**スクール**

**Rigidity School, Tokyo 2012/2013**

**日時：２０１３年１月７日（月）〜１月１０日（木）**

**場所：東京大学数理科学研究科 (駒場キャンパス)**

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

**宿泊施設は各自で手配して下さるようお願いいたします。**

**この研究集会は、東京大学大学院数理科学研究科GCOEプログラム「数学新展開の研究教育拠点」の援助のもとに開催されます。**

**講演者：**

**Bachir BEKKA (Universite de Rennes 1)**

**Andres NAVAS (Universidad de Santiago de Chile)**

**小澤登高 (京都大学数理解析研究所)**

**深谷友宏 (東北大学)**

**松崎克彦 (早稲田大学)**

**田中守 (東北大学)**

**田中亮吉 (東北大学)**

**問合せ先・世話人：**

**井関裕靖 (慶応大学) izeki@math.keio.ac.jp**

**金井雅彦 (東京大学) mkanai@ms.u-tokyo.ac.jp**

**納谷信 (名古屋大学) nayatani@math.nagoya-u.ac.jp**

**プログラム (暫定):**

**January 7 (Mon)**

**13:30-14:30 Narutaka Ozawa (1)**

**14:50-15:50 Andres Navas (1)**

**16:10-17:10 Bachir Bekka(1)**

**January 8 (Tue)**

**9:10-10:10 Andres Navas (2)**

**10:30-11:30 Bachir Bekka(2)**

**13:30-14:30 Tomohiro Fukaya**

**14:50-15:50 Narutaka Ozawa (2)**

**16:10-17:10 Mamoru Tanaka**

**January 9 (Wed)**

**9:10-10:10 Bachir Bekka(3)**

**10:30-11:30 Narutaka Ozawa (3)**

**13:30-14:30 Katsuhiko Matsuzaki**

**14:50-15:50 Andres Navas (3)**

**16:10-17:10 Ryokichi Tanaka**

**January 10 (Thu)**

**9:10-10:10 Narutaka Ozawa (4)**

**10:30-11:30 Bachir Bekka(4)**

**11:50-12:50 Andres Navas (4)**

タイトル、アブストラクト:

Narutaka Ozawa

Title: Fortifications of Kazhdan's property

Abstract: Kazhdan's property (T) is one of the most important properties in the analytic group theory, and has a numerous applications to many other fields of pure and applied mathematics. The prominent example of property (T) groups is SL(n,R). A group G is said to have property (T) if every affine isometric actions of G on a Hilbert space has a fixed point. Various fortifications of this property have been suggested by several researchers and proved for SL(n,R). In a series of lectures, I will talk about results in this direction of Lafforgue, Shalom, Mimura, and myself.

Andres Navas

Title: A geometric approach to cohomology in dynamics

Abstract: To each dynamical system one can associate a space of cocycles (test functions) as well as a subspace of coboundaries, so that the associated cohomology reflects some of the underlying dynamics. In this series of lectures, we will deal with a generalization of this framework: the associated dynamics will correspond to that of a group action, and the cocycles will take values in the isometry group of a space of nonpositive curvature. As we will see, most of the classical theorems admit generalized versions in this setting (e.g. Birkhoff ergodic theorem, Gotsschalk-Hedlund theorem). Moreover, this gives a unified view with other classical results (e.g. Oseledets theorem). More importantly, this allows obtaining new results, as for instance: 1) The space of orientation-preserving C^1 actions of every nilpotent group on a 1-dimensional compact space os connected (N). 2) C^2 circle diffeomorphisms of irrational rotation number admit no invariant 1-distribution other than the invariant measure (N-Triestino). 3) Every linear cocycle can be perturbed so that to become conformal along the Oseledets splitting. In case all Lyapunov exponents are zero, then it can be perturber so that to become cohomologous to a cocycle of rotations (Bochi-N). Several open questions will be addressed.

Bachir Bekka

Title: Rigidity of group actions on homogeneous spaces

Abstract: Given a group G acting on a probability space X by measure preserving transformations, one has a corresponding a unitary representation of G (Koopman representation); an important question, with diverse applications, is whether this action has a spectral gap, a rigidity property defined in terms of this representation. Actions of groups with property (T) always have such a spectral gap property. We will review some recent results as well as a few applications on this question in different cases: G is a subgroup of a Lie group H acting on X=H/L for a lattice L in H; G is a group of automorphisms of a torus or, more generally, of a nilmanifold X.

Tomohiro Fukaya

Title: Blow up of the boundary of relatively hyperbolic group

Abstract: A relatively hyperbolic group acts on a Gromov hyperbolic space. Its Gromov boundary has no information on peripheral subgroups since the boudnary of the orbit of any point by a peripheral subgroup consists only of one point, called cusp. We construct a blow up of cusp and give a nice boundary of a relatively hyperbolic group. We show that, under appropriate assumptions on peripheral subgroups, the $K$-homology of this boundary is isomorphic to the $K$-theory of the Roe-algebra of the group. As application, we give an explicit computation of the $K$-theory of the Roe-algebra of the fundamental group of the complement of a hyperbolic knot. If time permits, we will discuss a dual theory, that is, the $K$-theory of the stable Higson corona, coarse $K$-theory and coarse co-assembly map.

Mamoru Tanaka

Title: Coarse nonembeddability of a graph containing a sequence of expanders as induced subgraphs

Abstract: In this talk, we consider coarse nonembeddability of a graph containing a sequence of expanders as induced subgraphs into Hilbert spaces. Using this, we see that a graph containing a "generalized" sequence of expanders (a sequence of finite graphs which have uniformly bounded k-th eigenvalues of the Laplacians and uniformly bounded degrees and whose numbers of vertices diverge to infinity) is not coarsely embeddable into Hilbert spaces.

Katsuhiko Matsuzaki

Title: Conjugation of a group of circle diffeomorphisms

Abstract: By applying concepts in the quasiconformal Teichm\"uller theory, we give a necessary and sufficient condition for a non-abelian group $G$ of $(1+\alpha)$-diffeomorphisms of the circle with $\alpha>1/2$ to be conjugated to a group of M\"obius transformations by a diffeomorphism in the same class. In its argument, we also see certain rigidity of such a group $G$ in the deformation given by conjugation of symmetric self-homeomorphisms of the circle.

Ryokichi Tanaka

Title: The triviality of the boundary for nilpotent groups

Abstract: I will formulate a version of law of large number in the strong sense for random walks on Lie groups, and show this is useful to describe the boundary associated with a group and a random walk on it. More precisely, for example, for nilpotent Lie groups, we can show that the asymptotic direction of random walks on them is unique almost surely. But, even for some polycyclic groups, the asymptotic direction of random walks on them is not unique any more, due to Kaimanovich. I explain how the asymptotic direction describes the boundary. I will also give some questions around this construction.