School

**Rigidity School, Tokyo 2013/2014**

Date: January 6 (Mon) -- January 10 (Fri), 2014

Place: Graduate School of Mathematical Sciences, University of Tokyo

Room 123 (002 for morning sessions of 7(Tue) and 8 (Wed))

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:

Katsuhiko MATSUZAKI (Waseda University)

Denis OSIN (Vanderbilt University)

Richard WENTWORTH (University of Maryland)

Yuichi KABAYA (Tokyo Institute of Technology

Yohei KOMORI (Waseda University)

HIrokazu MARUHASHI (Kyoto Uniersity)

Ryosuke MINEYAMA (Osaka University)

Shin-ichi OGUNI (Ehime University)

Yuriko UMEMOTO (Osaka City University)

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 :

January 6 (Mon)

13:30--14:30 Wentworth (1)

15:00--16:00 Osin (1)

16:30--17:30 Matsuzaki (1)

January 7 (Tue)

10:00--11:00 Umemoto

11:30--12:30 Komori

14:30--15:30 Osin (2)

16:00--17:00 Wentworth (2)

17:30-- Party

January 8 (Wed)

10:00--11:00 Mineyama

11:30--12:30 Oguni

14:30--15:30 Matsuzaki (2)

16:00--17:00 Osin (3)

January 9 (Thu)

10:00--11:00 Wentworth (3)

11:30--12:30 Osin (4)

14:30--15:30 Matsuzaki (3)

16:00--17:00 Maruhashi

January 10 (Fri)

10:00--11:00 Kabaya

11:15--12:15 Wentworth (4)

12:30--13:30 Matsuzaki (4)

Title & Abstract:

Richard Wentworth

Title: The Bando-Siu Conjecture

Abstract: In these lectures I will give a complete description of the limiting behavior of the Yang-Mills flow of integrable unitary connections on hermitian vector bundles over Kaehler manifolds. The fundamental work of Donaldson and Uhlenbeck-Yau proves the convergence the smooth convergence of the flow in the case of stable vector bundles. This was generalized by Bando and Siu to incorporate certain (singular) hermitian structures on reflexive sheaves. Bando-Siu also conjectured what happens when the initial sheaf is unstable; namely, that the limiting behavior should be controlled by the Harder-Narasimhan filtration. Moreover, in a the case of Kaehler surfaces Daskalopoulos-Wentworth proved a stronger statement that includes information on the relationship between analytic and algebraic singularities that may occur. By work of several authors, the full result in higher dimensions has now been established. These lectures will provide background and outline the main ideas of the proof.

Denis Osin

Title: Acylindrically hyperbolic groups.

Abstract: A group is acylindrically hyperbolic if it admits a non-elementary acylindrical isometric action on a hyperbolic space. The class of acylindrically hyperbolic groups is surprisingly wide and includes all non-elementary hyperbolic and relatively hyperbolic groups, most 3-manifold groups, automorphism groups of certain free objects (such as free groups or free polynomial algebras), all but finitely many mapping class groups, and many other examples. On the other hand, geometric methods allow one to obtain many general results about acylindrical group actions, some of which are new even for well-studied particular classes. After discussing basic definitions and examples, we will survey recent results and open problems about acylindrically hyperbolic groups.

Katsuhiko Matsuzaki

Title: Circle diffeomorphisms and Banach structures on the universal Teichmueller space

Abstract: Symmetric homeomorphisms play the central role in the theory of symmetric structures on the circle, which is due to Gardiner-Sullivan. The space of symmetric homeomorphisms defines a Banach structure on the universal Teichmueller space. First we consider rigidity of a Fuchsian representation of symmetric deformation and a failure of the fixed point property for an isometry group acting on the Teichmueller space of symmetric homeomorphisms. This space includes the space of diffeomorphisms with Hoelder continuous derivatives. Then we characterize these mappings by the decay order of the Beltrami coefficients of quasiconformal extensions and Schwarzian derivatives of their Bers embeddings. This is used for proving certain rigidity of the representation of a Fuchsian group in the group of the diffeomorphisms. We also consider quasiconformal automorphisms of the disk whose Beltrami coefficients are integrable with respect to the hyperbolic metric. Cui and Takhtajan-Teo introduced the Hilbert structure and the Weil-Petersson metric on the universal Teichmueller space using these maps in the square integrable case. We generalize this for the p-integrable case and introduce the Banach structure on it. As an application, we investigate the fixed point property of isometry groups with bounded orbit in these spaces and give a condition for a group of diffeomorphisms with Hoelder continuous derivatives to be conjugate to a Fuchsian group in the same class.

