Schedule of Topology Seminars: 2007-08

Suppose that F is a foliation in M³ with Gromov hyperbolic fundamental group so that F is almost transverse to a quasigeodesic pseudo-Anosov flow. Then F has the continuous extension property. To prove this, one big ingredient is the structure of the intersection of the stable and unstable foliations of the flow intersected with the leaves of F, particularly in the universal cover. We describe this structure: including asymptotic behavior of individual (1-dim) leaves, the structure of non separated leaves and leaves with ideal points identified. We also analyse how many orbits can a leaf of F intersect, when lifted to the universal cover. This does not need geometric assumptions on F or M. If we have time, we will also discuss how this helps to prove the continuous extension property in the geometric situation.

Given a closed, irreducible 3-manifold, its complexity is the minimum number of tetrahedra in a triangulation of the manifold. The complexity is therefore known for all closed manifolds which appear in certain computer generated censuses, and the question of determining the complexity of a given closed 3-manifold has an algorithmic solution which, in general, is impractical. It is an open problem to determine the complexity of each member of an infinite family of closed 3-manifolds. In this talk, I will present a solution to this problem by showing that the lens space L(2n,1), n>1, has complexity 2n-3. If time permits I will discuss some other potential applications of the methods used. This is joint work with Bus Jaco and Hyam Rubinstein.

In 2004 Ruifeng Qiu announced a proof of the Gordon Conjecture: The sum of two Heegaard splittings is stabilized if and only if one of the two summands is stabilized. His proof (and Bachman's proof of the same vintage) have been difficult for topologists to absorb. Last summer I spent a week in Dalian, talking with Qiu and his graduate student Ming Xing Zhang. In this way I came to appreciate the core ideas of the proof, which are combinatorially intricate but quite beautiful and amazing. In this talk I will begin to present these core ideas.

If N is a fibered 3-manifold, then Thurston showed that S¹ x N supports a symplectic structure. It is an open question, whether the converse holds, i.e. whether S¹ x N symplectic implies N fibered. In our approach to the conjecture we study the Seiberg-Witten invariants of S¹ x N and its finite covers. Using this approach we can relate the conjecture to interesting questions regarding 3-manifold groups.

Braids can be placed one on top of the other, and hence form a group. Tangles can be glued together in all sorts of other ways, and hence form a planar algebra. I will describe a "representation" that maps a planar algebra to a space of functions on the set of paths in a graph. This simultaneously extends a representation of the braid group, and a knot invariant. I'll focus on the Temperley-Lieb algebra, but I might have time to mention A2, B2, and the Higman-Sims graph.

The Bieri-Neumann-Strebel invariant of a group G is an open subset U of the projectivization of the first cohomology. U cap (-U) corresponds to homomorphisms of G onto Z with finitely generated kernel. For 3-manifolds this corresponds to fibrations over the circle. We discuss an extension of these ideas. This is a report on work in progress with Stephan Tillmann.

A continuation of the 8 April talk, which explained the motivation of Qiu's proof. Here we present at least the outline of the proof itself.

Sutured manifold theory has long been used to study Dehn surgery on knots in 3-manifolds. It has not often been used to study 2-handle addition, a natural generalization of Dehn surgery. If a component F of a simple 3-manifold N has genus two, sutured manifold theory is particularly effective for studying degenerating separating curves on F. (A curve a is degenerating if attaching a 2-handle to it creates a non-simple 3-manifold N[a].) For example, suppose that the boundary of N consists of tori and the genus two surface F containing essential separating curves a and b. Then if N[a] is reducible and N[b] is non-simple, a and b are isotopic on F.

Similar sutured manifold theory techniques are useful for studying knots and links obtained by "boring" a split link or unknot. Such a perspective allows a theorem to be proved which is a generalization of two seemingly unrelated theorems. The first theorem generalized is the superadditivity of genus under band connect sum (Gabai, Scharlemann) and the second is the fact that a tunnel for a tunnel number one knot or link can be slid and isotoped to be disjoint from a minimal genus Seifert surface. As time permits, I will discuss other applications of sutured manifold theory to questions about bored split links and unknots.

Teichmuller space is the parameter space for the geometry of surfaces, endowed with its own geometry via the Teichmuller metric. In many ways, this metric space has complicated and interesting large-scale geometry. I'll show that the rate of divergence of geodesics is quadratic (and so intermediate between the rates occurring in flat spaces and in negative curvature) by constructing efficient "paths at infinity" which prescribe how to deform surfaces to look alike. This is joint work with Kasra Rafi.

Applying Seifert's Algorithm to an alternating projection produces a minimal genus orientable spanning surface for the projected link. I will present a natural generalization of Seifert's Algorithm for which the resulting surfaces - we call them layered surfaces - are mostly non-orientable. It turns out that for any alternating link this algorithm allows us to find a minimal genus orientable or non-orientable spanning surface for any particular (aggregate) boundary slope*. In particular, just as Seifert's Algorithm allows us to determine the (orientable) genus any alternating link, this algorithm allows us to determine the "non-orientable genus" for any alternating link. To prove this, we will employ William Menasco's geometric structure for alternating links. This was my undergrad research, with Colin Adams of Williams College as my advisor.

*We define the aggregate slope of a spanning surface to be the sum of its component-wise boundary slopes.

