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

## 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.## Other area seminars.

- Columbia Symplectic Geometry/Gauge Theory Seminar
- All Columbia Math Dept Seminars
- CUNY Geometry and Topology Seminar
- CUNY Complex Analysis & Dynamics Seminar
- CUNY Magnus Seminar
- Princeton Topology Seminar.

## 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.