School

**Rigidity School, Tokyo 2014 (2nd)**

Date: Nobember 22 (Sat) -- Nobember 24 (Mon), 2014

Place: Graduate School of Mathematical Sciences, University of Tokyo

Room: 002 on 22 (Sat), 123 on 23 (Sun) and 24(Mon)

http://www.ms.u-tokyo.ac.jp/access/index.html

Supported by Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, the University of Tokyo.

Speakers:

Uri BADER (Technion)

Fanny KASSEL (Universite Lille 1)

Hiroki SAKO (Niigata Univ.)

Ryokichi TANAKA (Tohoku Univ.)

Hidetoshi MASAI (Univ. of Tokyo)

Masato MIMURA (Tohoku Univ.)

Yosuke MORITA (Univ. of Tokyo)

Organizers:

Hiroyasu IZEKI (Keio University) izeki@math.keio.ac.jp

Masahiko KANAI (University of Tokyo) mkanai@ms.u-tokyo.ac.jp

Shin NAYATANI (Nagoya University) nayatani@math.nagoya-u.ac.jp

Program（Tentative） :

November 22 (Sat)

9:30--10:20 Morita

10:40--11:30 Masai

13:30--14:20 Tanaka

14:40--15:30 Bader(1)

16:10--17:00 Kassel(1)

17:30-- Party

November 23 (Sun)

9:30--10:20 Bader(2)

10:40--11:30 Kassel(2)

13:30--14:20 Sako

14:40--15:30 Bader(3)

16:10--17:00 Kassel(3)

November 24 (Mon)

9:30--10:20 Kassel(4)

10:40--11:30 Bader(4)

13:30--14:20 Mimura

14:40--15:30 Kassel(5)

16:10--17:00 Bader(5)

Title & Abstract:

Uri Bader

Title: Algebraic representations of Ergodic actions

Abstract: The basic setup which I would like to consider in my lectures is an algebraic representation of a locally compact group: a continuous homomorphism from a locally compact group S to GL_n(k), where k is a topological field. Our way of studying this setup would be by considering various invariants called "algebraic gates", relating the ergodic theory (measurable actions) of S and its algebraic actions via the given representation. Fixing an ergodic action of S, we will consider algebraic representations of the action groupoid, and study the category of such representations. The associated algebraic gate is by definition an initial object in that category. The presentation of the theory will be accompanied by the discussion of various examples. We will also discuss applications to rigidity theory. This lecture series is based on a joint work with Alex Furman.

Fanny Kassel

Title: Margulis spacetimes and their analogues in negative curvature

Abstract: The Auslander conjecture, open since 1964, states that the symmetry group of any affine tiling of R^n, with compact tiles, should be virtually solvable. In 1977, Milnor asked whether this holds for affine tilings with noncompact tiles, and Margulis answered the question negatively in 1983 by constructing examples of nonabelian free groups acting properly discontinuously on R^3, by affine transformations. Since then, Margulis spacetimes, namely quotients of R^3 by such free groups, have been the object of a rich literature. We will present some recent developments (joint with Jeffrey Danciger and Francois Gueritaud) in understanding their geometry and topology, moduli space, and fundamental domains. A key point in these developments is to see Margulis spacetimes as ? infinitesimal analogues ? of their negatively-curved analogues (quotients of the so-called anti-de Sitter 3-space), and to make strong links with 2-dimensional hyperbolic geometry. I will also discuss related problems and applications of the core ideas, for example on factor maps of projective actions of higher rank lattices.

Hiroki Sako

Title: Property A for discrete metric spaces and a generalization of expander sequence

Abstract: I will talk about property A, which is an amenability-type condition for discrete metric spaces defined by Yu. Amenability was originally defined for discrete groups and characterized by a geometric property called Folner condition. The geometric condition is not so interesting in discrete metric spaces, but its modification called property A is important in operator K-theory. Property A is characterized by several other properties. Some are geometric, others are analytic. I will explain that the ``operator norm localization property'' is equivalent to property A. As a corollary, it is shown that a space does not have property A, if and only if it includes highly connected parts.

Ryokichi Tanaka

Title: Random Dirichlet series arising from records

Abstract: We study the distributions of the random Dirichlet series with parameters (s, b) defined by S=?sum_{n=1}^{?infty}?frac{I_n}{n^s}, where (I_n) is an independent sequence of Bernoulli random variables taking value 1 with probability 1/n^b and 0 otherwise. The random series of this type is motivated by the record indicator sequence (and also by random walks on groups). By estimates of exponential sums by van der Corput, we specify the parameter region where the distributions have densities.

Hidetoshi Masai

Title: Fibered commensurability and arithmeticity of random mapping tori.

Abstract: In this talk we discuss fibered commensurability of random mapping classes. We prove that under certain condition, any random walk gives a minimal mapping class in its fibered commensurability class with exponentially high probability. If time permits, we also discuss how non-arithmeticity of cusped random mapping tori follows from minimality of random mapping classes.

Masato Mimura

Title: Metric Kazhdan constants

Abstract: We introduce a notion of "metric Kazhdan constants" for (marked) groups with target in a class of complete metric spaces. We show that if the class is stable under some operations (such as taking scaling limits), then the metric Kazhdan constant is lower semi-continuous on the space of marked groups, in the sense of Grigorchuk. We prove that if the class is that of all Hilbert spaces, then the metric Kazhdan constant coincides with the Kazhdan constant in the ordinary sense. As a corollary, the Kazhdan constant is lower semi-continuous in the space of marked groups. Some application of the metric Kazhdan constants to expander graphs is also discussed.

Yosuke Morita

Title: A necessary condition for the existence of compact manifolds locally modelled on homogeneous spaces

Abstract: A manifold is called locally modelled on G/H if it is covered by open sets that are diffeomorphic to open sets of G/H and the transition functions are locally given by translation by elements of G. In this talk, we provide an obstruction for that a given homogeneous space G/H admits a compact manifold locally modelled thereon. For instance, there does not exists a compact manifold locally modelled on SL(p+q, R)/SO(p, q) if p, q are odd. The key tool of the proof is the homomorphism constructed by Kobayashi-Ono that gives a "lower bound" of cohomology of a compact manifold locally modelled on G/H.