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Series: Geometry Topology Seminar

The fractional Dehn twist coefficient (FDTC), defined by Honda-Kazez-Matic, is an invariant of mapping classes. In this talk we study properties of FDTC by using open book foliation method, then obtain results in geometry and contact geometry of the open-book-manifold of a mapping class. This is joint work with Tetsuya Ito.

Series: Geometry Topology Seminar

By a classical result of Eliashberg, contact manifolds in dimension 3 come in two flavors: tight (rigid) and overtwisted (flexible). Characterized by the presence of an "overtwisted disk", the overtwisted contact structures form a class where isotopy and homotopy classifications are equivalent.In higher dimensions, a class of flexible contact structures is yet to be found. However, some attempts to generalize the notion of an overtwisted disk have been made. One such object is a "plastikstufe" introduced by Niederkruger following some ideas of Gromov. We show that under certain conditions, non-isotopic contact structures become isotopic after connect-summing with a contact sphere containing a plastikstufe. This is a small step towards finding flexibility in higher dimensions. (Joint with E. Murphy, K. Niederkruger, and A. Stipsicz.)

Series: Geometry Topology Seminar

The Heegaard Floer package provides a robust tool for studying contact 3-manifolds and their subspaces. Within the sphere of Heegaard Floer homology, several invariants of Legendrian and transverse knots have been defined. The first such invariant, constructed by Ozsvath, Szabo and Thurston, was defined combinatorially using grid diagrams. The second invariant was obtained by geometric means using open book decompositions by Lisca, Ozsvath, Stipsicz and Szabo. We show that these two previously defined invariant agree. Along the way, we define a third, equivalent Legendrian/transverse invariant which arises naturally when studying transverse knots which are braided with respect to an open book decomposition.

Series: Geometry Topology Seminar

It is a theorem of Bass, Lazard, and Serre, and, independently, Mennicke, that the special linear group SL(n,Z) enjoys the congruence subgroup property when n is at least 3. This property is most quickly described by saying that the profinite completion of the special linear group injects into the special linear group of the profinite completion of Z. There is a natural analog of this property for mapping class groups of surfaces. Namely, one may ask if the profinite completion of the mapping class group embeds in the outer automorphism group of the profinite completion of the surface group. M. Boggi has a program to establish this property for mapping class groups, which couches things in geometric terms, reducing the conjecture to determining the homotopy type of a certain space. I'll discuss what's known, and what's needed to continue his attack.

Series: Geometry Topology Seminar

Series: Geometry Topology Seminar

The hyperelliptic Torelli group SI(S) is the subgroup of the
mapping class group of a surface S consisting of elements which commute
with a fixed hyperelliptic involution and which act trivially on
homology. The group SI(S) appears in a variety of settings, for example
in the context of the period mapping on the Torelli space of a Riemann
surface and also as a kernel of the classical Burau representation of
the braid group. We will show that the cohomological dimension of SI(S)
is g-1; this result fits nicely into a pattern with other subgroups of
the mapping class group, particularly those of the Johnson filtration.
This is joint work with Leah Childers and Dan Margalit.

Series: Geometry Topology Seminar

There are two simple ways to construct new surface bundles over surfaces from old ones, namely, we can connect sum along the base or the fiber. In joint work with Inanc Baykur, we construct explicit surface bundles over surfaces that are indecomposable in both senses. This is achieved by first translating the problem into one about embeddings of surface groups into mapping class groups.

Series: Geometry Topology Seminar

I will talk briefly about how the study of fibred knots and Thurston's classification of automorphisms of surfaces in the 70's lead to Gabai and Oertel's work on essential laminations in the 80's. Some of this structure, for instance fractional Dehn twist coefficients, has implications in contact topology. I will describe results and examples, both old and new, that emphasize the special nature of S^3. This talk is based on joint work with Rachel Roberts.

Series: Geometry Topology Seminar

We prove the moduli space M_{g,n} of the surface of g genus with n punctures admits no complete, visible, nonpositively curved Riemannian metric, which will give a connection between conjectures from P.Eberlein and Brock-Farb. Motivated from this connection, we will prove that the translation length of a parabolic isometry of a proper visible CAT(0) space is zero. As an application of this zero property, we will give a detailed answer toP.Eberlein's conjecture.

Series: Geometry Topology Seminar

Let S_g be a closed orientable surface of genus g > 1 and C a simple closed nonseparating curve in S_g. Let t_C denote a left handed Dehn twist about C. A fractional power of t_C of exponent L/n is a h in Mod(S_g) such that h^n = t_C^L. Unlike a root of a t_C, a fractional power h can exchange the sides of C. We will derive necessary and sufficient conditions for the existence of both side-exchanging and side-preserving fractional powers. We will give some applications of the main result in both cases. Finally, we give a complete classification of a certain class of side-preserving and side-exchanging fractional powers on S_5.