Workshop on pseudoAnosovs with small dilatation
24–25 April 2010
Organizers: Jordan Ellenberg and JeanLuc Thiffeault
From the topological viewpoint, the most interesting
transformations of surfaces are the socalled pseudoAnosovs,
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.
Confirmed speakers
Joan Birman (Columbia) 
Spencer Dowdall (Chicago) 
Nathan Dunfield (Illinois) 
JiYoung Ham (Seoul) 
Eriko Hironaka (Florida State) 
Thomas Koberda (Harvard) 
Erwan Lanneau (Marseille) stuck in Europe! (probably here Sat.) 
Chris Leininger (Illinois) 
Dan Margalit (Tufts) 
ChiaYen Tsai (Illinois) 
Participants
Kyle Armstrong (Florida State) 
Nigel Boston (Wisconsin) 
Phil Boyland (Florida) 
Michael Childers (Wisconsin) 
Hao Fang (Iowa) 
Benson Farb (Chicago) stuck in Europe! 
Vaibhav Gadre (Illinois) 
Keiko Kawamuro (Iowa) 
Eiko Kin (Tokyo Inst. of Tech.) 
Sarah Matz (Wisconsin) 
Mitsuhiko Takasawa (Tokyo Inst. of Tech.) 
Aaron Valdivia (Florida State) 
Registration
Please send email to JeanLuc Thiffeault to
register your attendance.
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) 2794881
or (608) 2574881, 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.
General info on
traveling to Madison.
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 pseudoAnosov maps
We will discuss the twin themes of this conference: hyperbolic
3manifolds 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. pseudoAnosovs with small dilatation. (Joint work with Keiko
Kawamuro and Peter Brinkmann.)

10:30 
break 
11:00 
JiYoung Ham (Seoul) 
The minimal dilatation of a genus two surface

12:00 
lunch 
13:30 
ChiaYen Tsai (Illinois) 
Asymptotics of least pseudoAnosov dilatations
Let $l_{g,n}$ be the logarithm of least pseudoAnosov
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
pseudoAnosov 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 pseudoAnosov 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 JeanLuc Thiffeault)
concerning the behavior of smallest dilatations of (orientable) pseudoAnosov
mapping classes considered as a function of genus.

15:30 
break 
16:00 
Thomas Koberda (Harvard) 
PseudoAnosov homeomorphisms and homology
I will discuss some of the methods and difficulties
involved in trying to understand pseudoAnosov 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 pseudoAnosovs and 3 manifolds I
This is the first of two talks on our theorem which says
that all small dilatation pseudoAnosovs are obtained as the
monodromies of a finite list of fibered 3manifolds (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 pseudoAnosovs. We will also describe several
corollaries. This is joint work with Benson Farb and Dan Margalit.

10:00 
Dan Margalit (Tufts) 
Small dilatation pseudoAnosovs and 3 manifolds II
This is the second of two talks on our theorem that all
small dilatation pseudoAnosov maps come from a finite list of
3manifolds. 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 3manifolds is well
ordered, for each fixed g there is a genusg surface bundle over the
circle of minimal volume. I will describe an explicit family of
genusg 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
NeumannZagier 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 pseudoAnosovs 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 selfintersections for pointpushing pseudoAnosov
homeomorphisms
This talk is about the dilatations of pseudoAnosovs obtained by
pushing a marked point around a filling curve. After reviewing this
"pointpushing" construction, I will give both upper and lower bounds
on the dilatation in terms of the selfintersection number of the
filling curve. In addition, we will discuss explicit examples and
bound the least dilatation of any pseudoAnosov in the pointpushing
subgroup.

15:00 
Erwan Lanneau (Marseille) 
Dilatations of pseudoAnosov homeomorphisms and RauzyVeech induction
In this talk I will explain the link between pseudoAnosov
homeomorphisms and RauzyVeech 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
RauzyVeechYoccoz induction to give lower bound on dilatations.

16:00 
discussion, beer, tearful goodbyes 
Bibliography



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Preprint.


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