**スクール**

**Rigidity School, Tokyo 2011/2012**

**日時：２０１２****年３月１７日（土）午後〜３月２０日（火）**

**場所：東京大学数理科学研究科 (駒場キャンパス)００２号室**

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。**

**会場に関する注意: **

**注意１： 数理科学研究科 ００２号室の入り口は研究科棟の西側（井の頭線踏切寄）にあり，本棟の入り口（北側のメインエントランス，東西にある
ふたつの小さな入り口）とは異なります．School期間中は掲示を張り出す予定ですので，ご注意下さい．**

**注意２：１９日午後のみ，１２３号室を利用します．**

**講演者：**

**Michael KAPOVICH (University of California, Davis)**

**藤原 耕二 (東北大学)**

**糸健太郎 (名古屋大学)**

**奥田 隆幸 (東京大学)**

**中西敏浩 (島根大学)**

**松田能文 (東京大学)**

**矢吹康浩 (名古屋大学)**

**山形紗恵子 (明石高専)**

**吉野太郎 (東京大学)**

**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:**

**March 17 (Sat)**

**14:00-15:00 Koji Fujiwara (1)**

**15:20-16:20 Taro Yoshino**

**16:40-17:40 Misha Kapovich (1)**

**March 18 (Sun)**

**9:40-10:40 Misha Kapovich (2)**

**11:00-12:00 Toshihiro Nakanishi**

**14:00-15:00 Takayuki Okuda**

**15:20-16:20 Koji Fujiwara (2)**

**16:40-17:40 Saeko Yamagata**

**March 19 (Mon)**

**9:40-10:40 Koji Fujiwara (3)**

**11:00-12:00 Yasuhiro Yabuki**

**14:00-15:00 Yoshifumi Matsuda**

**15:20-16:20 Misha Kapovich (3)**

**16:40-17:40 Kentaro Ito**

**March 20 (Tue)**

**9:40-10:40 Koji Fujiwara (4)**

**11:00-12:00 Misha Kapovich (4)**

タイトル、アブストラクト:

Misha Kapovich

Title: Quasi-isometric rigidity in geometric group theory

Abstract: I will present several techniques in quasi-isometric rigidity and their applications: Asymptotic cones, coarse topology, quasi-conformal maps. Among applications are Morse lemma, Mostow and Tukia rigidity theorems, coarse surjectivity of quasi-isometric embeddings of Poincare duality groups.

Koji Fujiwara

Title: Group actions on quasi-trees

Abstract: The study on group actions on simplicial trees is classical. Bass-Serre theory gives a description of the group if the action is non-trivial, namely, there is no global fixed point. It also has some connection to property T.

A quasi-tree is a graph which is quasi-isometric to a simplicial tree. We will study group actions on quasi-trees by automorphisms. In this case, an action is non-trivial if there is an unbouded orbit. We give sufficient conditions for a group to have a non-trivial action on a quasi-tree, discuss standard examples, and study some consequence from non-trivial group actions on quasi-trees. The lecture stars with the basic of delta-hyperbolic geometry.

Kentaro Ito

Title: A family of slices of the Maskit slice for the twice punctured torus

Abstract: Via combination theorem, the Maskit slice $M(1,2)$ for the twice punctured torus can be regard as a subset of the product of the Maskit slice $M(1,1)$ for the once punctured torus (or the 4-puncture sphere). In this talk, we consider slices of the Maskit slice $M(1,2)$ which are obtained by fixing one variable in the boundary of $M(1,1)$ and show that these slices are just translations of $M(1,1)$.

Takayuki Okuda

Title: Semisimple symmetric spaces with properly discontinuous actions of surface groups

Abstract: We consider symmetric spaces M := G/H with connected linear simple Lie groups G. In this talk, we give a classification of such symmetric spaces for which there exist properly discontinuous actions of surface groups as isometries of M. Our method is based on the results of T. Kobayashi [Math. Ann. '89] and Y. Benoist [Ann. Math. '96] together with combinatorial techniques of nilpotent orbits.

Tosihiro Nakanishi

Title: Parametrization for Teichm¥"uller spaces and its application to a representation of the mapping class groups

Abstract: Every marked Fuchsian group of type $(g,m)$ is determined by the traces of 6g-5 matrices in the group. This result was first proved by P. Schmutz in 1994. We review this result and show that the mapping class group of the genus 2 surface can be represented by a group of rational transformations in 7 traces.

Yoshifumi Matsuda

Title: Blowing up and down compacta with geometrically finite convergence actions of a group

Abstract: We consider two compacta with minimal non-elementary convergence actions of a countable group. When there exists an equivariant continuous map from one to the other, we call the first a blow-up of the second and the second a blow-down of the first. It is known that if one is a blow-up of the other, then each maximal parabolic subgroup with respect to the first is a parabolic subgroup with respect to the second. We show that the converse holds if both actions are geometrically finite. We also provide some applications. This talk is based on a joint work with Shin-ichi Oguni and Saeko Yamagata.

Yasuhiro Yabuki

Title: Proper conjugation of isometry groups on Gromov hyperbolic spaces

Abstract: A discrete group G consisting of isometries of a Gromov hyperbolic space X is said to have a proper conjugation if the conjugate of G by some isometry of X is strictry contained in G. We give a sufficient condition for G to have no proper conjugation, which is a result of a joint work with Katsuhiko Matsuzaki.

Saeko Yamagata

Title: Commensurators of surface braid groups

Abstract: Let g and n be integers at least two. In this talk, we explain that the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components. This talk is based on a joint work with Yoshikata Kida.

Taro Yoshino

Title: Deformation space of Clifford-Klein form and topological blow-up

Abstract: The study of deformation spaces of Clifford-Klein forms in a general setting was initiated by T. Kobayashi as a generalization of the Teichmuller space.

The first part of this talk surveys some of my previous results and known facts on deformation spaces, including: (1) some rigidity theorem, (2) explicit form of some deformation spaces. We also see that deformation spaces may fail to be Hausdorff.

In the latter part, we introduce `Topological Blow-up' as a method of understand a non-Hausdorff space.