Back to WTC Summer 2007

Matt Bainbridge
Billiards in L-shaped tables with barriers
I'll discuss the computation of the volumes of some moduli spaces of holomorphic 1-forms on Riemann surfaces, and I'll give applications to counting closed billiards trajectories in certain L-shaped polygons with barriers.

Nathan Broaddus
Irreducible Sp-representations and distortion in the mapping class group
In this report on joint work with B. Farb and A. Putman I will show that the Torelli subgroup of the mapping class group is at least exponentially distorted, answering a question of Hamenst\"adt.  I will then outline a generalization (for the surface with one boundary component) that shows that nontrivial finitely generated normal subgroups of the mapping class group contained in the Torelli group are at least exponentially distorted.  To prove this kind of general result on normal subgroups, without actually knowing what these groups might be, we must rely on constraints from symplectic representation theory. A preprint of these and other distortion results is available at arXiv:0707.2262.

Kai-Uwe Bux
Finitenes properties of arithmetic groups and some relatives (with Amir Mohammadi and Kevin Wortman)
We showed that the group SL_n( Z[t] ) is not of type FP_{n-1}. The proof uses the action on an associated Euclidean building. Although SL_n( Z[t] ) is not an arithmetic group, the method of proof draws from methods used to study SL_n( F_q[t] ). I would like to give an overview of the
geometric ideas used to study finiteness properties of artihmetic groups and variations like SL_n( Z[t] ).

Jim Conant
The topology of sets of Boolean formulae
The k-SAT problem in computer science asks whether a given formula in a specified canonical form (a conjunction of disjunctions of k literals) can be satisfied by some assignment of truth values to the variables. It is well known that the 2-SAT problem can be solved in polynomial time, whereas the 3-SAT problem is NP-complete. The set of satisfiable formulae in k-SAT is a poset under logical implication, and in this talk we will study the topology of this poset. The goal of the project is to find topological differences between 2-SAT and 3-SAT with an eye toward separating the classes of P and NP problems. At this stage, the general topological behavior of these posets remains mysterious.

W. Patrick Hooper
An infinite surface with the lattice property and applications
We will describe a construction which yields a translation surface, infinite in both genus and area, which has the lattice property. The surface arises as a limit of some of Veech's original lattice surfaces, the regular 2n-gons with opposite sides identified.

A similar construction yields this infinite surface as a limit of translation surfaces corresponding to billiards in some irrational polygons. We will utilize this surface to study periodic billiard paths in certain irrational polygons. For instance, we provide lower bounds for the growth rate of periodic billiard trajectories in some irrational polygons.

Dan Margalit
Cohomological dimension of the Torelli group
We prove that the cohomological dimension of the Torelli group for a surface of genus g is 3g-5, and the cohomological dimension of the Johnson kernel is 2g-3.  We also show that the (3g-5)-th integral homology of the Torelli group is infinitely generated.  Finally, we give a topological proof of the theorem of Mess which gives a precise description of the Torelli group in genus 2.  The main tool is a new contractible complex on which the Torelli group acts.

Jessica Purcell
Volume bounds from knot and link diagrams
Given the diagram of a hyperbolic knot or link, one ought to be able to determine geometric information on the link complement, such as its volume.  However, it seems to be difficult to read geometry off a diagram. We recently proved a theorem giving bounds on the change of volume of a hyperbolic 3-manifold under Dehn filling. We will discuss how this result applies to give volume bounds on large classes knots and links. This work is joint with David Futer and Effie Kalfagianni.

Anne Thomas
Lattices for Fuchsian buildings and Platonic complexes
Let L be a graph.  A (k,L)-complex is a polygonal complex X such that each 2-cell is a regular k-gon, and the link of each vertex is the graph L. Examples include Euclidean buildings, products of trees and Fuchsian (hyperbolic) buildings.  A (k,L)-complex X is Platonic if Aut(X) acts transitively on the set of adjacent triples (vertex, edge, face) in X. An example is when L is the Petersen graph.  We study lattices in Aut(X), for X a Fuchsian building or a Platonic complex.  We give new constructions of uniform and nonuniform lattices, characterize the set of covolumes of lattices, and consider towers (infinite ascending sequences) of lattices.