## The Redbud Topology Conference

### Rank vs. Genus

March 3-4, 2012, Oklahoma State University

This conference is focussed around Tao Li's recent examples of hyperbolic 3-manifolds whose Heegaard genera are strictly greater than the rank of their fundamental groups.

Confirmed Speakers:
Marion Campisi, University of Texas
Alex Coward, University of California at Davis
Nathan Dunfield, University of Illinois, Urbana-Champaign
Richard Weidmann, Christian-Albrechts University
Tao Li, Boston College
Juan Souto, University of British Columbia
Maggy Tomova, University of Iowa

This conference is funded by a grant from the NSF. Travel support will be available for students and early career mathematicians. Women and members of traditionally underrepresented groups are particularly encouraged to apply for funding.

Talks will be held in room 514 of the MSCS building, which is number 111 on the campus map. Refreshments will be in the department lounge, room 423.

## Schedule:

 Time Speaker Title Saturday 8:30-9:00 Refreshments 9-9:50 Tao Li Rank and genus of 3-manifolds 9:50-10:30 Refreshments 10:30-11:20 Maggy Tomova Flipping Bridge Surfaces 11:30-12:20 Nathan Dunfield Integer homology 3-spheres with large injectivity radius 12:20-2:30 Lunch 2:30-3:20 Juan Souto Hyperbolic 3-manifolds with fixed rank 3:20-4:00 Refreshments 4:00-5:00 Alex Coward Topological and physical link theory are distinct Sunday 8:30-9:00 Refreshments 9:00-9:50 Tao Li Continuation of Saturday talk 9:50-10:30 Refreshments 10:30-11:20 Marion Campisi Bridge distance and exceptional surgeries 11:40-12:30 Richard Weidmann Distinguishing Nielsen classes

## Abstracts:

Tao Li - Rank and genus of 3-manifolds

We construct a counterexample to the Rank versus Genus Conjecture, i.e. a closed orientable hyperbolic 3-manifold with rank of its fundamental group smaller than its Heegaard genus. Moreover, we show that the discrepancy between rank and Heegaard genus can be arbitrarily large for hyperbolic 3-manifolds. We also construct toroidal such examples containing hyperbolic JSJ pieces.

Maggy Tomova - Flipping Bridge Surfaces

Recently several teams of mathematicians were able to construct a manifold with two different Heegaard splittings so that their common stabilization had high genus. In this talk we will discuss the natural generalization of this question to bridge surfaces.

Nathan Dunfield - Integer homology 3-spheres with large injectivity radius

Conjecturally, the amount of torsion in the first homology group of a hyperbolic 3-manifold must grow rapidly in any exhaustive tower of covers (see Bergeron-Venkatesh and F. Calegari-Venkatesh). In contrast, the first betti number can stay constant (and zero) in such covers. Here "exhaustive" means that the injectivity radius of the covers goes to infinity. In this talk, I will explain how to construct hyperbolic 3-manifolds with trivial first homology where the injectivity radius is big almost everywhere by using ideas from Kleinian groups. I will then relate this to the recent work of Abert, Bergeron, Biringer, et. al. In particular, these examples show a differing approximation behavior for L^2 torsion as compared to L^2 betti numbers. This is joint work with Jeff Brock.

Juan Souto - Hyperbolic 3-manifolds with fixed rank

In this talk I will describe a result describing the structure of those hyperbolic 3-manifolds whose fundamental group can be generated by $k$ elements and which also have injectivity radius at least $\epsilon$ for some positive but otherwise arbitrary $\epsilon$. The basic result asserts that such manifolds are obtained by glueing at most $c(k,\epsilon)$ pieces taken from a finite menagerie. In particular, their Heegaard genus is bounded from above by some constant $g(k,\epsilon)$. This is joint work with Ian Biringer.

Alexander Coward - Topological and physical link theory are distinct

Physical knots and links are one-dimensional submanifolds of R^3 with fixed length and thickness. In this talk, which exposes joint work with Joel Hass, we show that isotopy classes in this category can differ from those of classical knot and link theory. In particular, we produce a Gordian Split Link, a two component link that is a split link in classical knot theory but cannot be split through a physical isotopy.

Marion Campisi - Bridge distance and exceptional surgeries

In this talk I will discuss ongoing work which shows that links that have high distance bridge surfaces do not admit cosmetic surgeries. The distance d(T) of a bridge surface T is defined in terms of the arc and curve complex for the bridge surface and has become a standard way of measuring the "complexity'' of the link. This is joint work with Ryan Blair, Jesse Johnson, Scott Taylor and Maggy Tomova.

Richard Weidmann - Distinguishing Nielsen classes

In this talk we discuss the problem of distinguishing Nielsen equivalence classes of tuples of elements in a group. We will show in particular that knot groups of hyperbolic 2-bridge knots have infinitely many Nielsen classes of generating pairs. (partly joint work with Michael Heusener)

## Travel and Lodging:

If you're flying to the conference, you should fly in and out of Oklahoma City Airport (OKC). We will arrange shuttles between the airport and campus. You can also fly through the Tulsa Airport (TUL) but you may have to arrange your own transportation to Stillwater. To add your name to the shuttle information mailing list, please register for the conference by February 4th.

Plan on arriving in late afternoon/early evening Friday, March 2nd and leaving either Sunday evening (March 4th) or Monday morning. The last talk on Sunday will end between noon and 1pm. Leaving after 4pm on Sunday is ideal, but we will arrange shuttles for earlier departure times if necessary.

We have a block of rooms at the Atherton hotel on campus. The special rate of \$98 a night (plus taxes, etc.) will be available until February 3rd. You can reserve a room by calling (405)744-6835 and saying that you are with the topology conference or with the math department. (Note: Participants receiving travel funds from the conference do not need to make their own reservations.)

If you have further questions, please send an e-mail to jjohnson@math.okstate.edu.