# 2017 Lafayette-Lehigh Geometry-Topology Seminar

The Lafayette-Lehigh Geometry-Topology Seminar will be held Saturday March 25, 2017 at Lafayette College. The Mathematics Department is located in Pardee Hall (building # 37 on the Campus Map). The Directions on the Lafayette Website direct you to the Visitor Parking Deck behind Markle Hall. You may be able to park near Pardee Hall—there are spots adjacent to the building, between Pardee and Watson Courts. Please refer to the the Campus Map.

The talks will be held in Pardee Hall 217. We will gather for coffee and bagels, etc., around 9:15 AM, across the hall in Pardee 218 (the Saalfrank Common Room). We will also have a light lunch there between the morning and afternoon talks.

** Schedule **

9:45-10:45: ** Jason Manning ** (Cornell)

** Boundaries of hyperbolic and relatively hyperbolic groups **

* Abstract *: Hyperbolic and relatively hyperbolic groups admit proper actions on naturally associated Gromov hyperbolic spaces. Such a space has a compactification, which is called the Gromov or Bowditch boundary of the group, depending on whether it is hyperbolic or relatively hyperbolic. The limit set of a geometrically finite Kleinian group is an example of such a compactification. I’ll survey some results relating cut point structure of the boundary with splittings of the group, and talk about how the boundary can change (or not) after a group theoretic Dehn filling. This work is joint with Daniel Groves and Alessandro Sisto.

11:00-12:00: ** Thomas Church ** (Stanford/IAS)

** Stability and vanishing in the unstable cohomology of SL _{n}(Z) **

Abstract: Borel proved that in low dimensions, the cohomology of a locally symmetric space can be represented not just by harmonic forms but by invariant forms. This implies that the

*k*-th rational cohomology of SL

_{n}(

**Z**) is independent of n in a linear range

*n*≥

*ck*, and tells us exactly what this “stable cohomology” is. In contrast, very little is known about the

*unstable*cohomology, in higher dimensions outside this range.

In this talk I will explain a conjecture on a new kind of stability in the unstable cohomology of arithmetic groups like SL_{n}(**Z**). These conjectures deal with the “codimension-*k*” cohomology near the TOP dimension (the virtual cohomological dimension), and for SL_{n}(**Z**) they imply the cohomology *vanishes* there. Although the full conjecture is still open, I will explain how we proved it for the codimension-0 and codimension-1 cohomology. The key ingredient is a version of Poincare duality for these groups based on the algebra of modular symbols. Joint work with Benson Farb and Andrew Putman.

12:15-1:15: ** Lunch. **

1:15-2:15: ** Emily Dryden ** (Bucknell)

** Recovering geometric and topological information from oscillation frequencies **

* Abstract *: The Steklov problem models the vibrations of a free membrane that has all its mass concentrated along the boundary. The frequencies of oscillations encode certain information about the geometry and topology of the membrane, but not everything! We’ll explore this idea in the two-dimensional setting, allowing the boundaries of our surfaces to have mild singularities. Some simple computations will lead to surprising results.

This is based on joint work with Teresa Arias-Marco, Carolyn S. Gordon, Asma Hassannezhad, Allie Ray, and Elizabeth Stanhope.

2:30-3:30: **Renato Bettiol ** (University of Pennsylvania)

** Deforming flat manifolds and flat orbifolds **

* Abstract *: Compact flat orbifolds (respectively, manifolds) are quotients of Euclidean space by crystallographic groups (respectively, torsion-free crystallographic groups). In this talk, I will recall the construction of the moduli space of flat metrics on them to show that flat manifolds can always be deformed, while flat orbifolds may be rigid. I will also describe the moduli space boundary; establishing that the Gromov-Hausdorff limit of flat manifolds is a flat orbifold and, conversely, every flat orbifold is the Gromov-Hausdorff limit of flat manifolds. In particular, out of the 17 different flat 2-dimensional orbifolds (corresponding to the 17 wallpaper groups), 10 can be obtained as limits of flat 3-dimensional manifolds. This is joint work with Andrzej Derdzinski and Paolo Piccione.