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Boston College

HMI

NSF

SFI

# Knots, Surfaces and Three-manifolds

## Schedule of Talks

All talks will be held in the Maxwell Lecture Theatre in the Hamilton Building, TCD (see campus map).

 Time Speaker Talk Thursday 9.30 - 10.30 Matthew Day (Caltech) An analogue of the Birman exact sequence for automorphism groups of free groups 10.30-10.45 COFFEE 10:45 - 11:45 Graham Niblo (Southampton) Immersion rigidity 11:45 - 1:30 LUNCH 1:30 - 2:30 Cameron Gordon (UT Austin) Seifert fibered Dehn filling 2:45 - 3:45 Brendan Owens (Glasgow) Alternating links and rational balls 4:00 - 5:00 Sucharit Sarkar (Columbia) Knot cobordisms and knot Floer homology Friday 9:30 - 10:30 Brian Bowditch (Warwick) Models of 3-manifolds and hyperbolicity 10.30 - 10.45 COFFEE 10:45 - 11:45 Juan Souto (Michigan, Ann Arbor) Homomorphisms between mapping class groups 11:45 - 1:30 LUNCH 1:30 - 2:30 Jeremy Kahn (Stonybrook) Counting Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds 2:45 - 3:45 Alan Reid (UT Austin) Surface groups, 3-manifold groups and SL_3 4:00 - 5:00 Brendan Guilfoyle (Tralee) Mean curvature flow, holomorphic discs and umbilic points on surfaces Saturday 10:00 - 11:00 Andras Juhasz (Cambridge) Cobordisms of sutured manifolds 11.00 - 11.15 COFFEE 11.15- 12:15 Elisenda Grigsby (Boston College) Khovanov homology, Heegaard Floer homology, and gluing 12.15 - 2.00 DISCUSSIONS AND LUNCH END OF WORKSHOP

## Abstracts

Brian Bowditch (Warwick)
Models of 3-manifolds and hyperbolicity
Abstract: We give an outline of how various geometric models have been used to understand hyperbolic 3-manifolds in relation Teichmuller space.  It is possible to characterise models in terms Gromov hyperbolicity.  The situation is well understood in the bounded geometry case, and we suggest ways in which this can be generalised.

Matthew Day (California Institute of Technology)
An analogue of the Birman exact sequence for automorphism groups of free groups
Abstract: The classical Birman exact sequence relates mapping class groups of surfaces with different numbers of marked points to braid groups on surfaces.  We construct an analogue for automorphism groups of free groups.  In the automorphism group of the free group of rank $n$, we consider the pointwise stabilizer of the set of conjugacy classes of $k$ elements of a free generating set for the free group, for $n>k>0$.  This group has a natural map to the automorphism group of the free group of rank $n-k$.  We analyze the kernel of this map; we find an explicit finite generating set and an explicit infinite presentation for the kernel, but we show that in nontrivial cases this kernel is not finitely presentable.  This is joint work with Andy Putman.

Cameron Gordon (University of Texas, Austin)
Seifert fibered Dehn filling
Abstract: Much is known now about hyperbolic 3-manifolds with pairs of non-hyperbolic Dehn fillings. The situation that is least understood is when one of the fillings is a Seifert fibered space. We will discuss joint work in progress with Steve Boyer and Xingru Zhang  on the case where one of the fillings is Seifert fibered and the other is toroidal.

Elisenda Grigsby (Boston College)
Khovanov homology, Heegaard Floer homology, and gluing
Abstract: Ozsvath-Szabo's spectral sequence from Khovanov homology to Heegaard Floer homology has generated a number of interesting applications to questions in low-dimensional topology.  Furthermore, viewing the connection through the lens of sutured manifold theory has enhanced our understanding of its algebraic structure.  In particular, a generalization of Juhasz's surface decomposition theorem implies that the spectral sequence behaves "as expected" under natural geometric operations like cutting and stacking.

In this talk, I will describe joint work in progress with Denis Auroux and Stephan Wehrli aimed at understanding how the connection between Khovanov and Heegaard-Floer homology behaves under gluing.  More precisely, we will see how to recover (a portion of) the sutured version of Khovanov homology using bimodules over quiver algebras originally defined by Khovanov-Seidel.  Along the way, we will discuss an intriguing relationship between these Khovanov-Seidel bimodules and certain bimodules appearing in the bordered Floer package of Lipshitz-Ozsvath-Thurston.

