Jean-Luc Thiffeault's Homepage

Workshop on pseudo-Anosovs with small dilatation

24–25 April 2010

Organizers: Jordan Ellenberg and 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.

Confirmed speakers

Joan Birman (Columbia)
Spencer Dowdall (Chicago)
Nathan Dunfield (Illinois)
Ji-Young Ham (Seoul)
Eriko Hironaka (Florida State)
Thomas Koberda (Harvard)
Erwan Lanneau (Marseille) stuck in Europe! (probably here Sat.)
Chris Leininger (Illinois)
Dan Margalit (Tufts)
Chia-Yen Tsai (Illinois)


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)


Please send e-mail to Jean-Luc 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) 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.

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.



17:00 drinks at the Memorial Union Terrace (directions from University Inn)


9:00 welcome and introduction
9:30 Joan Birman (Columbia) Characteristic polynomials of pseudo-Anosov maps
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
14:30 Eriko Hironaka (Florida State) Families of small dilatation mapping classes
15:30 break
16:00 Thomas Koberda (Harvard) Pseudo-Anosov homeomorphisms and homology
17:00 discussion and beer
18:30 dinner at Fugu


9:00 Chris Leininger (Illinois) Small dilatation pseudo-Anosovs and 3 manifolds I
10:00 Dan Margalit (Tufts) Small dilatation pseudo-Anosovs and 3 manifolds II
11:00 break
11:30 Nathan Dunfield (Illinois) Hyperbolic surfaces bundles of least volume
12:30 lunch
14:00 Spencer Dowdall (Chicago) Dilatations and self-intersections for point-pushing pseudo-Anosov homeomorphisms
15:00 Erwan Lanneau (Marseille) Dilatations of pseudo-Anosov homeomorphisms and Rauzy-Veech induction
16:00 discussion, beer, tearful goodbyes


J. W. AABER AND N. M. DUNFIELD, Closed surface bundles of least volume, 2010. Preprint.

J. BIRMAN, P. BRINKMANN, AND K. KAWAMURO, Characteristic polynomials of pseudo-Anosov maps, 2010. Preprint.

P. BRINKMANN, A note on pseudo-anosov maps with small growth rates, Experiment. Math., 13 (2004), pp. 49-53.

J.-H. CHO AND J.-Y. HAM, The minimal dilatation of a genus-two surface, Experiment. Math., 17 (2008), pp. 257-267.

S. DOWDALL, Dilatation versus self-intersection number for point-pushing pseudo-Anosov homeomorphisms, 2010. Preprint.

B. FARB, C. J. LEININGER, AND D. MARGALIT, Small dilatation pseudo-Anosovs and 3 manifolds, 2009. Preprint.

J.-Y. HAM AND W. T. SONG, The minimum dilatation of pseudo-Anosov 5-braids, Experiment. Math., 16 (2007), pp. 167-179.

E. HIRONAKA, Small dilatation pseudo-Anosov mapping classes coming from the simplest hyperbolic braid, 2009. Preprint.

E. HIRONAKA AND E. KIN, A family of pseudo-Anosov braids with small dilatation, Algebraic & Geometric Topology, 6 (2006), pp. 699-738.

E. KIN AND M. TAKASAWA, Pseudo-Anosovs on closed surfaces having small entropy and the Whitehead sister link exterior, 2010. Preprint.

E. KIN AND M. TAKASAWA, Pseudo-Anosov braids with small entropy and the magic 3-manifold, 2010. Preprint.

E. LANNEAU AND J.-L. THIFFEAULT, On the minimum dilatation of pseudo-Anosov diffeomorphisms on surfaces of small genus, Ann. Inst. Fourier (2010), in press.

C. J. LEININGER, On groups generated by two positive multi-twists: Teichmüller curves and Lehmer's number, Geom. Topol., 8 (2004), pp. 1301-1359.

C. T. McMULLEN, Polynomial invariants for fibered 3-manifolds and Teichmüller geodesics for foliations, Ann. Sci. École Norm. Sup., 4 (2000), pp. 519-560.

C. T. McMULLEN, Entropy on Riemann surfaces and Jacobians of finite covers, 2010. Preprint.

H. MINAKAWA, Examples of pseudo-Anosov braids with small dilatations, J. Math. Sci. Univ. Tokyo, 13 (2006), pp. 95-111.

R. C. PENNER, Bounds on least dilatations, Proc. Amer. Math. Soc., 113 (1991), pp. 443-450.

W. T. SONG, Upper and lower bounds for the minimal positive entropy of pure braids, Bull. London Math. Soc., 37 (2005), pp. 224-229.

W. T. SONG, K. H. KO, AND J. E. LOS, Entropies of braids, J. Knot Th. Ramifications, 11 (2002), pp. 647-666.

W. P. THURSTON, On the geometry and dynamics of diffeomorphisms of surfaces, Bull. Am. Math. Soc., 19 (1988), pp. 417-431.

C.-Y. TSAI, The asymptotic behavior of least pseudo-Anosov dilatations, Geom. Topol., 13 (2009), pp. 2253-2278.

R. W. VENZKE, Braid Forcing, Hyperbolic Geometry, and Pseudo-Anosov Sequences of Low Entropy, PhD thesis, California Institute of Technology, 2008.

A. Y. ZHIROV, On the minimum dilation of pseudo-Anosov diffeomorphisms of a double torus, Russ. Math. Surv., 50 (1995), pp. 223-224.