We investigate the analog of the Thurston boundary of Teichmuller space in the context of convex real projective structures on closed manifolds. In particular we give a new interpretation of measured laminations in terms of non-standard hyperbolic structures over the hyper-reals.

We will construct an explicit lower bound for the volume of a hyperbolic orbifold dependent on dimension and the maximal order of torsion in the orbifolds' fundamental group. We will then discuss progress in developing a sharp bound in dimension 4.

Recently Sangbum Cho and Darryl McCullough discovered a remarkable classification of tunnel number one knots and their unknotting tunnels. In December, raging Oklahoma storms prevented McCullough from reaching an AIM conference where he was expected to talk about this work. Conference organizers asked me to impersonate him. This talk will be a reprise of that performance.

There is only one connected 3-manifold known not to have a real projective structure. We will sketch a proof.

In this talk I define a Coxeter group as a natural generalization of a finite (or Euclidean) reflection group and then discuss how Coxeter groups can be roughly classified by the symmetric spaces on which they act as reflection groups. The goal of the talk is to clarify the various things it can mean to be a hyperbolic reflection group.

Boring is a method for converting a knot or 2-component link in a 3-manifold into another knot or 2-component link. It generalizes many well-known operations in knot theory, including attaching a band or changing a crossing. I will discuss how combinatorial sutured manifold theory can be used to give bounds on the euler characteristic of certain essential surfaces (including Seifert surfaces and planar surfaces) in the exterior of knots and links obtained by boring a split link or unknot (subject to minor restrictions). The talk will include a whirlwind introduction to combinatorial sutured manifold theory. Hopefully, only the knots and links (and not the audience) will be bored!

The group SL(2,Z) acts on the curve complex of a Torus, yielding a presentation of SL(2,Z) through Basse-Serre theory. I will give a new palindromic presentation of SL(3,Z) by defining a simplicial complex, the 3rd Torus Complex, on which SL(3,Z) acts and using a theorem of K.S. Brown concerning the structure of groups acting on a CW-complex. As a corollary, I will show that the Torus Complex is simply connected.

I extend the notion of perturbation of wreath products from my thesis, and similar to the methods in my thesis, prove that certain perturbations are still quasi-isometric to the original wreath product. As an application, I show that a property of containing free subgroup or free subsemigroup, and many other algebraic properties of groups are not quasi-isometrically rigid.

We view braids as automorphisms of punctured disks and define a partial order on pseudo-Anosov braids called the "forcing order". The order measures whether one automorphism induces another given automorphism on the surface. Pseudo- Anosov growth rate decreases relative to the order and appears to give a good measure of braid complexity. Unfortunately it appears difficult computationally to determine explicitly the partial order structure by hand. We use several computer algorithms to study the bottom part of the partial order when the number of braid strands is fixed. From the algorithms, we build sequences of low entropy pseudo- Anosov n-strand braids that are minimal in the sense that they do not force any other pseudo-Anosov braids on the same number of strands. The sequences are an extension of work done by Hironaka and Kin, and we conjecture the sequences to achieve minimal entropy among certain nontrivial classes of braids. In general, the lowest entropy pseudo-Anosov braids appear to have mapping tori that come from Dehn surgery on very low volume hyperbolic 3-manifolds and we begin to analyze the relation between entropy and hyperbolic volume.

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. In joint work with R. Kent and S. Schleimer, we prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in the sense of Farb and Mosher. In particular, we obtain an affirmative answer to their question of local convex cocompactness of Whittlesey's group. I'll begin by explaining the relevance of convex cocompactness to the coarse geometry of surface bundles. Then I'll describe some of the ingredients of our proof of the theorem above, including an interesting relation between the action of the kernel on the curve complex and on a family of trees.

The interplay between topology and geometry has been an important topic in the study of 3-manifolds. We will discuss an approach to study hyperbolic 3-manifolds by using a Heegaard decomposition. The main problem is to start with a Heegaard diagram for the 3-manifold and find a combinatorial description of the hyperbolic metric.

This talk is intended to address the following question: Under what circumstances can one change a crossing in a knot yet not change the overall knot type? There is a conjecture in knot theory which attempts to classify the crossings which have this property. We will discuss the proof of this conjecture in the special case of closed 3-braid diagrams.

Right-veering surface homeomorphisms play an interesting part in 3-dimensional contact topology. We will discuss an algorithm driven by the unreduced Burau representation and an invariant known as the fractional Dehn twist coefficient which allows us to identify a pseudo-Anosov representative of an element of B₃ as right-veering, left-veering or neither.

Planar algebras give operations on graded vector spaces quite analogous to multiplication of polynomials in several variables. We will begin with the most evolved definition of planar algebras, at least those of relevance to the study of von Neumann algebras. Adopting the operadic point of view (though not all the formalism!) we see that the meaning of the identity for a planar algebra is a tangle without input discs. This gives the already rather rich structure of the so-called Temperley-Lieb algebra.

We discuss some joint work with Bestvina and Kleiner addressing the question of when two right angled Artin groups are quasi-isometric.

I will describe some old and new results on polygonal billiards and connections with pseudo-Anosov diffeomorphisms, groups of automorphisms of surfaces and scissors congruence.

I will define the generalized Conway product of links and give a tight lower bound for the bridge number of this product in terms of the bridge numbers of the two factor links.