I just heard that Jeremy Kahn has announced joint work with Vladimir Markovic showing that every hyperbolic 3-manifold contains an immersed, -injective surface. This is equivalent to showing that the fundamental group of every hyperbolic 3-manifold contains a subgroup isomorphic to the fundamental group of a surface. The abstract for Jeremy’s talk is on the conference web page. This is a long standing open problem that Henry Wilton wrote about on this blog back in February. I’m not at the conference, so I didn’t see the talk, but if any of you readers were ther and have anything to add, feel free to write in in the comments.

Update: Danny Calegari has posted a discussion of some aspects of the proposed proof here and an outline written by Jeremy here.

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I would be very curious to know whether the proof also shows that every homologically trivial geodesic in a hyperbolic 3-manifold bounds an injective (non-closed) surface.

Also, how is 3-dimensionality used? (or is it?) It’s not clear from the linked abstract.

Comment by Danny Calegari — August 6, 2009 @ 11:24 pm |

I wasn’t at Kahn’s talk, but my guess is their approach may be related to the one Lewis Bowen tried in [math.GT/0411662], especially as Kahn and Markovic have worked on the Ehrenpreis Conjecture which Bowen also attacks in that preprint. That is, one pieces together a large number of nearly-geodesic pairs of pants with very long cuff lengths. I think the tricky bit is that if the cuff lengths are large, the distances between cuffs is very small, which means the angles between adjacent pants have to be very very small it the resulting surface is going to be nearly geodesic.

If this is the case, then I doubt that KM’s method would also answer your above question. I have no clue about the whether 3-dimensionality would be essential, though.

Comment by Nathan Dunfield — August 7, 2009 @ 2:32 pm |

Hi Nathan – I just heard from Ian that they glue pants with an almost-fixed “twist” of some definite length (so that the very short arcs joining pairs of long cusps do not match up) and so one doesn’t need the angle to be so very very small (just very small :).

There is still the question of where the pants come from. At the risk of indulging in pure speculation, let me propose a method. One way to construct a pair of pants with edge cuffs of length close to R is to put a little geodesic letter H in your manifold (where the horizontal edge is perpendicular to the vertical one), then look for a pair of closed geodesics of complex length very close to R (i.e. with imaginary component of length very close to zero) and which are almost tangent to the two vertical sides of the letter H. For big R, such geodesics will be very close to wherever you look for them. The homotopy class of the H is a pair of pants with all three cuff lengths close to R (I think). One can choose in advance which the first cuff will be (providing R is big enough).

The next obvious problem is pairing up the cuffs; if they have a way to do that, then I would guess it also lets one first prescribe some set of (sufficiently long) boundary curves. So maybe one can fill an arbitrary homologically trivial chain, providing the geodesics are sufficiently long . . .

Anyway, this (completely speculative and uninformed) sketch does not use dimension 3. In fact, maybe it also works in dimension 2, and for finite volume but cusped manifolds, in which case I must have gotten something wrong . . .

Comment by Danny Calegari — August 7, 2009 @ 5:47 pm

In fact, the crossbar of the “H” is the most delicate part of the pants (that’s where it’s thinnest) so this would not be a good place to try to pin down the geometry. I guess this is why they use tripods instead, centered at the most robust part of the pants (see Ian’s post below).

Comment by Danny Calegari — August 7, 2009 @ 7:17 pm

Do I infer correctly that, following this result, every LERF hyperbolic 3-manifold is virtually fibered?

The implications would go as follows. Kahn and Markovic give you an essential immersed surface. LERF allows you to find a cover where the surface is embedded. So M is virtually Haken. Then, Ian Agol’s recent theorem tells you that that M is actually virtually fibered.

Reducing virtual fibering to the LERF conjecture seems extremely strong…

Comment by Dave Futer — August 7, 2009 @ 12:54 pm |

@ Dave: Yes, you’re correct that [KM] + [A] + LERF = virtual fibering.

Comment by Nathan Dunfield — August 7, 2009 @ 2:19 pm |

Here’s a bit of a report on their proof. I can scan my notes, if anyone is interested.

With regard to Dave Futer’s comment, their result actually implies that hyperbolic manifolds are cubulated, so it supersedes my result (which required Haken & LERF).

With regard to Danny’s comment, Kahn claims that 3-dimensionality is not important, although they still can’t prove the Ehrenpreis conjecture. But their method should produce surface subgroups in any rank 1 lattice of dimension > 2.

