# Columbia Geometric Topology Seminar

## Fall 2011

The GT seminar meets on Fridays in Math. 520, at 1:15PM.
Organizer: Walter Neumann.
Other area seminars. Our e-mail list. Archive of previous semesters

### FALL 2011

Date Speaker Title
September 09 Everyone Organizational meeting joint with SGGTC seminar
September 16
Room 622
Kristen Hendricks A rank inequality for knot Floer homology of branched double covers
September 23 Dan Margalit (Georgia Tech) Mapping class groups, symplectic groups, and the Burau representation
September 30 Anastasiia Tsvietkova (UTK) An alternative approach to hyperbolic structures on link complements
October 7 Jenny Wilson (Chicago) The cohomology groups of the pure string motion group are uniformly representation stable
October 14  Vaibhav Gadre (Harvard) Curve complex translation lengths
Thursday October 20, 11-12am, Room 622 Matthias Kreck (Bonn) From cutting and pasting to topological quantum field theories
October 21  Feng Luo (Rutgers) Solving Thurston's equation in the real numbers
October 28  Danny Ruberman (Brandeis) Applications of Heegaard-Floer theory to link concordance
November 4, Double header! 9:30am Tali Pinsky (Technion) Knotted geodesics on Hecke triangles
November 4, 1:15pm Joan Licata Legendrian knot invariants in Seifert fibered spaces
November 11  Anne Pichon (Marseille) Bilipschitz classification of singularities of complex surfaces
November 18  Greg Kuperberg Denseness and Zariski denseness of Jones braid representations
November 25  No Seminar University Holiday
December 2 Jeff Brock (Brown U) The Weil-Petersson metric and the geometry of hyperbolic 3-manifolds
December 9  Harold Sultan Large Scale Geometry of Teichmuller Space; Thickness and Divergence
December 16  Martin Bridson (Oxford) Profinite completions of groups, geometry, and decision problems

## Abstracts.

#### September 16. Note room change: Math 622

Kristen Hendricks, “A rank inequality for knot Floer homology of branched double covers”

Abstract: Given a knot K in the three sphere, we compare the knot Floer homology of (S^3, K) with the knot Floer homology of (Sigma(K), K), where Sigma(K) is the double branched cover of the three-sphere over K. By studying an involution on the symmetric product of a Heegaard surface for (Sigma(K), K) whose fixed set is a symmetric product of a Heegaard surface for (S^3, K), and applying recent work of Seidel and Smith, we produce an analog of the classical Smith inequality for cohomology for knot Floer homology. To wit, we show that the rank of the knot Floer homology of (S^3,K) is less than or equal to the rank of the knot Floer homology of (Sigma(K), K).

#### September 23

Dan Margalit, (Georgia Tech), “Mapping class groups, symplectic groups, and the Burau representation”

Abstract: There is a natural map from the braid group on 2g strands to the mapping class group of a surface of genus g. By considering the action on the homology of the surface, we obtain a representation of the braid group. This representation is nothing other than the Burau representation evaluated at t = -1. We are interested in the kernel of this representation, as this group is isomorphic to the fundamental group of the branch locus of the period mapping from Torelli space to the Siegel upper half-plane. Hain has conjectured that this kernel is generated by Dehn twists. After explaining the necessary background, we will provide evidence for and progress towards this conjecture. We will also discuss other cohomological properties of the kernel. This is joint work with Tara Brendle, Leah Childers, and Allen Hatcher.

#### September 30

Anastasiia Tsvietkova (UTK), “An alternative approach to hyperbolic structures on link complements”

Abstract: Thurston demonstrated that every link in $S^3$ is a torus link, a satellite link or a hyperbolic link and these three categories are mutually exclusive. It also follows from work of Menasco that an alternating link represented by a prime diagram is either hyperbolic or a $(2,n)$--torus link. A new method for computing the hyperbolic structure of the complement of a hyperbolic link, based on ideal polygons bounding the regions of a diagram of the link rather than decomposition of the complement into ideal tetrahedra, was suggested by M. Thistlethwaite. Although the method is applicable to all diagrams of hyperbolic links under a few mild restrictions, it works particularly well for alternating (non-torus) links. The talk will introduce the basics of the method. Some applications will be discussed, including a surprising rigidity property of certain tangles, a new numerical invariant for tangles, and formulas that allow one to calculate the volume of 2--bridged links directly from the diagram.

