Jean-Luc Thiffeault

From the topological viewpoint, the most interesting transformations of surfaces are the so-called pseudo-Anosovs, one of the three types arising from the Thurston–Nielsen clasification theorem. These stabilize a pair of transverse foliations, but they change the measure on these foliations by a positive real factor l, called the dilatation or expansion constant. For a given surface, it is known that dilatations are algebraic units, and that there exists a minimum value. This minimum is related to the shortest geodesic of Teichmüller flow, so it is an important number from many points of view: topological, dynamical, algebraic, and geometrical. Until recently, only one such nontrivial minimum was known, for a closed surface of genus 2 (Zhirov, 1995). However, in the past few years there has been a flurry of activity as new tools are developed. Moreover, the known smallest dilatations are Salem numbers, familiar to number theorists, which suggests intriguing connections.

This workshop aims to bring together several researchers interested in this problem for two days of talks and discussions.

Registration

Accomodations and Travel

We've reserved a block of rooms at the University Inn, near campus. The rooms are held for arrival on April 23, departure on April 26. However, you must confirm your arrival and departure date directly with the hotel. You can do so by calling (800) 279-4881 or (608) 257-4881, and identify yourself as part of the "Math Department Spring Workshop Block." This must be done before March 23, 2010. We will be covering accommodations for speakers, but you will need to provide a credit card number to hold the room and for incidental expenses.

Getting to the Math Building

The Mathematics Department is in Van Vleck Hall. You can easily walk from University Inn (A) to Van Vleck (B), though be aware that it's uphill all the way. (Ignore Google maps' detour around Bascom Hall at the end of the route: you can easily go left of the Hall.)

The talks will be in room B231, on the B2 level of Van Vleck. This means you have to go down two floors from the entrance.

Schedule

Friday

Saturday

9:00	welcome and introduction
9:30	Joan Birman (Columbia)	Characteristic polynomials of pseudo-Anosov maps We will discuss the twin themes of this conference: hyperbolic 3-manifolds of small "complexity", and pA maps of small dilatation. In the process of trying to understand whether they are related, in one explicit infinite sequence of examples, we were lead to study the factorization of the characteristic polynomial of a pA map. We will explain the relevance of that matter to the title of this workshop, i.e. pseudo-Anosovs with small dilatation. (Joint work with Keiko Kawamuro and Peter Brinkmann.)
10:30	break
11:00	Ji-Young Ham (Seoul)	The minimal dilatation of a genus two surface
12:00	lunch
13:30	Chia-Yen Tsai (Illinois)	Asymptotics of least pseudo-Anosov dilatations Let $l_{g,n}$ be the logarithm of least pseudo-Anosov dilatations. We will prove in detail that for fixed genus >2, $l_{g,n}$ converges to zero like $\log \|\chi(S)\|/\|\chi(S)\|$. If time is allowed, we will describe a general method of constructing pseudo-Anosov mapping classes with small dilatations along some $(g,n)$-rays such that $l_{g,n}$ converges to zero like $1/\|\chi(S)\|$ along these rays.
14:30	Eriko Hironaka (Florida State)	Families of small dilatation mapping classes We will talk about a general method for constructing families of pseudo-Anosov mapping classes with easily computed dilatations. This method uses Thurston's theory of fibered faces and McMullen's Teichmüller polynomials. Our main result supports a conjecture (suggested by work of Erwan Lanneau and Jean-Luc Thiffeault) concerning the behavior of smallest dilatations of (orientable) pseudo-Anosov mapping classes considered as a function of genus.
15:30	break
16:00	Thomas Koberda (Harvard)	Pseudo-Anosov homeomorphisms and homology I will discuss some of the methods and difficulties involved in trying to understand pseudo-Anosov homeomorphisms of a surface and their dilatations by looking at tractable subgroups and quotients of the fundamental group of the surface.
17:00	discussion and beer
18:30	dinner at Fugu

Sunday

9:00	Chris Leininger (Illinois)	Small dilatation pseudo-Anosovs and 3 manifolds I This is the first of two talks on our theorem which says that all small dilatation pseudo-Anosovs are obtained as the monodromies of a finite list of fibered 3-manifolds (up to removing singularities). In this talk, we will state the main theorem and explain in detail it's prototype, which is McMullen's construction of small dilatation pseudo-Anosovs. We will also describe several corollaries. This is joint work with Benson Farb and Dan Margalit.
10:00	Dan Margalit (Tufts)	Small dilatation pseudo-Anosovs and 3 manifolds II This is the second of two talks on our theorem that all small dilatation pseudo-Anosov maps come from a finite list of 3-manifolds. In this talk, we give the idea of the proof. This is joint work with Benson Farb and Chris Leininger.
11:00	break
11:30	Nathan Dunfield (Illinois)	Hyperbolic surfaces bundles of least volume Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. I will describe an explicit family of genus-g bundles which we conjecture are the unique such manifolds of minimal volume. Conditional on a very plausible assumption, I will show prove that this is indeed the case when g is large. The proof combines a soft geometric limit argument with a detailed Neumann-Zagier asymptotic formula for the volumes of Dehn fillings. The examples are all Dehn fillings on the sibling of the Whitehead manifold, and one can also analyze the dilatations of all closed surface bundles obtained in this way, identifying those with minimal dilatation. This gives new families of pseudo-Anosovs with low dilatation, including a genus 7 example which minimizes dilatation among all those with orientable invariant foliations. (Joint work with John W. Aaber.)
12:30	lunch
14:00	Spencer Dowdall (Chicago)	Dilatations and self-intersections for point-pushing pseudo-Anosov homeomorphisms This talk is about the dilatations of pseudo-Anosovs obtained by pushing a marked point around a filling curve. After reviewing this "point-pushing" construction, I will give both upper and lower bounds on the dilatation in terms of the self-intersection number of the filling curve. In addition, we will discuss explicit examples and bound the least dilatation of any pseudo-Anosov in the point-pushing subgroup.
15:00	Erwan Lanneau (Marseille)	Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction In this talk I will explain the link between pseudo-Anosov homeomorphisms and Rauzy-Veech induction. We will see how to derive properties on the dilatations of these homeomorphisms (I will recall the definitions) and as an application, we will use the Rauzy-Veech-Yoccoz induction to give lower bound on dilatations.
16:00	discussion, beer, tearful goodbyes

Workshop on pseudo-Anosovs with small dilatation

Confirmed speakers

Participants