A rough plan of each talk is as follows:

(1) The universal Teichmueller space and its Bers embedding in the linear space of bounded projective structures. Sections of Teichmueller spaces to the corresponding spaces of Beltrami differentials (Douady-Earle, Beurling-Ahlfors, Ahlfors-Weill).

(2) Symmetric homeomorphisms and its Teichmueller space. Non-triviality of asymptotic Teichmueller spaces for Fuchsian groups.

(3) Teichmueller spaces of diffeomorphisms with Hoelder continuous derivatives. Characterization by using Beltrami coefficients, Schwarzian derivatives and conformal barycentric extension.

(4) Teichmueller spaces of integrable quasiconformal homeomorphisms. The Banach structure and generalization of the Weil-Petersson metric on the universal Teichmueller space. The fixed point property of isometry groups. Conjugation problem of diffeomorphisms.

Yuriko Umemoto

Title: The growth function of Coxeter dominoes and 2?Salem numbers

Abstract: In general, if we consider a group G with ?nite generating set S, we obtain the length function on G with respect to S. Then the formal power series whose coef?cients are determined by the length function, called the growth series is de?ned. For the case of a hyperbolic Coxeter group which is a discrete group generated by re?ections with respect to facets of a hyperbolic Coxeter polytope, it is known that the growth series is a rational function. In this talk, I will present the growth functions of hyperbolic Coxeter groups with respect to 4-dimensional compact polytopes constructed by successive gluing of Coxeter polytopes which we call Coxeter dominoes, and that their growth rates are 2-Salem numbers.

Yohei Komori

Title: On degenerate families of Riemann surfaces over elliptic curves

Abstract: Cut a 1-dimensional complex torus T along a path from the origin to a point p of T, take g copies of T and paste them along their slits. Then we have a Riemann surface of genus g with g-cyclic symmetry having p of T as its parameter. In this talk we make this family of Riemann surfaces parametrized by T as the degenerate family over T, and describe its singular fibers and holomorphic sections explicitly.

Ryosuke Mineyama

Title: Limit sets of Coxeter groups

Abstract: Recentry Hohlweg, Labbe, Ripoll introduced a normalized action of Coxeter groups on a hyperplane of a vector space to investigate asymptotic behavior of their roots. The normalized action turns out to be a discrete action on a CAT(0) space in the case that associating bilinear form of a Coxeter group has signature (n-1,1). From this we can define a "limt set" of such a Coxeter group. I am interested in how geometric aspects of Coxeter groups are mirrored on their limit sets. In this talk we discuss about the existence of Cannon-Thurston maps from Gromov boundaries of Coxeter groups to their limit sets.

Shin-ichi Oguni

Title: Coarse Baum-Connes conjecture and coarse algebraic topology

Abstract: I discuss the coarse Baum-Connes conjecture from the viewpoint of the coarse algebraic topology, which is a coarse geometric counterpart of the algebraic topology. First I explain a few applications of the conjecture. Second I give proofs of some partially positive results

for the conjecture by using the coarse algebraic topology. This talk is based on joint-works with Tomohiro Fukaya (Tohoku university).

Hirokazu Maruhashi

Title: Vanishing of cohomology and parameter rigidity of actions of solvable Lie groups

Abstract: We study smooth actions of noncompact connected Lie groups on closed manifolds. An action ?rho of G on M is called parameter rigid if any action of G on M with the same orbit decomposition as that of ?rho is conjugate to ?rho. Namely an action is parameter rigid if it is determined only by its orbit structure. Rigidity is often related to vanishing of some first cohomology. This is the case for parameter rigidity of actions of nilpotent Lie groups as we have shown before. In this talk we discuss a generalization of this to actions of solvable Lie groups.

Yuichi Kabaya

Title: Parametrization of PGL(n,C)-representations of surface groups

Abstract: Let S be a surface of genus g with b boundary components. The interior of S is ideally triangulated into 2(2g-2+b) ideal triangles. Fock and Goncharov gave a parametrization of (framed) PGL(n,C)-representations of the fundamental group of S by (n-1)(n-2)/2 parameters for each ideal triangle and (n-1) parameters for each edge of the triangulation. In this talk, we will give a parametrization of PGL(n,C)-representations as an analogue of the Fenchel-Nielsen coordinates using the Fock-Goncharov coordinates. This is joint work with Xin Nie.