Research projects

He made computation of Gelfand-Fuks cohomology of diffeomorphism groups and their homogeneous spaces especially keeping in his mind possible applications of it to rigidity problems. A new perspective on infinitesimal rigidity of Anosov actions of higher-rank abelian groups that arise from semisimple Lie groups of rank greater than one has also been obtained by him. It is known that the domain of discontinuity of a convex-cocompact Kleinian group is compact. Conversely, he proved that a Kleinian group is convex cocompact provided the Hausdorff dimension of the limit set is less than n/2. They introduced a combinatorial notion of harmonic map of a simplicial complex into singular space of nonpositive curvature, and showed an existence theorem under an appropriate assumption. A fixed point theorem for an isometric action of a discrete group on a nonpositively curved space has been established as an application. She investigated, in a joint work of T.Sunada, the large deviation principle. Another achievement by her is the Lipschitz continuity of the bounds of the spectrum of some self-adjoint operator with magnetic, effect. A projective Anosov flow is said to be regular if its stable and unstable plane fields are integrated by smooth foliations. He proved that for a regular projective Anosov flow on a Seifert fibered space if the associated foliations have no compact leaf then it is a regular Anosov flow, and in consequence is quasi-Fuksian, due to a theorem of Ghys. In the joint work with Bestvina, he made a computation of the second bounded cohomology of the mapping class groups. It follows that a discrete subgroup of a Lie group can never be realized as a subgroup of the mapping class groups.

The group is an algebraic object which also gives a description of symmetries of spaces. Some important and interesting groups often admits a property called "superrigidity", which we tried to understand as an extremal property among that involving infinite discrete groups. We could show that a fixed-point property, which should be considered to be an important aspect of superrigidity, is shared by finitely presented groups with overwhelming probability.

A group G is said to have rigidity, if the space accepting the action of G is very restrictive. A group G is said to have fixed-point property for a space Y, if every action of G fixes a point in Y. Recent development of mathematical research suggests that there are plenty of groups having rigidity. We proved that, in a certain model, groups have strong fixed-point property with high probability.

We developed a theory of combinatorial harmonic maps and obtained a fixed-point theorem for isometric discrete-group actions on nonpositively curved metric spaces. Suppose that a discrete group Γ acts on a simplicial complex X properly discontinuously and cofinitely. Let Γ acts isometrically on a nonpositiovely curved metric space Y. We defined an energy for maps from X into Y which are equivariant with respect to the actions of Γ on both spaces. A combinatorial harmonic map is defined to be a critical point of this energy functional Y. We gave a sufficient condition for the gradient flow of the energy functional to converge to a constant map. This means that the given action of Γ on Y admits a global fixed point. Here the condition is independent of the action of Γ on Y, we conclude that Γ has a fixed-point property for Y. Namely, every isometric action of Γ on Y admits a global fixed-point. If Y is a Hilbert space, then this fixed-point property is known to be equivalent to Kazhdan's property (T). The sufficient condition is expressed in terms of a spectral invariant of a finite graph, a link of a vertex of X. We introduced a local invariant of nonpositively curved metric space, and gave an estimate of the spectral invariant in terms of the eigenvalue of the Lapalacian of the link and this local invariant.

ある種の超エルミート多様体を底とするトーラス束における超CR構造の構成法を導き、様々な擬凸性や他の類似の構造との違いを与える例を構成した。また、超CR多様体におけるCR構造の族とそのツイスター空間上のCR構造の積分可能性の関連を調べた。定スカラー曲率ケーラー構造に関連して、ハミルトンS^1-作用をもつコンパクト不定値ケーラー曲面は、ある線織面に双正則であり、さらに、全スカラー曲率が零ならばあるヒルツェブルフ曲面に双正則であることを示した。

The aim is to study the non-commutative objects in Number theory, algebraic geometry, differential geometry, topology and mathematical physics including particle physics and to propose new idea in geometry. The features of this study is to hire the theory of deformation quantization and non-commutative geometry and develop them to apply several research topics. In particluar, we construct a new method of non-formal deformation quantization to cooperate with theoretical physics, including particle physics and string theory. We also establish an international research network in this research area.

