The GT seminar meets on Fridays
in Math. 507,
at 1:15PM.

Organizer:
Walter Neumann.

Other
area seminars. Our e-mail
list. Archive of previous semesters

## Fall 2013

## Abstracts.

**September 27**:
Spencer Dowdall
“Average distances in the curve complex and in Teichmuller space”
**Abstract**:This talk will investigate the concept of average distance both in the curve complex and in
Teichmuller space. More precisely, I will aim to address the question "What is the average distance
between a pair of points in the sphere of radius R?" in both settings and to explain why the answer
reflects a statistical sort of hyperbolicity. The bulk of the talk will be devoted to the curve complex
setting, but time permitting I will also discuss related results for Teichmuller space. This is joint
work with Moon Duchin and Howard Masur.

**October 4**:
Timothy Susse
“Stable commutator length in cyclic amalgams”
**Abstract**:
We show that stable commutator length is rational
on free products of free abelian groups amalgamated over Z^k, a
class of groups containing the fundamental groups of all torus knot
complements. We consider a geometric model for these groups
and parameterize all surfaces with specified boundary mapping to
this space. Using this work we provide a topological algorithm to
compute stable commutator length in these groups.

**October 11**:
Tetsuya Ito
“Overtwisted discs in planer open books”
**Abstract**:
For a planar open book, by using open book foliation technique,
which is a generalization of Birman-Menasco's braid foliation machinery,
we show that one can put an overtwisted disc in a (topologically) nice
position. Such an overtwisted disc tells us how the monodromy of an open
book twists curves. As a corollary, we give a topologicial argument for
certain planar open books to support a tight contact structure.

**October 18**:
Gaven Martin
“The solution to Siegel's Problem on small covolume lattices”
**Abstract**:We outline the history and the proof of the identification of
the minimal covolume lattice of hyperbolic 3-space as the 3-5-3 Coxeter
group extended by the involution preserving the symmetry of this diagram.
This solves (in three dimensions) the problem posed by Siegel in 1945
(Siegel solved this problem in two dimensions by deriving the signature
formula identifying the (2,3,7)-triangle group as having minimal co-area).
There are strong connections with arithmetic hyperbolic geometry in the
proof and the result has applications in the maximal symmetry groups of
hyperbolic 3-manifolds (in much the same way that Hurwitz 84g-84 theorem
and Siegel's result do).

**November 8, 9:30am**:
Ben Burton
“Exploring parameterised complexity in computational topology”
**Abstract**:
Topological decision problems in three dimensions are notoriously difficult:
the mere existence of an algorithm is often a major result, and many
important algorithms have yet to be studied from a complexity viewpoint.
Even "simple" problems, such as unknot recognition (testing whether a loop
of string is knotted), or 3-sphere recognition (testing whether a
triangulated 3-manifold is topologically trivial), have best-known
algorithms that are worst-case exponential time. In practice, however, some
of these algorithms run surprisingly well in experimental settings, and we
discuss some reasons why parameterised complexity now looks to be the
"right" tool to explain this behaviour. We outline some initial forays into
the parameterised complexity of topological problems, including both
tractability and hardness results.

**November 8, 1:15pm**:
Andrew Putman
“Vanishing and nonvanishing in the high-dimensional cohomology of
SL_n(O_k)”
**Abstract**:
For O_k a ring of integers and p close to the vcd of SL_n(O_k), I will
discuss a sequence of results and conjectures about H^p(SL_n(O_k);Q). This
is joint work with Tom Church and Benson Farb.

**November 15, 9:30am**:
Saul Schleimer
“Compressed words in Gromov hyperbolic groups”
**Abstract**:
Markus Lohrey defined the compressed word problem for
groups and gave a polynomial-time solution in the case of free
groups.
I'll explain how this leads to a polynomial-time algorithm to the
usual word problem for automorphism groups of free groups. Time
permitting, I will sketch a few of the ideas needed to go from the
free group to Gromov hyperbolic groups in general. As a nice
application, this gives a polynomial-time solution to the word
problem
for the mapping class group; this new solution is very different from
the usual one!

**November 15, 1:15pm**:
Tara Brendle
“Congruence subgroups of braid groups”
**Abstract**:
The integral Burau representation gives a symplectic representation of the braid group. In this talk we will discuss the resulting congruence subgroups of braid groups, that is, preimages of the principal congruence subgroups of the symplectic group. In particular, we will show that the level 4 congruence braid group is generated by squares of Dehn twists. One key tool is a "squared lantern relation" amongst Dehn twists. This is joint work with Dan Margalit.

**November 22**:
John Parker
“Constructing non-arithmetic lattices”
**Abstract**:I will survey the relationship between arithmetic groups and lattices
in semi-simple Lie groups; the construction of non-arithmetic lattices
in SU(n,1) for n=2, 3 by Mostow and Deligne in the 1980s. I will then
describe a joint project with Martin Deraux and Julien Paupert to
construct new examples in SU(2,1).

**December 6**:
Ingrid Irmer
“A family of curve complexes and Chillingworth's winding numbers”
**Abstract**:
Curve complexes have traditionally been used to study the mapping class
group of a surface. For the Torelli group, i.e. the subgroup of the mapping class
group that acts trivially on homology, standard techniques do not apply. In this
talk it will be shown that a family of oriented curve complexes give a "linear
approximation" to the Torelli group in the following way: the stable lengths
of an element of the Torelli group acting on the family of curve complexes
define a cohomology class on the surface, proportional to the Chillingworth
class. The Chillingworth class is the dual of a tensor contraction of the Johnson
homomorphism.

**December 13**:
Alessandro Sisto
“Relative hyperbolicity of (random) right-angled Coxeter groups”
**Abstract**:
Right-angled Coxeter groups (RACGs) form a rich class of groups inspired
by groups generated by reflections across orthogonal axes, while
relative hyperbolicity can be defined in terms of the existence of an
"interesting" proper action on a Gromov-hyperbolic space.
I will discuss the fact that relative hyperbolicity of a given RACG can
be detected algorithmically reasonably quickly.
Also, I will present a randomized model for RACGs that, depending on a
certain parameter, generates either relatively hyperbolic or
non-relatively hyperbolic RACGs with asymptotic probability 1.
Joint with Jason Behrstock and Mark Hagen.

# Other relevant information.

## Previous semesters:

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.