trends in 3-manifolds

Trends in 3-manifolds - Sujets branchés en 3-variétés

WHEN: May 1-2, 2004

WHERE: CIRGET, Département de mathématiques, Université du Québec à Montréal (UQAM)

SPEAKERS: Ian Agol, Danny Calegari, Nathan Dunfield, Stavros Garoufalidis, Marc Lackenby, Saul Schleimer and Genevieve Walsh.

The research group in Geometry and Topology here in Montréal regularly organises 2-day mini-workshops in the spring. This year, we have been invited by Steve Boyer (director of CIRGET) to organise a workshop with the theme "Trends in 3-manifolds." The format of the workshop is as follows. We expect people to arrive on Friday so that the mathematical activities may start early Saturday. On Saturday evening, an outing is planned. Talks will resume Sunday morning and finish around lunchtime, which means people can get home in the afternoon or take the opportunity to explore the city. Montreal has fantastic, affordable restaurants and other pleasant activities. The weather in Montréal in early May should be cool and comfortable.

Marc Lackenby will also give a special CIRGET Colloquium talk at 2pm on Friday, April 30. Other mathematical events that may be of interest to people arriving on Friday are the CIRGET Seminar at 11am and the CRM-ISM Colloquium at 4pm.

Registration for the workshop will not be necessary. However, it is important that you let the organisers know by April 20 whether you intend to join us for dinner on Saturday night.

If you have questions, feel free to contact us at the addresses given below. We have included some links below which may prove useful in finding your way to and accommodation in Montréal.

We hope to see you in May!

The Organisers:
Ben Klaff (klaff@math.uqam.ca)
Stephan Tillmann (tillmann@math.uqam.ca)

DIRECTIONS AND LINKS
CIRGET is located in the Département de mathématiques of UQAM. This is not on the UQAM main campus. For visitors making their way there within Montréal by metro: use the green line and get off at the metro station Place des Arts. On the platform, follow the signs to Jeanne-Mance; then the signs to UQAM, which will take you to the right building (President Kennedy, also known as The Boat). Take the elevator to the fifth floor. The CIRGET coffee lounge is in room PK 5214. The street address is 201 Avenue du Président-Kennedy.

Information for visitors, including accommodation suggestions, the airports and the metro system can be found here on the CRM web-site. Please note that the CRM and CIRGET are in different parts of town. Hotels near CIRGET include: Le jardin d'Antoine and the Hôtel Casa Bella. Mention CIRGET and UQAM to get better rates. Nearby is also the Holiday Inn Midtown.

THE SCHEDULE
All talks will be held in Room PK 5115. Refreshments will be served during the breaks.

Saturday, 1 May Sunday, 2 May

9-10 Saul Schleimer 9.30-10.30 Danny Calegari

10.30-11.30 Genevieve Walsh 11-12 Ian Agol

1.30-2.30 Nathan Dunfield

3-4 Stavros Garoufalidis

4.30-5.30 Marc Lackenby

7 Conference dinner

THE TALKS

Ian Agol (University of Illinois at Chicago)

TITLE Tameness of hyperbolic 3-manifolds

ABSTRACT We show that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact manifold with boundary. In fact, we prove this more generally for PNC manifolds (with ``hyperbolic cusps''). This answers affirmatively the Marden conjecture (which was already known for manifolds with indecomposable fundamental group by Bonahon). Applications to various conjectures will be given.

Danny Calegari (California Institute of Technology)

TITLE Quasigeodesic flows and universal circles

ABSTRACT The most well-known examples of quasigeodesic flows on hyperbolic 3-manifolds are pseudo-Anosov flows, which are typically constructed dually to codimension one objects like taut foliations or sutured manifold hierarchies. We show that all quasigeodesic flows on hyperbolic 3-manifolds give rise to an abstract dynamical package with many features in common with pseudo-Anosov flows, including a universal circle, and a pair of invariant laminations. As a corollary, we show that the Weeks manifold does not admit a quasigeodesic flow. In the case that a quasigeodesic flow is transverse to a taut foliation, we discuss the relationship between the dynamical packages arising from the foliation and from the flow.