Jeremy spoke at the conference on Tuesday and Thursday since he only got thru part of the argument on Tuesday. The argument looks

pretty good to me, modulo several technical details which

I haven’t checked.

The idea is similar to Bowen’s approach (and I’m

told Thurston also suggested gluing pants together).

Of course, the obstructions to making this approach

work are well-known: if one has a bunch of almost

geodesic pants, one must make sure that they are glued

together to form an almost geodesic surface, and one

must make sure that there are enough of them to glue

up to form a closed surface. They seem to have overcome

both of these obstacles.

Jeremy first described a one parameter family of

closed genus 2 surfaces. Take two pants where all three

boundary curves have length 2R, and glue them together

along the boundary so that the seams are shifted by 1

(this could be expressed in Dehn-Thurston coordinates).

Jeremy claims that this family of genus 2 surfaces

stays in a bounded subset of moduli space of genus 2

surfaces (as opposed to doubling the pants, which

creates short geodesics from the doubled seams).

As R gets large, the seams of the pants get

short. But with the twist, the genus 2 surface will still have

injectivity radius bounded below. He points out that

any transverse geodesic with length 1 will cross

the cuffs at most R times. I haven’t checked these claims, but they allow them to control the quasi-isometry of the embedding of surfaces.

Now, he makes some claims about having pants in

a hyperbolic 3-manifold which are almost geodesic and

symmetric, up to some tolerances. I’m not going to write

down the exact details, but the lengths of the boundary

curves should be close to 2R, and the rotational part

should be close to zero, measured by some small constant

epsilon.

They want to find a lot of these almost symmetric

and almost geodesic pants in a closed hyperbolic 3-manifold.

They construct these by taking pairs of tripods, i.e.

a triple of geodesics which make angles of 120 degrees

with each other. One should think of these as the

vertices of the spine of a symmetric pants with cuffs of

length 2R. They shoot out geodesic rays of length r

(the length of the edges in said spine), and measure

how close they get to the other tripod, within some

epsilon’ (depending on epsilon). If they are close

(actually, this is done in the frame bundle), they

get a nearly geodesic pants whose cuff lengths are within

epsilon of 2R. For a fixed epsilon, this probability

is non-zero, and gets distributed quite evenly because

of exponential mixing of the geodesic flow (cf. Moore or

Pollicott).

Now one considers for a geodesic of length

close to 2R and with small twist number, the collection

of such almost geodesic symmetric pants with this geodesic

as a cuff. Because the geodesic flow is exponentially

mixing, these get equidistributed around the geodesic

as R -> infinity (this is seen by considering the tripods, and noticing that one may rotate the tripods around the core of the geodesic by its centralizer. Exponential mixing implies that the tripods will close up within epsilon in an evenly distributed fashion about the geodesic).

Thus, there is a pants very close to differing

by 1+i*pi translation along a boundary curve of the

pants. One uses a version of

Hall’s Marriage Theorem to pair these pants up, and

show that they can be glued together to form an almost

geodesic closed surface.

As I said, I haven’t checked all the details, but

they certainly seem to have overcome some of the obstacles

that Lewis Bowen came up against. First, by gluing with

a twist, the tolerances for gluing pants together to

get a nearly geodesic surface become manageable. Second,

they use exponential mixing of the geodesic flow to

show that nearly geodesic pants are equidistributed

about a geodesic of length 2R, and get the required

symmetry to allow one to glue the pants together.

The reason they can’t prove the 2-dimensional case (i.e. Ehrenpreis) is that the nearly symmetric pants on both sides of a closed geodesic of length close to 2R won’t necessarily match up in number exactly.

This isn’t a problem in 3-D, because the normal bundle to a geodesic is a torus, and one only needs a matching pants within some tolerance.

-Ian

Comment by Ian Agol — August 7, 2009 @ 5:56 pm |

Since the argument does not (apparently) discriminate between closed and complete, it is good to hear that there is a clear reason why it doesn’t work in dimension 2!

Comment by Danny Calegari — August 7, 2009 @ 6:06 pm |

[…] pants, surface groups, Waldhausen | by Danny Calegari I just learned from Jesse Johnson’s blog that Vlad Markovic and Jeremy Kahn have announced a proof of the surface subgroup conjecture, that […]

Pingback by Surface subgroups in hyperbolic 3-manifolds « Geometry and the imagination — August 7, 2009 @ 8:34 pm |

[…] his argument with Vlad Markovic, that I blogged earlier about here (also see Jesse Johnson’s blog for other commentary). With his permission, this is reproduced below in its […]

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