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The 4th William Rowan Hamilton Geometry and Topology Workshop

August 28-30, 2008

The Hamilton Mathematics Institute, Trinity College Dublin

Heegaard splittings, mapping class groups,

curve complexes and related topics

Schedule of Talks

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

9.30 - 10.30
Cameron Gordon Heegaard genus and Dehn surgery
10:45 - 11:45
Darryl McCullough Disk complexes, arc complexes, and knots
11:45 - 1:30
1:30 - 2:30
Joel Hass Families of harmonic maps and stabilization of Heegaard splittings
2:45 - 3:45
Yair Minsky
Dynamics of Out(F_n) in character varieties
4:00 - 5:00
Danny Calegari
Stable commutator length is rational in free groups

9:30 - 10:30
Martin Scharlemann  Reconfiguring Qiu's proof of the Gordon Conjecture
10.30 - 10.45
10:45 - 11:45 Saul Schleimer
The geometry of train track splitting sequences
11:45 - 1:30 LUNCH
1:30 - 2:30 Vladimir Markovic Random ideal triangulations and the Weil-Petersson distance between finite degree  covers of punctured Riemann surfaces
2:45 - 3:45 Joan Porti Collapsing three-manifolds
4:00 - 5:00 Jeff Brock Bounded geometry and combinatorics for hyperbolic Heegaard splittings

9:30 - 10:30 Moon Duchin
Curves on flat surfaces
10.30 - 10.45
10:45 - 11:45 Juan Souto Some remarks about the mapping class group
11.45 - 1.30
1.30 - 2.30
Richard Weidmann Maps onto hyperbolic groups
2.45 - 3.45
Brian Bowditch


Speaker: Cameron Gordon (Texas)
Title: Heegaard genus and Dehn surgery
We will discuss the behavior of Heegaard genus under Dehn filling, focusing on the case of knots in S^3. We will also apply our results to the conjecture that any Seifert fibered Dehn surgery on a  hyperbolic knot in S^3 must be integral. (Joint work with Ken Baker andJohn Luecke.)

Speaker: Darryl McCullough (Oklahoma)
Title: Disk complexes, arc complexes, and knots
We will discuss: (1) The tree of knot tunnels, a classification of all tunnels of all tunnel number 1 knots (or equivalently of the genus-2 Heegaard splittings of the exteriors of knots in the 3-sphere), using the disk complex of the genus-2 handlebody, (2) Depth and bridge numbers, the "depth" invariant of a knot tunnel, and its application to bridge numbers of tunnel number 1 knots, and (3) Level position of knots, an application of arc complexes to knot theory. These are joint work with Sangbum Cho, and part (3) is also joint with Arim Seo.

Speaker: Joel Hass (Davis)
Title:  Families of harmonic maps and stabilization of Heegaard splittings
We give examples for each  genus greater than one of a 3-manifold with two genus g Heegaard splittings  that require g stabilizations to  become equivalent. Previously known examples required at most one stabilization. Control of families of Heegaard surfaces is obtained through a deformation to harmonic maps.
This is joint work with Abigail Thompson and Bill Thurston.

Speaker: Yair Minsky (Yale)
Title: Dynamics of Out(F_n) in character varieties
We study the PSL(2,C) character variety X(F_n) of the free group, with its natural action by Out(F_n). We make the observation that there is an open, Out(F_n)-invariant subset on which Out(F_n) acts properly discontinuously, but which is strictly larger than the Schottky space. In particular it contains degenerate representations, and representations with dense image. This suggests a dynamical dichotomy different from the usual discrete/indiscrete one, but leaves open many questions such as the what the maximal domain of discontinuity is, and what the ergodic decomposition of the action looks like.

Speaker: Danny Calegari (Caltech)
Title: stable commutator length is rational in free groups
Let G be a group, and g an element of its commutator subgroup. The commutator length of g, denoted cl(g), is the smallest number of commutators in G whose product is g. The stable commutator length, denoted scl(g), is the limit scl(g) = cl(g^n)/n. We show that in a free group, stable commutator length is rational, by an explicit constructive argument. As a corollary, one can show that every rational homology class in H_2 of a graph of free groups amalgamated over cyclic subgroups is (virtually) represented by an embedded surface group. Along the way we obtain a (polynomial time) algorithm to compute scl in a free group. We describe some of the surprising results about the values of this function that this algorithm reveals.

