Rigidity School ― The Final Meeting

Dates & Place
September 16-19, 2018
Nagoya University, Mathematics Building, room 509
http://www.math.nagoya-u.ac.jp/en/direction/nagoya.html#a
http://www.math.nagoya-u.ac.jp/ja/direction/campus.html

Organizers
IZEKI, Hiroyasu (Keio U)
KANAI, Masahiko (U Tokyo)
NAYATANI, Shin (Nagoya U)

Lecturers
BROWN, Aaron (U. Chicago)
FUKAYA, Tomohiro (Tokyo Metropolitan U)
HASSELBLATT, Boris (Tufts U)
MARUHASHI, Hirokazu (U Tokyo)
MATSUMOTO, Shigenori (Nihon U)
MIMURA, Masato (Tohoku U)
OGUNI, Shinichi (Ehime U)
TOYODA, Tetsu (NIT, Suzuka)

Banquet (New)

We are planning to have dinner together in the evening of 16th September.  Those participants who would like to join, please contact Hiroyasu Izeki ((izeki@math.keio.ac.jp) by 9th September.

Time Table
16th September
13:30-14:30 Matsumoto
15:00-16:00 Oguni
16:30-17:30 Brown 1

17th September
10:00-11:00 Brown 2
11:30 –12:30  Fukaya
14:30-15:30 Brown 3
16:00-17:00 Maruhashi

18th September
10:00-11:00 Brown 4
11:30 –12:30 Toyoda
14:30-15:30 Brown 5
16:00-17:00 Mimura

19th September
10:00-11:00 Hasselblatt
11:30 –12:30 Brown 6

Titles & Abstracts

BROWN, Aaron
Smooth ergodic theory and rigidity of lattice actions
The main result of this course is the resolution of Zimmer’s Conjecture for cocompact lattices in SL(n,R):
Theorem 1: for n>= 3 and any cocompact lattice in Sl(n,R)
1. any C^2 action on a compact manifold of dimension at most  (n-2) is finite
2. any C^2 volume-preserving action on a compact manifold of dimension at most  (n-1) is finite
The main theorem of the course, however, is a theorem about Lyapunov exponents.
Theorem 2: for n>= 3, any C^2 action (resp. vol-preserving action) of a cocompact lattice in Sl(n,R) on a manifold of dimension at most (n-2) (resp. at most (n-1)) has uniform subexponential growth of derivatives.
We will see that Theorem 2, when combined with Strong Property (T) and Margulis Superrigidity, immediately implies the above result.
The goal of the course will be to motivate the above results, present sufficient background in smooth ergodic theory, lattice actions, and superrigidity theorems, and to give a complete proof of Theorem 1.  If there is time, I will also explain some ideas used to prove the result for non-uniform lattices.
A provisional outline for the course is the following:
1. Background on lattices and Lie groups, Motivation, statement of results and recent developments
2. Margulis and Zimmer superrigidity theorems; Statement of Theorem 2 and Proof of Theorem 1
3. Lyapunov exponents for diffeomorphisms and actions of abelian groups
4. Unstable manifolds, entropy, relationships between entropy, exponents and geometry of conditional measures
5. Begin proof of Theorem 2
6. Conclude proof of Theorem 2

FUKAYA, Tomohiro
Coarsely convex spaces
In a joint-work with Shin-ichi Oguni, we introduced a new class of metric spaces, called coarsely convex spaces, which can be regarded as a coarse geometric analogue of non-positively curved spaces.  In this talk, I explain the definition of a coarsely convex space and a construction of its boundary.  Then I give an outline of the proof of the coarse Cartan-Hadamard theorem.  If time permits, we also talk on relations with semihyperbolic spaces by Alonso-Bridson, and combing corona by Engel-Wulff. As a corollary, we show some group-theoretic property of groups acting on coarsely convex spaces, like isoperimetric inequality and cohomological dimension.

HASSELBLATT, Boris
New contact flows on 3-manifolds
Foulon-Ding-Geiges-Handel-Thurston surgery produces contact flows that are unusual and interesting in several ways. The periodic fiber flow in the unit tangent bundle becomes parabolic, and the geodesic flow becomes a nonalgebraic contact Anosov flow with larger orbit growth. An idea by Vinhage promises a quantification of the gap between the Liouville and topological entropies.

MARUHASHI, Hirokazu
A computation of the de Rham cohomology of certain foliations
It's quite straightforward to define the de Rham cohomology of a smooth foliation of a manifold if you know the definition of the de Rham cohomology of a manifold. What's not straightforward is its computation. It's been more than four decades since this concept was first investigated with regard to the deformation theory of foliations, but not much computations have been done to this day in my opinion. (Probably because people haven't paid much attention to it.)  For example, let $\mathcal{F}$ be the weak stable foliation of the geodesic flow of a compact hyperbolic surface, which is a good example of a foliation you can find in any textbook on foliations. The first cohomology of $\mathcal{F}$ was computed by Matsumoto and Mitsumatsu in 2003.  But the second cohomology has remained unknown as far as I know. In this talk I will show you how to compute the cohomology of $\mathcal{F}$ in all degrees at the same time. A Kodaira-Spencer theory for parameter deformations of a locally free action was a motivation for the computation. This is an ongoing work (in the sense that the method is being generalized) with Mitsunobu Tsutaya from Kyushu University.

MATSUMOTO, Shigenori
Left orders on countable groups
Left invariant total orders on a countable group have strong connections with its actions on the real line. The set of such orders form a totally disconnected compact metrizable spaces. For example, if the group is a free group on two generators or more, then this space is homeomorphic to  a Cantor set.  However there are groups which admit isolated points of  this set. Isolation of a left invariant order has relations with the rigidity of the corresponding actions on the real line.  Likewise left invariant circular orders of a group are connected with its actions on the circle.  My talk will be a survey of these topics.

MIMURA, Masato
Profinite actions on a common set
We give an answer to the question asking how different group properties of two finitely generated dense subgroups of a common compact group can be. Our constructions allow us to take one dense subgroup to be amenable (in fact, a locally-finite-lift of Z) and the other to contain a given countable residually finite group. These provide examples of profinite actions on a common set with considerably different properties.

OGUNI, Shinichi
Under the assumption that a metric space X is geodesic, many simple conditions for X which are equivalent to the condition that X is CAT(0) have been known. On the other hand, when we characterize metric spaces which embed isometrically into CAT(0) spaces, we have to omit the geodesicness assumption.  M. Gromov remarked that a four-point metric space embeds isometrically into a CAT(0) space if and only if it satisfies the weighted quadruple inequalities.  In this talk, we present another proof of Gromov's remark and show that a five-point metric space embeds isometrically into a CAT(0) space if and only if it satisfies the weighted quadruple inequalities.