Columbia Geometric Topology Seminar

Fall 2014

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

Date Speaker Title
September 5 Organizational meeting 1:15pm Room 520
September 12 No seminar Eilenberg Lecture
September 19 Abhijit Champanerkar Geometrically and diagrammatically maximal knots
September 26 BoGwang Jeon Hyperbolic three-manifolds of bounded volume and trace field degree, III
October 3 Samuel Taylor Hyperbolic extensions of free groups
October 10 Mark Hagen Cubulating hyperbolic free-by-Z groups
October 17 Jane Gilman The hyperbolic geometry of skew convex hexagons and PSL(2,C) discreteness sequences
October 24 No Seminar MSP board meeting
October 31 Moira Chas Computer Driven Theorems and Questions in Geometry
November 7 Anne Pichon Lipschitz geometry of minimal singularities
November 14 Ruth Charney Morse Boundaries
November 21 Neil Fullarton Palindromic automorphisms of free groups
November 28 No Seminar Thanksgiving holiday
December 5 David Shea Vela-Vick A refinement of the Ozsvath-Szabo contact invariant
December 12 Matthew Durham Convex cocompactness and stability in mapping class groups

Abstracts.

Abhijit Champanerkar: September 19, “Geometrically and diagrammatically maximal knots”
Abstract: The ratio of volume to crossing number of a hyperbolic knot is known to be bounded above by the volume of a regular ideal octahedron, and a similar bound is conjectured for the knot determinant per crossing. In this paper, we investigate a natural question motivated by these bounds: For which knots are these ratios nearly maximal? We show that many families of alternating knots and links simultaneously maximize both ratios. One such family is weaving knots, which are alternating knots with the same projection as a torus knot, and which were conjectured by Lin to be among the maximum volume knots for fixed crossing number. For weaving knots, we provide the first asymptotically correct volume bounds. This is joint work with Ilya Kofman and Jessica Purcell.

BoGwang Jeon: September 26, “Hyperbolic three-manifolds of bounded volume and trace field degree, III”
Abstract: In the previous talks, I proposed a way to attack the conjecture that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree. In this talk, I present the key ideas of my recent proof of this conjecture.

Samuel Taylor: October 03, “Hyperbolic extensions of free groups”
Abstract: Every subgroup $G$ of the outer automorphism group of a finite-rank free group $F$ naturally determines a free group extension $1\to F \to E_G \to G\to 1$. In this talk, I will discuss geometric conditions on the subgroup $G$ that imply its corresponding extension $E_G$ is hyperbolic. These conditions are in terms of the action of G on the free factor complex of $F$ and allow one to easily build new examples of hyperbolic free group extensions. This is joint work with Spencer Dowdall.

Mark Hagen: October 10, “ Cubulating hyperbolic free-by-Z groups”
Abstract: I'll discuss what it means to "cubulate" a group, explain one of several related strategies for doing this, and give some indication of why cubulating a group is a useful step in understanding it.Â  I'll then discuss recent joint work with Dani Wise, in which we prove that every word-hyperbolic free-by-Z group acts geometrically on a CAT(0) cube complex.

Jane Gilman: October 17, “The hyperbolic geometry of skew convex hexagons and PSL(2,C) discreteness sequences”
Abstract: (Click for pdf). We present a new approach to the PSL(2,C) discreteness problem. A subgroup, G, of PSL(2,C) (equivalently Isom(H^3)) is not discrete if there exists an infinite sequence of distinct elements of the group that converges to the identity. To date there are only ad hoc techniques for finding such a sequence of primitive elements in any given G. When G is the image of a non-elementary representation of a rank two free group, it may or may not be discrete or free, but the representation determines a set of so called core points in H^3 and a new ordering of the rational numbers, the representation ordering. We use the hyperbolic geometry of H^3 as applied to palindromes in G and the representation ordering of the rationals to construct a unique sequence of primitive elements corresponding to a given representation. Theorem: (i) The sequence of core points is finite if and only if the group is discrete and free (ii) if the sequence is infinite and converges to a point on the boundary of H^3 , the group is either not geometrically finite or not discrete and (iii) if the sequence is infinite and converges to an interior point of H^3 , the group is not discrete. The proof involves an extension of Fenchel's theory of right angled hexagons in H^3 to skew-convex hexagons. This is joint work with Linda Keen.