Speaker: Martin Scharlemann (Santa Barbara)
Title:   Reconfiguring Qiu's proof of the Gordon Conjecture
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.  In this talk I hope to present some core ideas of his proof.

Speaker: Saul Schleimer (Warwick)
Title: The geometry of train track splitting sequences
The final step in the proof that the disk complex is Gromov
hyperbolic requires a delicate analysis of the geometry of train track
splitting sequences.  We will sketch a proof that a splitting sequence in
a surface S gives an unparametrized quasi-geodesic in the curve complex
C(Y), for any essential subsurface Y of S.  As a consequence, a
Masur-Minsky hierarchy and a splitting sequence have bounded Hausdorff
distance in every C(Y).  This is joint work with H. Masur and, in part,
with L. Mosher.

Speaker: Vladimir Markovic (Warwick)
Title: Random ideal triangulations and the Weil-Petersson distance between finite degree  covers of punctured Riemann surfaces
We show that for any two punctured Riemann surfaces $S$ and $R$ (of  finite  hyperbolic volumes) and every $\epsilon>0$ one can explicitely construct finite  degree unbranched covers $S_{\epsilon} \to S$  and $R_{\epsilon} \to R$ such  that the Weil-Petersson distance  between  $S_{\epsilon}$ and $R_{\epsilon}$ is less than $\epsilon$. This  is joint work with Jeremy Kahn.

Speaker: Joan Porti (Barcelona)
Title: Collapsing three-manifolds
We propose a proof of Perelman's collapsing theorem, used at the end of geometrization of three manifolds. Our proof does not use Alexandrov spaces,  but simplicial volume and Thurston's hyperbolization  for Haken manifolds. This is joint work with L. Bessières, G. Besson,  M. Boileau and S. Maillot.

Speaker: Jeff Brock (Brown)
Title: Bounded geometry and combinatorics for hyperbolic Heegaard splittings
Given a hyperbolic 3-manifold with a Heegaard splitting, one may consider geometric implications of the combinatorics of meridians for each side of the splitting in the curve complex of the Heegaard surface.  In his thesis, H. Namazi gave a sufficient condition for bounded geometry, i.e. a lower bound to the injectivity radius, in terms of Minsky's notion of bounded combinatorics.  In fact a hyperbolic 3-manifold with bounded geometry can admit splittings with somewhat more general combinatorial properties; in this talk I will discuss a more general necessary and sufficient combinatorial condition for bounded geometry in hyperbolic 3-manifolds with splittings of Heegaard distance at least 3.  This is joint work with Yair Minsky, Hossein Namazi, and Juan Souto.

Speaker: Moon Duchin (Michigan)
Title: Curves on flat surfaces
Quadratic differentials on surfaces induce metrics that are flat away from a finite number of cone points.  We investigate the length spectrum in this class of singular flat metrics, studying the extent to which a metric is determined by length data for curves.  This is joint work with Chris Leininger and Kasra Rafi.

Speaker: Juan Souto (Michigan)
Title: Some remarks about the mapping class group.
I will sketch the proof that geodesics length functions are convex with respect to suitably chosen Fenchel-Nielsen coordinates. This permits to use Kerckhoff's strategy to prove that finite subgroups of the  mapping class group lift to the group of diffeomorphism. Once this is  done, I will sketch a quite simple proof of the theorem of Morita asserting that the mapping class group itself does not lift; the proof  only uses Milnor's bound for the euler-number of flat circle bundles. Finally, I will describe how to construct injective homomorphisms between  mapping class groups which do not preserve the property of being pseudo-Anosov and have other curious properties. These results are joint work with combinations of J. Aramayona, M. Bestvina, K. Bromberg, T. Church, K. Fujiwara and C. Leininger.

Speaker: Richard Weidmann (Heriot-Watt)
Title: Maps onto hyperbolic groups
Sela gives a complete description of Hom(H,G) where H is an arbitrary finitely presented group and G is a torsion-free hyperbolic group. Despite this fact it is known that there is no uniform algorithm that decides whether Hom(H,G) contains an epimorphism; this is not even possible if H is a free group. In this talk we discuss how methods similar to the ones used to prove the various accessibility results can yield a complete description of all epimorphisms from H to G where G is a good hyperbolic group.


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