#### October 7

Jenny Wilson (Chicago) “The cohomology groups of the pure string motion group are uniformly representation stable”

Abstract: The string motion group Sn, the group of motions of n disjoint, unlinked, unknotted circles in 3-space, is a generalization of the braid group. It can be identified with the symmetric automorphism group of the free group on n letters. The pure string motion group PSn, which is the analogue of the pure braid group, admits an action by the hyperoctahedral group Wn. The rational cohomology of PSn is not stable in the classical sense -- the dimension of the degree k cohomology group tends to infinity as n grows -- however, Church and Farb have recently developed a notion of stability for a sequence of vector spaces with a group action, which they call representation stability. Inspired by their recent work on the cohomology of the pure braid group, they conjectured that for each k>0, the degree k rational cohomology of PSn is uniformly representation stable with respect to the induced action of Wn, that is, the description of the decomposition of the cohomology group into irreducible Wn-representations stabilizes for n>>k. In this talk, I will give an overview of the theory of representation stability, and briefly outline a proof verifying this conjecture. This result has implications for the cohomology of the string motion group, and the permutation-braid group.

#### October 14

Vaibhav Gadre (Harvard), “Curve complex translation lengths”

Abstract: The curve complex C(S) of a closed orientable surface S of genus g is an infinite graph with vertices isotopy classes of essential simple closed curves on S with two vertices adjacent by an edge if the curves can be isotoped to be disjoint. By a celebrated theorem of Masur-Minsky, the curve complex is Gromov hyperbolic. Moreover, a pseudo-Anosov map f of S acts on C(S) as a hyperbolic isometry with an invariant quasi-axis. This allows one to define an asymptotic translation length for f on C(S). In joint work with Chia-yen Tsai, we prove that the minimal pseudo-Anosov asymptotic translation lengths on C(S) are of the order 1/g^2. We shall also outline related interesting results and questions.

#### October 20, 11-12am

Matthias Kreck (Bonn) “From cutting and pasting to topological quantum field theories”

Abstract: A cut and paste invariant (SK-invariant) is an invariant on closed smooth manifolds M (oriented, Spin or with other structure) which is unchanged if one cuts M along a codimension 1 submanifold into two pieces and glues them differently by a structure preserving diffeomorphism. A weaker condition is that the difference of the invariants only depends on the gluing diffeomorphism, not on the pieces. These are called cut and paste controlled invariants (SKK-invariants). Both the cut and paste invariants and the cut and paste controlled invariants were determined long ago in joint work with Karras, Neumann and Ossa. The partition function of an invertible topological quantum filed theory is a cut and paste controlled invariant. In joint work with Stolz and Teichner we investigate the question, which cut and paste controlled invariants are such partition functions. Using deep work by Galatius, Madsen, Tillmann and Weiss one can give a complete answer. We also try to construct such field theories explicitely. First examples in dimension 3 are explained.

#### October 21

Feng Luo (Rutgers) “Solving Thurston's equation in the real numbers”

Abstract: Thurston's equation defined on triangulated 3-manifolds tends to find hyperbolic structures. It is usually solved in the complex numbers. We are interested in solving Thurston's equation in the real numbers and we establish a variational principle associated to such solutions.

#### October 28

Danny Ruberman (Brandeis) “Applications of Heegaard-Floer theory to link concordance”

Abstract: Invariants derived from Heegaard-Floer theory have proved to be very powerful in investigating smooth knot concordance. I will discuss some applications of these invariants, especially the so-called correction term (or d-invariant) to problems of link concordance. For example, answering a question of Jim Davis, I will describe links with trivial (two-variable) Alexander polynomial and unknotted components that are not concordant to the Hopf link. I will also discuss concordance aspects of a simple cabling construction of links.

#### November 4 9:30am

Tali Pinsky, “Knotted geodesics on Hecke triangles”

Abstract: Ghys discovered in 2006 that the periodic orbits of the modular flow are identical to the set of orbits of the Lorenz flow, studied by Birman and Williams. Thus, for example, these orbits are all prime knots, fibered, and have positive signature. This is the first result regarding the topological properties of closed geodesics on Hecke triangles, which are otherwise a well studied class. The results were obtained using a "template" for the flow, which reduces the original three dimensional flow to a much simpler two dimensional system. In the talk I will present a method for computing templates for geodesic flows on an infinite class of Hecke triangle groups. We will then use the arising templates to show that as knots, all closed geodesics corresponding to any of these groups are prime.

#### November 4

Joan Licata, “Legendrian knot invariants in Seifert fibered spaces”

Abstract: Knot theory beyond the three-sphere has seen increased attention in recent years, and in this talk I will focus on knot theory in Seifert fibered spaces. In particular, we'll consider Legendrian knots in Seifert fibered spaces equipped with a special contact form. This setting gives rise to both topological and contact geometric questions, and I'll describe some of the ingredients used to prove the Legendrian non-simplicity of an infinite family of knot types representing torsion homology classes. This is joint work with J. Sabloff.

