スクール

Rigidity School, Nara 2010/2011

Ghani ZEGHIB (Ecole Normale Superieure de Lyon)

プログラム :

March 5 (Sat)

13:30-14:30 Ghani Zeghib: Around Zimmer program: rigidity of actions of higher rank lattices (1)

15:00-16:00 Masayuki Asaoka: Parameter deformation of locally free actions and leafwise cohomology (1)

16:30-17:30 Tetsuya Hosaka: On equivariant homeomorphisms of boundaries of CAT(0) groups

March 6 (Sun)

9:00-10:00 Ghani Zeghib: Around Zimmer program: rigidity of actions of higher rank lattices (2)

10:30-11:30 Masayuki Asaoka: Parameter deformation of locally free actions and leafwise cohomology (2)

13:30-14:30 Hirokazu Maruhashi: Parameter rigid actions of simply connected nilpotent Lie groups

15:00-16:00 Jun-ichi Mukuno: Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds

16:30-17:30 Masato Mimura: Property (TT)/T and homomorphism rigidity into mapping class groups

Evening 　 Party

March 7 (Mon)

9:00-10:00 Masayuki Asaoka: Parameter deformation of locally free actions and leafwise cohomology (3)

10:20-11:20 Ghani Zeghib: Around Zimmer program: rigidity of actions of higher rank lattices (3)

11:40-12:40 Shin-ichi Oguni: Relatively hyperbolic groups and relatively quasiconvex subgroups

Afternoon 　 Discussion

March 8 (Tue)

9:00-10:00 Masayuki Asaoka: Parameter deformation of locally free actions and leafwise cohomology (4)

10:30-11:30 Ghani Zeghib: Around Zimmer program: rigidity of actions of higher rank lattices (4)

タイトル、アブストラクト:

Ghani Zeghib

Title: Around Zimmer program: rigidity of actions of higher rank lattices

Abstract: Lattices in semi-simple Lie groups, e.g. SL(n, Z) are central examples of discrete groups due to their role in interplays between many mathematical domains. They split into hyperbolic and higher rank ones. The linear representation theory of higher rank lattices in completely solved by the Margulis super-rigidity theorem. The Zimmer program is a tentative to understand actions of these groups on compact manifolds.

A precise piece of this program is the conjecture that if a lattice in a Lie group $G$ acts on $M$, then the dimension of M equals at least the rank of G, and the rank + 1, if the action is volume preserving.

The standard examples are affine actions of subgroups of SL(n, R) on n-tori, and their projective actions of (n-1)-spheres. Most of the existing results state local rigidity of these actions.

In these lectures we start reviewing known results, and then focus attention on the (real) analytic case (and also the holomorphic case, if time allows). We recall classical results on super-rigidity and Ratner Theorem. We then discuss an approach to rigidity using a technique of global linearization.

Masayuki Asaoka

Title: Parameter deformation of locally free actions and leafwise cohomology

Abstract: Parameter deformation problem asks how an action can be deformed into another action without changing the orbits. For locally free actions of abelian groups, parameter deformation is completely described by the first cohomology of the orbit foliation. For non-abelian actions, the description is not established in general because of non-linearity. Recently, some people overcame this difficulties and showed that parameter rigidity is equivalent to the vanishing of the cohomology for several cases.

In this lecture, we discuss the above relation between parameter deformation and leafwise cohomology in both abelian and non-abelian cases. We also compute the leafwise cohomology of several explicit examples since the computation itself is non-trivial and interesting.

Tetsuya Hosaka

Title: On equivariant homeomorphisms of boundaries of CAT(0) groups

Abstract: We investigate an equivariant homeomorphism of the boundaries $\partial X$ and $\partial Y$ of two proper CAT(0) spaces $X$ and $Y$ on which a CAT(0) group $G$ acts geometrically. We provide a sufficient condition to obtain a $G$-equivariant homeomorphism of the two boundaries $\partial X$ and $\partial Y$ as a continuous extension of the quasi-isometry $\phi:Gx_0\rightarrow Gy_0$ defined by $\phi(gx_0)=gy_0$, where $x_0\in X$ and $y_0\in Y$. As an application, we introduce some examples of equivariant rigid CAT(0) groups.

Hirokazu Maruhashi

Title: Parameter rigid actions of simply connected nilpotent Lie groups

Abstract: A locally free action $\rho$ of a connected Lie group on a closed manifold is said to be parameter rigid if each action which has the same orbits as $\rho$ is conjugate to $\rho$. There are not so many known parameter rigid actions of noncommutative groups. In this talk we give a criterion for parameter rigidity of nilpotent group actions and construct parameter rigid actions of nilpotent groups.

Jun-ichi Mukuno

Title: Properly discontinuous isometric group actions on inhomogeneous Lorentzian manifolds

Abstract: If a homogeneous space $G/H$ is acted properly discontinuously upon by a subgroup $\Gamma$ of $G$, the quotient space $\Gamma \backslash G/H$ is called a Clifford--Klein form. In 1962, E. Calabi and L. Markus proved that no infinite group acts isometrically, effectively, and properly discontinuously on the de Sitter space $O(n+1, 1)/O(n, 1)$. It follows that no compact Clifford--Klein form of the de Sitter space exists. In this talk, we will present a new extension of the theorem of E. Calabi and L. Markus to a certain class of Lorentzian manifolds that are not necessarily homogeneous.

Masato Mimura

Title: Property (TT)/T and homomorphism rigidity into mapping class groups

Abstract: The following result of ours will be seen: Let Gamma be a finite index subgroup of the universal lattice SL_m(Z[x1,..,xk]) (m at least 3); or that of the symplectic universal lattice Sp_{2m}(Z[x1,...,xk]) (m at least 2). Then every homomorphism, from Gamma into the mapping class group of a compact oriented connected surface with finite (possibly zero) genus and punctures (; or into the outer automorphism group of a fintely generated free group), has finite image." To prove this theorem, we introduce the notion of property (TT)/T" for groups, which is a strengthening of Kazhdan's property (T) and a weaking of Monod's property (TT).

Shin-ichi Oguni

Title: Relatively hyperbolic groups and relatively quasiconvex subgroups

Abstract: We give an introduction to relatively hyperbolic groups and relatively quasiconvex subgroups, and also discuss recent development containing our results of a joint work with Yoshifumi Matsuda and Saeko Yamagata.