Nathan Dunfield (California Institute of Technology)

TITLE Does a random 3-manifold fiber over the circle?

ABSTRACT I'll discuss the question of when a tunnel number one 3-manifold fibers over the circle. In particular, I will discuss a criterion of Brown which answers this question from a presentation of the fundamental group. I will describe how techniques of Agol, Hass and W. Thurston can be adapted to calculate this very efficiently by using that the relator comes from an embedded curve on the boundary of a genus 2 handlebody. I will then describe some experiments which strongly suggest the answer to the question: Does a random tunnel-number one 3-manifold fiber over the circle? (Joint work with Dylan Thurston.)

Stavros Garoufalidis (Georgia Institute of Technology)

TITLE The geometry of the Jones polynomial

ABSTRACT We will discuss the relation between the Jones polynomial and the SL2(C) character variety of a knot complement. This relation is obtained by the means of a q-difference equation of the Jones polynmial, and is formulated in terms of a "Characteristic versus Deformation Variety Conjecture". We will discuss evidence for this conjecture, and extensions to higher rank groups. Finally we will discuss positive and negative evidence for the Hyperbolic Volume Conjecture.

Marc Lackenby (St. Catherine's College and the University of Oxford)

TITLE A characterisation of large finitely presented groups

ABSTRACT A group is known as `large' if it has a finite index subgroup that admits a surjective homomorphism onto a non-abelian free group. Such groups have many interesting properties, for example super-exponential subgroup growth. Possibly the strongest form of the virtually Haken conjecture asserts that any hyperbolic 3-manifold has large fundamental group. This is known to be true in the cusped case. In my talk, I will give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a nested sequence of finite index subgroups where successive quotients are abelian groups with sufficiently large rank and order. The proof is topological in nature, using a version of thin position for Cayley graphs of finite groups.

Saul Schleimer (University of Illinois at Chicago)

TITLE Heegaard splittings and Haken's program

ABSTRACT Haken's program (extended by Jaco and Oertel) studies the topology of three-manifolds via a linear structure on the set of incompressible surfaces. It is natural to try and extend these ideas to Heegaard splittings. A recent result: If H and K are surfaces in M with a fixed Haken sum, and if H + nK is a strongly irreducible Heegaard splitting for arbitrarily large values of n, then K is a pi_1 injective surface. This is joint work with Yoav Moriah and Eric Sedgwick.

Genevieve Walsh (University of Texas at Austin)

TITLE Virtually Haken fillings of fibered knot complements

ABSTRACT (This is joint work with Daryl Cooper, UCSB.) Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along the boundary of M are virtually Haken. Combined with a result of Cooper and Long, this gives the corollary that infinitely many fillings of the complement of any non-trivial knot in the three-sphere are virtually Haken.

http://www.labmath.uqam.ca/~tillmann/cirget/
26 April 2004

WHEN:	May 1-2, 2004
WHERE:	CIRGET, Département de mathématiques, Université du Québec à Montréal (UQAM)
SPEAKERS:	Ian Agol, Danny Calegari, Nathan Dunfield, Stavros Garoufalidis, Marc Lackenby, Saul Schleimer and Genevieve Walsh.

Saturday, 1 May		Sunday, 2 May
9-10	Saul Schleimer	9.30-10.30	Danny Calegari
10.30-11.30	Genevieve Walsh	11-12	Ian Agol
1.30-2.30	Nathan Dunfield
3-4	Stavros Garoufalidis
4.30-5.30	Marc Lackenby
7	Conference dinner

	Ian Agol (University of Illinois at Chicago)
TITLE	Tameness of hyperbolic 3-manifolds
ABSTRACT	We show that a hyperbolic 3-manifold with finitely generated fundamental group is homeomorphic to the interior of a compact manifold with boundary. In fact, we prove this more generally for PNC manifolds (with ``hyperbolic cusps''). This answers affirmatively the Marden conjecture (which was already known for manifolds with indecomposable fundamental group by Bonahon). Applications to various conjectures will be given.

