Speaker | Time | Title |

Hossein Namazi, Yale | 10:30-11:30 am | Heegaard Splittings and Hyperbolic Geometry |

Mladen Bestvina, Utah | 1:30-2:30 pm | The Torelli subgroup of Out(F)_{n} |

Pete Storm, Stanford | 2:45-3:45 pm | Lower volume bounds for hyperbolic 3-manifolds with boundary |

Hyam Rubinstein, Melbourne | 4-5 pm | Separating incompressible surfaces are abundant in 3-manifolds. |

**Hossein Namazi:** *Heegaard Splittings and Hyperbolic
Geometry*

It is well known that every compact orientable 3-manifold admits a Heegaard splitting, but using the "combinatorics" of these splittings to describe the topology of the manifold has proved elusive. Hempel introduced a "handlebody distance" for Heegaard splittings and conjectured that when this distance is sufficiently large the manifold should be hyperbolic. Examples of 3-manifolds with large handlebody distance were constructed by Kobayashi and Luo. Using hyperbolic geometry, we prove that these and in fact a bigger class of 3-manifolds which satisfy a more general combinatorial property are "almost hyperbolic", i.e. they admit Riemannian metrics with sectional curvature close to -1. We will also mention some topological corollaries of this construction which were subject of a joint work with Juan Souto.

**Mladen Bestvina:** *The Torelli subgroup of Out(F _{n})*

The group in the title is the kernel of the natural map
Out(*F*_{n})-> GL(n, **Z**). As for the classical
mapping class group counterparts (except for the work of Mess in low
genus), the basic features such as the dimension and finiteness
properties are unknown. I will describe an approach to an
understanding to these groups that leads to a complete success when
*n*=3 and to at least a partial success when *n*=4. This
work is joint with Kai-Uwe Bux and Dan Margalit, and is still in
progress.

**Pete Storm: ***Lower volume bounds for hyperbolic
3-manifolds with boundary*

Nontrivial lower volume bounds can sometimes be obtained in terms of the decomposition of a hyperbolic 3-manifold with boundary into its geometric pieces. The main technique is the barycenter method, which is used like a replacement for harmonic maps. Perelman's work also raises the possibility of obtaining new volume information about closed Haken hyperbolic 3-manifolds via their infinite covers.

**Hyam Rubinstein:** *Separating incompressible surfaces are
abundant in 3-manifolds.*

This is joint work with Kazuhiro Ichihara and Makoto Ozawa. We give a construction showing that "most" closed orientable irreducible or compact orientable irreducible 3-manifolds with incompressible tori boundary have embedded closed separating non peripheral incompressible surfaces. The key idea is to first attached all the separating handles for a minimal genus Heegaard splitting. The resulting 3-manifold is shown to always have many non peripheral separating incompressible surfaces. These survive most ways of attaching the non separating handles. Moreover, the method gives an interesting test (but not an algorithm) to find such surfaces. For 3-manifolds of Heegaard genus 2, we discuss in detail how separating incompressible surfaces arise in most cases.