School

**Rigidity School, Nagoya 2016**

Date: July 26 (Tue) -- July 29 (Fri), 2016

Place: Graduate School of Mathematics, Nagoya University Room 207, Science A Building

http://www.math.nagoya-u.ac.jp/ja/direction/index.htm

Speakers:

Ursula HAMENSTAEDT (University of Bonn)

Koji FUJIWARA (Kyoto University )

Masato MIMURA (Tohoku University)

Takayuki OKUDA (University of Tokyo)

Takahiro KITAYAMA (University of Tokyo)

Mitsuteru KIMURA (University of Tokyo)

Yosikata KIDA (University of Tokyo)

Yosuke MORITA (University of Tokyo)

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 :

July 26 (Tue)

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

14:50--15:50 Morita

16:10--17:10 Mimura(2)

18:30-- Party

July 27 (Wed)

9:20--10:20 Mimura(3)

10:40--11:40 Hamenstaedt(1)

13:30--14:30 Kimura

14:50--15:50 Mimura(4)

16:10--17:10 Hamenstaedt(2)

July 28 (Thu)

9:20--10:20 Hamenstaedt(3)

10:40--11:40 Fujiwara(1)

13:30--14:30 Hamenstaedt(4)

14:50--15:50 Fujiwara(2)

16:10--17:10 Okuda

July 29 (Fri)

9:20--10:20 Fujiwara(3)

10:40--11:40 Kida

13:30--14:30 Kitayama

14:50--15:50 Fujiwara(4)

Title & Abstract:

Ursula Hamenstaedt

Title: Random walks, boundaries and rigidity

Abstract: The goal of these lectures is to discuss random walks on groups with hyperbolic properties and applications to rigidity.

Lecture one: We introduce random walks on groups and discuss some examples. We also introduce some boundaries of random walks on groups in some explicit cases.

Lecture two: For random walks on the fundamental group of a negatively curved manifold we relate the geometric boundary of the universal covering to the Poisson boundary of the group and discuss examples and open question.

Lecture three: For fundamental groups of closed hyperbolic 3-manifolds we relate properties of the random walk to geometric invariants (volume).

Lecture four: We explain which of the properties discussed in Lecture 3 can also be verified for random walks on hyperbolic groups with two-sphere boundary. The lecture will end with open questions.

Koji Fujiwara

Title: 1. Quasi-homomorphisms into non-commutative groups.

2. Acylindrically hyperbolic groups are not invariably generated.

Abstract: I will discuss two independent subjects.

1. A function from a group G to integers Z is called a quasi-morphism if there is a constant C such that for all g and h in G, |f(gh)-f(g)-f(h)| < C. Surprisingly, this idea has been useful. I will discuss a recent work with M. Kapovich when we replace the target group from Z to a non-commutative group, for example, a free group.

2. A group G is "invariably generated" , IG, if there is no proper subgroup that meets all conjugacy classes of G. For example finite groups and abelian groups are IG, but non-abelian free groups are not IG. We prove that acylindrically hyperbolic groups are not IG. This is a joint work with M. Bestvina.

Masato Mimura

Title: Synthesis in property (T): The Part and the Whole

Abstract: Kazhdan's property (T) for finitely generated groups is equivalent to the fixed point property with respect to all affine isometric group actions on arbitrary Hilbert spaces. For a class X of Banach spaces, the fixed point property (F_X), relative to X, is defined in a similar manner. This property has been drawing more and more attention in relation to rigidity in various aspects, including superexpanders, an obstruction to the Gromov hyperbolicity, and to certain variants of the coarse Baum--Connes conjecture.

One major way of establishing fixed point properties directly for discrete groups is a, so to speak, ``The Part and the Whole" strategy: first, show relative fixed point properties for a family of well-understood subgroups; and finally, synthesize them into the fixed point property for the whole group. In contrast to that the first step is almost in functional analytic flavor, there has been a variety of approaches to the last step, different in nature.

In this series of talks, I focus on and overview the latter ``Synthesis" step in the strategy above. I may plan to treat the following topics, as long as time permits: - Property (T) and (F_X): examples and applications; - Bounded Generation and synthesis by ``algebraization" (Y. Shalom); - Synthesis by ``almost orthogonality" (M. Ershov, A. Jaikin-Zapirain, et al.); - And some recent progress (myself).

Takayuki Okuda

Title: Splittings of singular fibers into Lefschetz fibers

Abstract: A degeneration of Riemann surfaces is defined as a holomorphic fibration of a complex surface over a disk allowed to have a central singular fiber. There are many examples where such a singular fiber splits into several simpler singular fibers by deforming the fibration. In this talk, we show the splittability of singular fibers with certain periodic monodromies into Lefschetz fibers. Furthermore we determine the associated decompositions of the monodromies, which give us Dehn-twist expressions of periodic mapping classes. This is partially based on a joint work with Shigeru Takamura (Kyoto University).

Takahiro Kitayama

Title: Representation varieties detect splittings of 3-manifolds

Abstract: Culler and Shalen established a method to construct nontrivial actions of a group on trees from ideal points of its SL_2-character variety. The method, in particular, gives essential surfaces in a 3-manifold corresponding to such actions of its fundamental group. Essential surfaces in some 3-manifold are known to be not detected in the classical SL_2-theory. We show that every essential surface in a 3-manifold is given by an ideal point of the SL_n-character variety for some n. The talk is partially based on joint works with Stefan Friedl and Matthias Nagel, and also with Takashi Hara.

Yoshikata Kida

Title: Inner amenable groups, stable actions, and central extensions

Abstract: Inner amenable groups arise from a probability-measure-preserving action whose orbit equivalence relation absorbs the hyperfinite equivalence relation under direct product. I will discuss recent progress around the structure of linear inner amenable groups (due to Tucker-Drob) and connection between inner amenability and existence of group-actions having the absorption property, especially focusing on central group-extensions.

Mitsuaki Kimura

Title: Conjugation-invariant norms on the commutator subgroup of the infinite braid group

Abstract: For a group G, we study the difference between ``G admits a stably unbounded conjugation-invariant norm'' and ``G admits a non-trivial quasimorphism'' (equivalentlly, the commutator length is stably unbounded, i.e. scl is non-trivial).There are few examples of groups which is known that the two conditions above are different. The commutator subgroup of the infinite braid group is such an example, it is shown by Brandenbursky and Kedra.In this talk, we give an another proof of their result by using a ``norm-controlled'' quasimorphism.

Yosuke Morita

Title: A cohomological obstruction to the existence of compact Clifford-Klein forms

Abstract: A Clifford-Klein form is a quotient of a homogeneous space G/H by a discrete subgroup Gamma of G acting properly and freely on G/H. It admits a natural structure of a manifold locally modelled on G/H. There is a natural homomorphism from relative Lie algebra cohomology to de Rham cohomology of a Clifford-Klein form. Relating this homomorphism with an upper-bound estimate for cohomological dimensions of discontinuous groups, we give a new obstruction to the existence of compact Clifford-Klein forms of a given homogeneous space. We obtain some examples of a homogeneous space that does not have a compact Clifford-Klein form, such as the "pseudo-Riemannian sphere" SO_0(p+1, q)/SO_0(p, q) with p, q > 0, q: odd.