# Columbia Geometric Topology Seminar

## Spring 2012

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

### SPRING 2012

Date Speaker Title
January 19, 3pm
Room 528
Dani Wise Quasiconvex Subgroups of Relatively Hyperbolic groups
January 20 Everyone Organizational meeting joint with SGGTC seminar
January 27 Jose Luis Cisneros-Molina Geometric elements in algebraic K-Theory
February 3 Tudor Dimofte Non-Abelian Torsion and the Neumann-Zagier Equations
February 10 David Futer The Jones polynomial and surfaces far from fibers
February 17 Craig Hodgson Angle structures on cusped hyperbolic 3-manifolds
February 24 Joseph Maher Growth rates for stable commutator length
March 2 Elizabeth Denne Transversality theorems for configuration spaces and applications to the "square peg" problem.
March 9 No seminar (too many people at Tampa AMS meeting or elsewhere)
March 16 Spring Break Spring Break
March 23 Ryan Blair Width is Not Additive
March 30 Room 507 Harriet Moser Upper Bound on Distance in the Pants Complex
April 6 Matthias Kreck Representing cohomology classes by submanifolds, revisited
April 13 Ina Petkova On the gradings and decategorification of Bordered Heegaard Floer homology
April 20 Yi Liu Virtual cubulation of nonpositively curved graph manifolds
April 27 Haydee Aguilar Cabrera New open book decompositions in Singularity Theory
May 4 No seminar: Cornell Topology Festival May 4-7
May 11 Christian Zickert: Seminar postponed to the Fall Princeton BrowderFest May 8-11

### Fall 2012

Date Speaker Title
September 7 Organizational meeting
September 21 No Seminar: Group Theory on the Hudson
Christian Zickert TBA
November 9 Chris Leininger TBA

November 23 No Seminar Thanksgiving

December 14 Last seminar of the semester

## Abstracts.

#### January 19. Note room change: Math 528

Dani Wise, “Quasiconvex Subgroups of Relatively Hyperbolic groups ”

Abstract: I will review Bowditch's treatment of a relatively hyperbolic group, and in that context, describe an elegant reformulation of the notion of "relatively quasiconvex subgroup" introduced with Eduardo Martinez-Pedroza. I will then describe work with Hadi Bigdely that applies this criterion to recognize the relative quasiconvexity of a subgroup H in a group G splitting as a graph of groups. The simplest applications revisit beautiful work of Dahmani and others.

#### January 27

Jose Luis Cisneros-Molina “Geometric elements in algebraic K-Theory”

Abstract: In this talk we show how to represent elements in $K_3(C)$ using homology 3-spheres and hyperbolic 3-manifolds endowed with a representation of their fundamental group. We review some results of Jones and Westbury and the relation of such elements with the Bloch invariant defined by Neumann and Yang.

#### February 3

Tudor Dimofte “Non-Abelian Torsion and the Neumann-Zagier Equations”

Abstract: I will present an extraordinarily simple formula for the torsion of a hyperbolic 3-manifold M in terms of shape parameters and gluing matrices involved in an ideal triangulation. The torsion appears (for example) as the first subleading correction to the large-N asymptotics of the Kashaev invariant of M. The formula immediately shows that the torsion belongs to the invariant trace field of M. (This is ongoing work with S. Garoufalidis.)

#### February 10

David Futer “The Jones polynomial and surfaces far from fibers”

Abstract: This talk explores relations between colored Jones polynomials and the topology of incompressible spanning surfaces in knot and link complements. Under mild diagrammatic hypotheses, we prove that the growth of the degree of the colored Jones polynomials is a boundary slope of an essential surface in the knot complement. We also show that certain coefficients of the Jones and colored Jones polynomials measure how far this surface is from being a fiber in the knot complement. This is joint work with Effie Kalfagianni and Jessica Purcell.

#### February 17

Craig Hodgson “Angle structures on cusped hyperbolic 3-manifolds”

Abstract: It is conjectured that every cusped hyperbolic 3-manifold has a "geometric" ideal triangulation, i.e. a decomposition into positive volume ideal hyperbolic tetrahedra. Under a mild homology assumption on the manifold we construct topological ideal triangulations which admit a strict angle structure, which is a necessary condition for the triangulation to be geometric. In particular, every hyperbolic knot or link complement in the 3-sphere has such a triangulation. We will also discuss some ongoing work on when ideal triangulations with a strict angle structure are actually geometric.