Moira Chas: October 31, “Computer Driven Theorems and Questions in Geometry”
Abstract: Consider an orientable surface S with negative Euler characteristic, a minimal set of generators of the fundamental group of S, and a hyperbolic metric on S. Then each unbased homotopy class C of closed oriented curves on S determines three numbers: the word length (that is, the minimal number of letters needed to express C as a cyclic word in the generators and their inverses), the minimal geometric self-¿½intersection number, and finally the geometric length. On the other hand, the set of free homotopy classes of closed directed curves on S (as a set) is the vector space basis of a Lie algebra discovered by Goldman. This Lie algebra is closely related to the intersection structure of curves on S. These three numbers, as well as the Goldman Lie bracket of two classes, can be explicitly computed (or approximated) using a computer. These computations led us to counterexamples to existing conjectures, to formulate new conjectures and (sometimes) to subsequent theorems.

Anne Pichon: November 07, “Lipschitz geometry of minimal singularities”
Abstract: (Click for pdf). It is a classical fact that the topology of a germ of a complex variety (X, 0) \subset (C^n,0) is locally homeomorphic to the cone over its link X^(\epsilon) = S^{2n-1}\cap X, where S^{2n-1} denotes a sphere of radius \epsilon centered at the origin in C^n. Much richer classifications are obtained by taking into account the metric properties of (X, 0). Any germ of complex analytic space is equipped with two natural metrics: the outer metric induced by the hermitian metric of the ambient space and the inner metric, which is the associated Riemannian metric on the germ. These two metrics are in general nonequivalent up to bilipschitz homeomorphism. In fact, if (X, 0) is an irreducible germ of curve, its two metrics are bilipschitz equivalent if and only if (X, 0) is smooth. I will present a recent joint work with Walter Neumann and Helge Moller Pedersen in which we show that it doesn't remain true in higher dimension: any minimal surface singularity has its two metrics bilipschitz equivalent.

Ruth Charney: November 14, “Morse Boundaries”
Abstract: In joint work with H. Sultan, we defined a "contracting boundary" for a CAT(0) space consisting of equivalence classes of contracting rays and proved that the contracting boundary is a quasi-isometry invariant. In a CAT(0) space, the contracting property for a ray is equivalent to the Morse property (quasi-geodesics with endpoints on the ray stay bounded distance from the ray) and this fact is a key ingredient in the proof. More generally, using the Morse property instead of the contracting property, one can define a quasi-isometry-invariant Morse boundary for any proper geodesic metric space. I will discuss recent work of M. Cordes on these generalized Morse boundaries.

Neil Fullarton: November 21 “Palindromic automorphisms of free groups”
Abstract: The palindromic automorphism group of a free group is the group of automorphisms that take each member of some fixed free basis to a word that reads the same backwards as forwards. This group is an obvious free group analogue of the hyperelliptic mapping class group of an oriented surface. I will discuss some elementary properties of palindromes and palindromic automorphisms, and introduce a newï¿½complexï¿½on which the palindromic automorphism group acts. In particular, we will discuss how the action on thisï¿½complexï¿½can be used to find a generating set for the so-called palindromic Torelli group. I will also discuss recent joint work with Anne Thomas on generalisations of these results to the right-angled Artin group setting.

David Shea Vela-Vick: December 05, “A refinement of the Ozsvath-Szabo contact invariant”
Abstract: We introduce a refinement of Ozsvath and Szabos contact invariant in Heegaard Floer homology. It assigns to a closed 3-manifold, an element b" in the positive integers union infinity. By construction, b is infinite precisely when the usual contact invariant is nonzero, and b = 1 when the contact structure under consideration is overtwisted. We further show that if (Y,\xi) is a contact structure supported by an open planar book with fractional Dehn twist coefficients all greater than two, then \xi is tight, reproving a result originally due to Ito and Kawamuro. This is going work with John Baldwin.

Matthew Durham: December 12, “Convex cocompactness and stability in mapping class groups”
Abstract: Originally defined by Farb-Mosher to study hyperbolic ¿½extensions of surface subgroups, conv xï¿½cocompact subgroups of mapping class groups have deep ties to the geometry of Teichmuller space and the curve complex.¿½ In joint work with Sam Taylor, we define a strong notion of quasiconvexity called stability and prove it coincides with convex cocompactness in mapping class groups.¿½ Stability characterizes convex cocompactness solely in terms of the intrinsic geometry of the ¿½mapping class group and defines a new class of subgroups of finitely generated groups. Time permitting, I will also discuss ¿½some work in progress and a few motivating open questions.

Other relevant information.

Previous semesters:

Spring 2014, Fall 2013, Spring 2013, Fall 2012, Spring 2012, Fall 2011, 2010/11, Spring 2010, Fall 2009, Spring 2009, Fall 2008, Spring 2008, Fall 2007, Spring 2007, Fall 2006.

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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.