研究代表者・納谷信は、昨年度に引き続き、連携研究者・小林俊公(摂南大学工学部・講師)、研究協力者・豊田哲(名古屋大学大学院多元数理科学研究科、博士後期課程3年)とともに、井関と納谷が論文Combinatorial harmonic maps and discrete-group actions on Hadamard spaces, Geometriae Dedicata 114 (2005), 147--188において導入した不変量δを、できるだけ一般的なCAT(0)空間の接錐に対して上から評価するという問題について研究を行った。豊田哲は、方向空間全体のなす距離空間族がグロモフ・ハウスドルフ距離に関してプレコンパクトであるようなCAT(0)空間の族に対して、そのすべての接錐のδが1より小さい定数によって上から一様に押さえられることを示した。 また、小林俊公と納谷は、あるクラスの1次元球面的ビルディングに対して、その上の距離錐の動径的歪み係数(ユークリッド空間に、原点からの距離を保ちながらできる限り等長的に近くなるように埋め込んだときの等長写像からのずれをはかる不変量)のよい評価を与え、それによりそれらの距離錐のδの上からの評価を得た。さらに、求めた値が最適値を与えるか否かという問題を解決することを念頭において、歪みの評価の問題への表現論を用いたアプローチについて考察した。

The aim of this research was capturing the higher dimensional structure of discrete metric spaces and getting some rigidity results as applications of this structure. We could not obtain a satisfactory result describing the higher dimensional structure of discrete sets, however, using an observation and results obtained in the course of this research, we proved that certain random groups have strong fixed-point property for a wide class of nonpositively curved metric spaces. Also we gave some good estimates of nonlinear spectral gap for some embeddings of discrete metric spaces into nonpositively curved metric spaces.

We proved that Gromov's group containing quasi-isometric image of expanders has fixed-point property for a large class of CAT(0) spaces. In relation to the possible geometric proof of the Margulis superrigidity theorem, we studied a certain geometric invariant of the 2-dimensional Euclidean building associated to PGL(3,Q_p). In the case p=2, we made progress on the related deformation problem of polyhedral. Motivated by a certain rigidity problem on discrete groups, we studied conformal-geometric structures such as strongly pseudoconvex CR structures and quaternionic CR structures, and obtained some results.

We discuss ed a long time behavior of periodic random walks on a crystal lattice in view of geometry, a large deviation property in particular, and relate it to a rational convex polyhedron in the first homology group of a finite graph, which, as remarkable combinatorial features,. A crystal lattice has a metric structure with the graph distance. By changing scale of the distance, we obtain a one-parameter family of metric spaces. The Gromov-Hausdorff limit of the sequence is called the asymptotic cone at the infinity of the crystal lattice. As the scale go to zero., because of the periodicity of the crystal lattice, the asymptotic cone exists and we determinded its unit ball explicitely in terms of combinatorial data. We also published a survey article on discrete geometric analysis of crystal lattice from Sugaku Expository, Amer.Math.Soc. In there, we discussed spectral properties and geometry of random walks on a crystal lattice, such as the law of large number, the central limit theorem, large deviation and spectrum of magnetic Schroedinger operators from non-commutative geometry.

組合せ調和写像をランダム群の固定点性質の解明に応用する研究を行った。成果として、三角モデルのランダム群があるクラスに属するCAT(0)空間のすべてに対して固定点性質を持つという結果を得た。また、グラフモデルのランダム群がより広いクラスに属するCAT(0)空間のすべてに対して固定点性質をもつことを証明した。これらCAT(0)空間のクラスは、ある不変量δを用いて記述されるので、不変量δを上から評価する問題に取り組んだ。CAT(0)空間の接錐の歪み係数を用いて上から評価できることを明らかにし、ユークリッド的ビルディングの場合にこの評価を実行した。

空間の対称性を数学的に記述するときに用いられる概念が「群」および「群の空間への作用」である。本研究では、どのような群がどのような空間に作用し得るかを、幾何学的な手法で研究した。その際に用いられるのは、群の空間への作用の様子のもっとも効率のよい実現を与える調和写像と呼ばれる写像である。この研究成果から、ある種の空間に非自明な仕方で作用し得ない群が非常に多く存在することが明らかになった。

We studied toward determi ning the rigorous values of the invariant δof the tangent cones at vertices of the Euclidean building associated with the algebraic group PGL(3,Q_p), where p are primes, and observed the p=2 case in detail. The problem is formulated as that of deforming a certain real symmetric matrix properly, and this matrix has 9 unknown parameters (functions of 1 variable). We determined 6 parameters of these.