Brendan Guilfoyle (Tralee Institute of Technology)
Mean curvature flow, holomorphic discs and umbilic points on surfaces
Abstract: A conjecture of Constantin Caratheodory dating from the 1920's states that the number of umbilic points on a closed convex body in Euclidean 3-space must be at least 2. In this talk we outline the proof of this conjecture obtained in collaboration with Wilhelm Klingenberg which utilizes neutral Kaehler geometry, mean curvature flow and holomorphic curves. Along the way we open up the geometric relationship between surfaces in a 3-dimensional space-form and surfaces in the 4-dimensional space of oriented geodesics.

Andras Juhasz (Cambridge)
Cobordisms of sutured manifolds
Abstract: Sutured manifolds are compact oriented 3-manifolds with boundary, together with a set of dividing curves on the boundary. Sutured Floer homology is an invariant of balanced sutured manifolds that is a common generalization of the hat version of Heegaard Floer homology and knot Floer homology. I will define cobordisms between sutured manifolds, and show that they induce maps on sutured Floer homology groups, providing a type of TQFT. As a consequence, one gets maps on knot Floer homology groups induced by decorated knot cobordisms.

Jeremy Kahn (Stonybrook University)
Counting Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds
Abstract: Let M be a closed hyperbolic 3-manifold; we let s(M, g) be the number of essential immersed surfaces of genus at most g. Building on our previous work, and improving on the work of Joseph Masters, we show that s(M, g) is on the order of g^2g, and we discuss theorems and conjectures on the precise nature of the growth of this function.
This is joint work with Vladimir Markovic.

Graham Niblo (Southampton University)

Immersion rigidity
Abstract: (Joint work with Aditi Kar) The classical torus theorem for 3 manifolds inspired an industry of generalizations including Kropholler's algebraic torus theorem for Poincaré duality groups. We outline a new result in this spirit which may be viewed as a topological analogue of the  geometric superrigidity theorem of  Ngaiming Mok, Yum-Tong Siu and Sai-Kee Yeung.

Starting from a generalisation of Stallings' Theorem concerning groups with more than one end we prove the following:

Theorem (Kar, GAN)
Let $\Gamma$ be  the fundamental group of a closed orientable manifold $N$ with universal cover $\tilde{N}$ which is a quaternionic hyperbolic space, the Cayley hyperbolic plane, or an irreducible symmetric space of real rank at least 2. Then any $\pi_1$-injective map from $N$ to an aspherical manifold $M$ with $dim(M)=dim(N)+1$ is homotopic to a finite cover of an embedding.
If  $\tilde{N}$ is either quaternionic hyperbolic space of quaternionic dimension at least 2 or is  the Cayley hyperbolic plane then one can show that the resulting embedding is non-separating so we obtain:
Corollary (Kar, GAN)
Let $\Gamma$ be  the fundamental group of a closed orientable manifold $N$ with universal cover $\tilde{N}$ either quaternionic hyperbolic space of quaternionic dimension at least 2 or the Cayley hyperbolic plane. If $M$ has trivial first Betti number then there are no  $\pi_1$-injective maps from $N$ to $M$.

Brendan Owens (Glasgow University)

Abstract:  For a slice knot K in the 3-sphere it is well known that the double branched cover Y_K bounds a smooth rational homology 4-ball.  Paolo Lisca has shown that this condition is sufficient to determine sliceness for 2-bridge knots, and that this generalises to 2-bridge links.  I will discuss the problem of determining whether Y_L bounds a rational ball when L is an alternating link.

Alan Reid (University of Texas, Austin)
Surface groups, 3-manifold groups and SL_3
Abstract: This talk will describe recent work on using representations of 3-manifold groups and surface groups to study the subgroup structure (both finite and infinite index) of SL(3,Z).

Sucharit Sarkar (Columbia University)
Knot cobordisms and knot Floer homology
Abstract: A pointed knot cobordism in $\mathbb{R}^3\times I$ induces a map on knot Floer homology. We would like to explore this map in a purely combinatorial setting, using grid diagrams.

Juan Souto (University of Michigan, Ann Arbor)
Homomorphisms between mapping class groups
Abstract: Suppose that $X$ and $Y$ are surfaces of finite type, that $X$ has genus $g(X)\ge 4$ and that $Y$ has genus $g(Y)\le 2g-2$. Then every homomorphism from the mapping class group of $X$ to the mapping class group of $Y$ is geometric; more concretely, induced by an embedding of $X$ into $Y$. This is joint work with Javier Aramayona.

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