Random 3-manifoldsIgor Rivin (Temple U and Brown U),

AMS Meeting, Baltimore March 29-30

Random 3-manifolds

What is a random 3-manifold?

Many definitions, most too hard to deal with

Random 3-manifolds

The most tractable (to date) definition is due to N. Dunfield and W. P. Thurston:

Fix an integer g, take two handle bodies of genus g, glue them by a random self-map [whatever that means] of surface of genus g.

Random fibered 3-manifolds

Even simpler: take a (random) surface automorphism T, construct the mapping torus of T.

Random surface automorphism?

Again, many ways to define, the most tractable (to date) is: take a (nice, finitely generated) subgroup of Mod(g), look at random words in generators of increasing length.

Random surface automorphisms

What is a “nice subgroup”? Depends on what you want. An example:

Non-elementary subgroup: contains two non-commuting pseudo-Anosovs.

Torelli-fat subgroup: Image under the Torelli homomorphism is Zariski dense in Sp(2g, Z).

Random fibered manifolds

J. Maher and I. Rivin: random automorphism is pseudo-Anosov, so random fibered manifold is hyperbolic, by Thurston’s theorem.

Random fibered manifold

What can we say?

About: homology, volume, bottom eigenvalue of the Laplacian, Cheeger constant, rank of the fundamental group…

Random fibered manifold: homology

The first betti number is generically equal to 1.

If the subgroup is Torelli-fat, log of torsion grows linearly with length of monodromy; there is also a central limit theorem for the distribution of the log torsion.

Random fibered 3-manifold: homology

Random fibered 3-manifolds homology

Random fibered 3-manifolds: volume

Two different ways (both using bi-Lipschitz models), to see that the volume of random fibered manifold grows at most and at least linearly. BUT, the truth is more interesting.

Random Fibered 3-manifolds: volume

Random fibered 3-manifolds: volume

Random fibered 3-manifolds

Rank of fundamental group: 2g+1 (mod Biringer-Souto)

Cheeger constant: C g/N

injectivity radius 1/log squared N

Bottom eigenvalue between C/N and C/N2.

Random 3-manifolds

First betti number is generically zero (already known to Dunfield-Thurston).

log torsion grows linearly (with central limit and large deviation thms, which gives exponentially small probability of nonzero first betti number.

Random 3-manifolds

Complexity grows linearly

generically hyperbolic (J. Maher)

Volume grows linearly (as before, can show linear lower and upper bounds).

Random 3-manifolds

Random 3-manifolds

Probability of having Galois cover with a fixed solvable deck group have nontrivial first Betti number goes to zero (uses Grunewald/Larsen/Lubotzky/Malestein).

Random 3-manifolds

Question: is a random Heegaard splitting Haken?

By K. Hartshorn, we know that the minimal genus of an essential surface grows linearly, but that is neither here nor there.