A pseudo-Riemannian metric on a manifold is called a neutral metric if it has neutral signature, and a family of local neutral metrics that conicide, except for multiplication by -1, on the overlaps, is called a neutral structure. Davidov et al. obtained examples of compact complex surfaces with a quaternion-like structure (parahypercomplex structure) that admit compatible neutral structures, but never admit any compatible neutral metric. Then we show that their example of a hyperelliptic surface can be deformed to a compatible neutral structure, which is not locally conformal parahyperkahler. Also, ｗe introduce the notion of strong integrability for a quaternionic CR manifold (of dimension greater than 7), and show that, under ultra pseudoconvexity and strong integrability, a partially integrable almost CR structure (called the twistor almost CR structure) is defined naturally on its twistor space.

Aim of this research proposal is to develop new methods to study geometric objects with singularities, or discrete spaces, which are not accessible by traditional differential geometrical technics. Our idea is to apply probability theory to those geometric objects. Some results are obtained and published from international journals.

We have studied the following : (1)Study of Yamabe Invariants We estimated and determined the Yamabe invariant of some positive 3-manifolds, by using the inverse mean curvature flow and families of Green's functions. In especial, we classified completely all 3-manifolds with Yamabe invariant greater than that of RP^3. We also studied the positive Yamabe constants of Riemannian products and the behavior of them under magnifying one factor. We are now studying Aubin's type lemma for the positive Yamabe constants of infinite coverings, with some new results. (2)Study of Conformal/Affine Geometry and group C^*-algebra We gave new developments on conformal and projective geometry. In particular, we obtained an interesting varitional characterization of affine connections induced from Einstein metrics. We studied on twisted K-theory and groupoid C^*-algebras, and then proved a generalization of Gromov-Lawson theorem for foliated spaces. (3)Study on Non-linear Analysis in Geometry We studied on eigenvalue problem on complete manifolds, discrete groups and valiational problems on mean curvature. We then obtained results on non-existence of eigenvalues on complete manifolds of non-positive curvature, and a fixed-point theorem for discrete-group actions on Hadamard spaces.

コンパクト多様体と言う空間に対して, そこに実現される形(リーマン計量) の在り様全体のなかで最も自然な形を求めることを目標とし, そのため導入された指標が山辺不変量と呼ばれる(共形幾何を経由して)定義された微分位相不変量である.この不変量を求めること, およびその振舞いを調べることが目標である.本研究では, 正の3次元多様体の山辺不変量の計算及びその振舞いに関して成果を得た.また正の山辺不変量を持つ直積多様体に関して, その下からの評価を与えた.さらに, この下からの評価と山辺不変量の手術理論の間に成立すると予想される, 山辺不変量に関する不等式の提出とこの予想の部分的結果を得た.

The cohomology of discrete groups, such as mapping class groups of closed surfaces, Coxeter groups and Artin groups, is one of the important objects in topology as well as geometry In this research project, we studied the cohomology of discrete groups. The following three subjects were emphasized in the project: (1) Relations with the cohomology of finite subgroups (2) Actions of discrete groups on manifolds/complexes and combinatorial structures (3) Algebraic methods (such as combinatorics and free resolutions) Concerning of (1), Akita and Kawazumi proved integral Riemann-Roch formulae for cyclic subgroups of mapping class groups, which are variants of Grothedieck-Riemann-Roch theorem for integral cohomology. Concerning of (2), Izeki and Hosaka obtained various results concerning of group actions and geometric structures Finally, concering of (3), Akita proved alternative formulae for the Euler characteristics of even dimensional triangulated manifolds. The key ingredient to prove the formulae is generalized Dehn-Sommerville equations obtained by Klee. In addition, Akita showed that mod p Riemann-Roch formulae hold for various cases, by using Kummer's congruences in classical number theory.

研究代表者金井は, 高階非コンパクト半単純リー群およびその余コンパクト格子に付随して現れるワイル領域流 (Weyl chamber flow) の剛性とヴェイユや松島の古典的な消滅定理の間に予期せぬ関係があることを指摘する新たな結果を得た. 剛性問題に新たな進展をもたらすことが期待される. このほかにも, 連携研究者等により剛性問題に対る新たな知見が多数得られた