	Danny Calegari (California Institute of Technology)
TITLE	Quasigeodesic flows and universal circles
ABSTRACT	The most well-known examples of quasigeodesic flows on hyperbolic 3-manifolds are pseudo-Anosov flows, which are typically constructed dually to codimension one objects like taut foliations or sutured manifold hierarchies. We show that all quasigeodesic flows on hyperbolic 3-manifolds give rise to an abstract dynamical package with many features in common with pseudo-Anosov flows, including a universal circle, and a pair of invariant laminations. As a corollary, we show that the Weeks manifold does not admit a quasigeodesic flow. In the case that a quasigeodesic flow is transverse to a taut foliation, we discuss the relationship between the dynamical packages arising from the foliation and from the flow.

	Nathan Dunfield (California Institute of Technology)
TITLE	Does a random 3-manifold fiber over the circle?
ABSTRACT	I'll discuss the question of when a tunnel number one 3-manifold fibers over the circle. In particular, I will discuss a criterion of Brown which answers this question from a presentation of the fundamental group. I will describe how techniques of Agol, Hass and W. Thurston can be adapted to calculate this very efficiently by using that the relator comes from an embedded curve on the boundary of a genus 2 handlebody. I will then describe some experiments which strongly suggest the answer to the question: Does a random tunnel-number one 3-manifold fiber over the circle? (Joint work with Dylan Thurston.)

	Stavros Garoufalidis (Georgia Institute of Technology)
TITLE	The geometry of the Jones polynomial
ABSTRACT	We will discuss the relation between the Jones polynomial and the SL2(C) character variety of a knot complement. This relation is obtained by the means of a q-difference equation of the Jones polynmial, and is formulated in terms of a "Characteristic versus Deformation Variety Conjecture". We will discuss evidence for this conjecture, and extensions to higher rank groups. Finally we will discuss positive and negative evidence for the Hyperbolic Volume Conjecture.

	Marc Lackenby (St. Catherine's College and the University of Oxford)
TITLE	A characterisation of large finitely presented groups
ABSTRACT	A group is known as `large' if it has a finite index subgroup that admits a surjective homomorphism onto a non-abelian free group. Such groups have many interesting properties, for example super-exponential subgroup growth. Possibly the strongest form of the virtually Haken conjecture asserts that any hyperbolic 3-manifold has large fundamental group. This is known to be true in the cusped case. In my talk, I will give a necessary and sufficient condition for a finitely presented group to be large, in terms of the existence of a nested sequence of finite index subgroups where successive quotients are abelian groups with sufficiently large rank and order. The proof is topological in nature, using a version of thin position for Cayley graphs of finite groups.

	Saul Schleimer (University of Illinois at Chicago)
TITLE	Heegaard splittings and Haken's program
ABSTRACT	Haken's program (extended by Jaco and Oertel) studies the topology of three-manifolds via a linear structure on the set of incompressible surfaces. It is natural to try and extend these ideas to Heegaard splittings. A recent result: If H and K are surfaces in M with a fixed Haken sum, and if H + nK is a strongly irreducible Heegaard splitting for arbitrarily large values of n, then K is a pi_1 injective surface. This is joint work with Yoav Moriah and Eric Sedgwick.

	Genevieve Walsh (University of Texas at Austin)
TITLE	Virtually Haken fillings of fibered knot complements
ABSTRACT	(This is joint work with Daryl Cooper, UCSB.) Suppose that M is a fibered three-manifold whose fiber is a surface of positive genus with one boundary component. Assume that M is not a semi-bundle. We show that infinitely many fillings of M along the boundary of M are virtually Haken. Combined with a result of Cooper and Long, this gives the corollary that infinitely many fillings of the complement of any non-trivial knot in the three-sphere are virtually Haken.