable also, but this is open, as far as I know.

What we really want to know is...whether H=<A, B, C, ...> is of

finite index.

This is roughly equivalent to the model question by the congruence subgroup property (it is usually not difficult to tell which congruence subgroup H would be if it were a congruence subgroup...), then check if H generates the congruence subgroup.

Question 4:

In our applications, we have two-generator subgroups. Is that any easier than the general case? That’s canonical, since:

Question 4, continued:

Sharma-Venkataramana: every lattice contains a three-generator lattice, so three is probably as hard as the general case.

One generator seems tractable...