#### November 11

Anne Pichon (Marseille), “Bilipschitz classification of singularities of complex surfaces”

Abstract: This is a joint work with Lev Birbrair and Walter Neumann.
We study the geometry of a normal complex surface $$X$$ in a neighbourhood of a singular point $$p \in X$$. It is well known that for all sufficiently small $$\epsilon > 0$$ the intersection of $$X$$ with the sphere $$S^{2n-1}_\epsilon$$ of radius $$\epsilon$$ about $$p$$ is transverse, and $$X$$ is therefore locally "topologically conical," i.e., homeomorphic to the cone on its link $$X\cap S^{2n-1}_\epsilon$$. However, as shown by Birbrair and Fernandez, $$(X,p)$$ need not be "metrically conical", i.e., bilipschitz equivalent to a standard metric cone when $$X$$ is equipped with the Riemanian metric induced by the ambient space. In fact, it was shown by Birbrair, Fernandez and Neumann that it rather rarely is.
I will present a complete classification of the bilipschitz geometry of $$(X,p)$$. It starts with a decomposition of a normal complex surface singularity into its "thick" and "thin" parts. The former is essentially metrically conical, while the latter shrinks rapidly in thickness as it approaches the origin. The thin part is empty if and only if the singularity is metrically conical. Then the complete classification consists of a refinement of the thin part into geometric pieces. I will describe it on an example, and I will present a list of open problems related with this new point of view on classifying complex singularities.

#### November 18

Greg Kuperberg (Davis), “Denseness and Zariski denseness of Jones braid representations”

Abstract: For several applications in computer science, it is useful to know that various braid group representations in quantum algebra are dense. The first and still most important result of this type is the theorem of Freedman, Larsen, and Wang, that the Jones braid representations at principal roots of unity are dense as unitary representations. However, the original proof was in the setting of compact groups, and depended on the classification of finite simple groups. I will discuss another approach which is successful in both compact and non-compact cases. One idea is to work at the level of Lie algebras and their representations whenever possible. A second idea is to change the question to the easier case of the Zariski topology, and postpone the analytic topology until the end. A third idea is to rely on a powerful surjectivity lemma of Phillip Hall, adapted to the setting of Lie groups or algebraic groups.
If time permits, I can also discuss an important negative consequence of the denseness results, that any fair approximation to a dense value of the Jones polynomial is #P-hard.

#### December 2

Jeff Brock (Brown), “The Weil-Petersson metric and the geometry of hyperbolic 3-manifolds”

Abstract: Many coarse relationships support the existence of a deep connection between the synthetic geometry of the Weil-Petersson metric and the geometry of hyperbolic 3-manifolds.  In this talk, I'll elaborate on some new, finer examples of such connections, as well as some contrary evidence to a fundamental link. This talk will present joint work with Yair Minsky and Juan Souto.

#### December 9

Harold Sultan (Columbia), “Large Scale Geometry of Teichmuller Space; Thickness and Divergence”

Abstract: I will talk about the asymptotic geometry of Teichmuller space equipped with the Weil-Petersson metric. In particular, I will give a criterion for determining when two points in the asymptotic cone of Teichmuller space can be separated by a point; motivated by a similar characterization in mapping class groups by Behrstock-Kleiner-Minsky-Mosher and in right angled Artin groups by Behrstock-Charney. I will also explain two new ways to uniquely characterize the Teichmuller space of the genus two once punctured surface amongst all Teichmuller spaces: one is that it is thick of order two and the other is that it has a divergence function which is superquadratic yet subexponential.

#### December 16

Martin Bridson (Oxford), “Profinite completions of groups, geometry, and decision problems”

Abstract: I'll discuss the geometric background to the questions: what can one say about groups with the same profinite completion, and which properties of a finitely presented group are visible in its profinite completion? I shall describe recent results in this direction.
I'll then sketch a proof of a recent theorem with Henry Wilton that implies, among other things, that the following problems for finitely presented discrete groups G (even CAT(0) groups) are undecidable:
1. Does G have a non-trivial finite quotient?
2. Is every subgroup of finite index in G perfect?
Moreover, the isomorphism problem for finitely presented residually-finite groups with a fixed profinite completion is unsolvable.

# Other relevant information.

## Previous semesters:

2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

## Our e-mail list.

Announcements for this seminar, as well as for related seminars and events, are sent to the GT seminar mailing list. You can subscribe directly or by contacting Walter Neumann.