Lectures Uri Bader An ergodic method for group linearity The use of ergodic theory for the study of linear groups is a powerful tool. A reason for this is the tension between the chaotic behavior of some discrete group actions and the tameness of algebraic group actions. In my talks I will explain this circle of ideas and derive some applications. I will assume no prior familiarity with neither ergodic theory nor algebraic groups. Kate Juschenko Groups of dynamical origin I will present an introduction on groups of dynamical origin, their algebraic and analytic properties. The properties that I will discuss are amenability, growth, simplicity, finite generacy. Amos Nevo Representation theory, discrete lattice subgroups, effective ergodic theorems, and applications (Slides I Slides II Slides III) Our first goal is to show how aspects of the representation theory of algebraic groups can be utilized to derive effective ergodic theorems for their actions and the actions of their discrete lattice subgroups. Our second goal is to demonstrate some the many applications that ergodic theorems with a rate of convergence have in a variety of problems, including some which appear completely unrelated to dynamics.   We will start by a discussion of property T and show how to extend the spectral estimates it provides considerably beyond their usual formulations, particularly for discrete lattice subgroups. We will then explain how the spectral estimates derived via representation theory can be used to derive effective ergodic theorems first for algebraic groups and then for their discrete lattice subgroups. Finally we will show how the rate of convergence in the ergodic theorem implies effective solutions in a host of natural problems, including non-Euclidean lattice point counting problems, fast equidistribution of lattice orbits on homogenous spaces, and exponents of Diophantine approximation on homogeneous algebraic varieties, as time permits. Talks Tomohiko Ishida Unboundedness of groups of Hamiltonian diffeomorphisms The notion of boundedness of groups was introduced by Burago, Ivanov and Polterovich. In many cases, groups of Hamiltonian diffeomorphisms are unbounded while those of ordinary ones are bounded. In this talk, I will introduce some metric properties of the former groups. Motoko Kato Higher dimensional Thompson groups have Serre's property FA The Thompson group $V$ is a subgroup of the homeomorphism group of the Cantor set $C$. Brin defined higher dimensional Thompson groups $nV$ as generalizations of $V$. $nV$ is a subgroup of the homeomorphism group of $C^n$. In this talk, we prove that $nV$ has property FA, for every $n$. This is a generalization of the corresponding result of Farley, who studied the Thompson group $V = 1V$. Yoshikata Kida Stable actions and central extensions A probability-measure-preserving action of a countable group is called stable if its transformation-groupoid absorbs the ergodic hyperfinite equivalence relation of type $\mathrm{II}_1$ under direct product. Some conditions for groups to have a stable action involve inner amenability and property (T). Among other things, I will discuss a characterization of a central group-extension having a stable action. Mitsuaki Kimura Conjugation-invariant norms on the commutator subgroup of the infinite braid group In 2008, Burago-Ivanov-Polterovich introduced the notion of conjugation-invariant norms. They asked whether there exists a group which admits a stably unbounded norm although the commutator length is stably bounded,and the existence of such group was shown by Kawasaki and Brandenbursky-Kedra. In this talk, we give an another answer of the problem. We show that the commutator subgroup of infnite braid group admits stably unbounded norms by using Kawasaki's method. To prove this, we observe that the signature of braids is a "norm-controlled" quasimorphism. Rostyslav Kravchenko Invariant and characteristic random subgroups We will discuss the notions of invariant and characteristic random subgroups and their connections with dynamical systems and geometric group theory. Pierre Mathieu Deviation inequalities and Central Limit Theorem for random walks on acylindrically hyperbolic groups Deviation inequalities quantify how much successive positions of a random walk deviate from a 'straight line'. In joint work with A. Sisto, we established deviation inequalities for random walks on hyperbolic and acylindrically hyperbolic groups. The Central Limit theorem, as well as the regularity of the rate of escape in terms of the driving measure, follow. Yoshifumi Matsuda Ziggurat and rigidity Ziggurats describe rotation number of compositions of circle homeomorphisms. We show rigidity phenomena for group actions on the circle corresponding to certain points of a ziggurat. Hiroki Matui Topological full groups of etale groupoids I will begin with the definition of topological full groups and explain various properties of them. Especially, I will discuss finiteness properties, abelianization, homology groups of groupoids, connection to K-theory of $\mathrm{C}^*$-algebras, etc. As a typical and important example, I will introduce the topological full groups of one-sided shifts of finite type. They are viewed as generalization of the Higman-Thompson groups. Yosuke Morita A cohomological obstruction to the existence of compact Clifford-Klein forms 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 compact 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 homogneous space that does not have a compact Clifford-Klein form, such as "pseudo-Riemannian sphere" $SO_0(p+1, q)/SO_0(p, q)$ with $p,q \geq 1$, $q$: odd. John Parker A complex hyperbolic Riley slice (Joint work with Pierre Will) In the late 1970s Robert Riley investigated subgroups of $\mathrm{SL}(2,\mathbf{C})$ generated by two parabolic transformations. The conjugacy classes of such groups may be parametrised by one complex number. Riley investigated for which values of this complex number the group is discrete. In our work we investigate subgroups of $\mathrm{SU}(2,1)$ generated by two unipotent parabolic maps whose product is also unipotent. The conjugacy classes of such groups may be parametrised by two real numbers and we investigate values of these parameters where the group is discrete. There are many similarities to Riley's work -- we find a copy of the figure $8$ knot complement, we use Ford domains to show discreteness -- and many differences -- our discreteness region conjecturally has a piecewise smooth boundary. Moreover, a consequence of our work is that we prove a conjecture of Schwartz for complex hyperbolic $(3,3,\infty)$ triangle groups.
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