#### February 24

Joseph Maher “Growth rates for stable commutator length”

Abstract: We give a gentle introduction to stable commutator length and shows that it grows as n / log n for both random walks and words of length n in hyperbolic groups. This is joint work with Danny Calegari.

#### Mar 2

Elizabeth Denne “Transversality theorems for configuration spaces and applications to the "square peg" problem.”

Abstract: In joint work with Jason Cantarella and John McCleary, we prove a transversality "lifting property" for compactified configuration spaces. Namely, the submanifold of configurations of points on an arbitrary submanifold of Euclidean space may be made transverse to any submanifold of the configuration space of points in Euclidean space by an arbitrarily small variation of the initial submanifold, as long as the two submanifolds of configuration space are boundary-disjoint. We use this setup to provide proofs of a number of "special inscribed configurations" using differential topology. For instance, there are an odd number of inscribed squares in a generic plane curve, while there is a family of inscribed regular (n+1)-simplices in a generic (n-1)-sphere.

#### Mar 23

Ryan Blair “Width is Not Additive”

Abstract: Width is an invariant of knots closely related to the notion of bridge number. Width minimizing embeddings of knots carry useful information about the topology of the knot complement. Although width has been an integral tool in several celebrated results in 3-manifold topology, some of its most basic properties are still unknown. In this talk, we will discuss how width behaves with respect to connected sum.

#### March 30 Room 507 (Note room change)

Harriet Moser “Upper Bound on Distance in the Pants Complex”

Abstract: Our goal is to establish an upper bound on the distance between $2$ distinct pants decompositions in the pants complex. As part of the process, an upper bound on distance in the pants complex, modulo the action of the mapping class group, is also found. This is done by use of graph theory together with techniques introduced by Putnam in his proof of connectivity of several curve complexes, but adapted here to study length in the pants graph.

#### Apr 6

Matthias Kreck “Representing cohomology classes by submanifolds, revisited”

Abstract: Thom (1954) has proved that for each cohomology class x of a closed oriented manifold M there is a multiple representable by an embedding. More precisely he shows that for classes of odd degree one can always represent a multiple by an embedding with trivial normal bundle and that the same holds if the degree of the class is even and larger than half the dimension of M. He doesn't say anything about the normal bundle in the other cases. Peter Teichner and I asked for conditions which give more information on the normal bundle of the embedding. For example we prove (as was earlier observed by Sullivan) that if x^2 = 0 one can always find an embedding with trivial normal bundle. What happens if x^3, or x^4 or .. vanish? Related problems are discussed if time permits.

#### Apr 13

Ina Petkova “On the gradings and decategorification of Bordered Heegaard Floer homology”

Abstract: Bordered Heegaard Floer homology is a Floer theory over Z/2 for manifolds with boundary. After gluing, it recovers Heegaard Floer homology for closed manifolds. I will describe the bordered Floer package and some applications to knot theory. Then I will define a Z/2 differential grading, and discuss the decategorification of bordered Heegaard Floer homology with this grading.

#### April 20

Yi Liu “Virtual cubulation of nonpositively curved graph manifolds”

Abstract: In this talk, I will show that a nontrivial compact graph manifold is nonpositively curved if and only if its fundamental group virtually embeds into a right-angled Artin group. As a consequence, nonpositively curved graph manifolds have linear fundamental groups.

#### Apr 27

Haydee Aguilar Cabrera “New open book decompositions in Singularity Theory”

Abstract: Given a real analytic $d$-regular function from $\mathbb{R}^n$ to $\mathbb{R}^p$, ($n \geq p$), with an isolated singularity at the origin, a refinement of the Milnor fibration Theorem associates an open book decomposition of $\mathbb{S}^{n-1}$ to the singularity.
We present a family of real analytic $d$-regular functions $f$ from $\mathbb{R}^4$ to $\mathbb{R}^2$ with isolated singularity at the origin. Let $L_f$ be the link of the singularity of $f$. From $f$ we define a function $F$ from $\mathbb{R}^6$ to $\mathbb{R}^2$, with link $L_F$, such that $F$ is $d$-regular, $F$ has an isolated singularity at the origin and the knot $(\mathbb{S}^{5},L_F)$ is a cyclic suspension of $(\mathbb{S}^{3},L_f)$.

#### May 11

Christian Zickert: Seminar postponed to the Fall

# Other relevant information.

## Previous semesters:

Fall 2011, 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.