# Groups, Dynamics and Related Topics[ Edit ]

## Moderator: Uri Shapira

*Abstract:*

joint with Yair Hartman, Kate Juschenko and Pooya Vahidi-Ferdowsi.

The notion of a proximal topological action was introduced by Glasner in the 1970's, together with the related notion of a strongly amenable group. Only a handful of new insights have been gained since then, and much remains mysterious. For example, it is known that all virtually nilpotent groups are strongly amenable, but it is not known if all strongly amenable groups are virtually nilpotent (within the class of discrete groups). We will introduce the definitions, survey what is known, and show that Thompson's infamous group F is not strongly amenable.

*Abstract:*

In the theory of Diophantine approximations, singular points are ones for which Dirichlet’s theorem can be infinitely improved. It is easy to see that all rational points are singular. In the special case of dimension one, the only singular points are the rational ones. In higher dimensions, points lying on a rational hyperplane are also obviously singular. However, in this case there are additional singular points. In the dynamical setting the singular points are related to divergent trajectories. In the talk I will define obvious divergent trajectories and explain the relation to rational points. In addition, I will present the more general setting involving Q-algebraic groups. Lastly I will discuss results concerning classification of divergent trajectories in Q-algebraic groups.

*Abstract:*

Continued fraction expansion (CFE) is a presentation of numbers which is closely related to Diophantine approximation and other number theoretic concepts. It is well known that for almost every x in (0,1), the coefficients appearing in the CFE of x obey the Gauss-Kuzmin statistics. This claim is not true for all x, and in particular it is not true for rational numbers which have finite CFE. In order to still have some statistical law, we instead group together the rationals p/q in (0,1) for q fixed and (p,q)=1 and ask whether their combined statistics converges as q goes to infinity. In this talk I will show how this equidistribution problem can be reformulated and solved using the language of dynamics of lattices in SL_2(Z)\SL_2(R) (and given time, how it extends naturally to the Adelic setting). This will in turn imply a stronger equidistribution of the CFE of rational numbers. This is a joint work with Uri Shapira.

*Abstract:*

The spectral gap conjecture for compact semisimple Lie groups stipulates that any adapted random walk on such a group equidistributes at exponential speed. In the first part of the talk, we shall review results of Bourgain and Gamburd, which relate this conjecture to diophantine properties of subgroups in Lie groups. Then, we shall study this diophantine problem in nilpotent Lie groups.

*Abstract:*

A geodesic conjugacy between two Riemannian manifolds is a diffeomorphism of the unit tangent bundles which commutes with the respective geodesic flows. A natural question to ask is whether a conjugacy determines a manifold up to isometry. In this talk we shall briefly explain the development of the geodesic conjugacy problem and describe some recent results.

*Abstract:*

Euclidean tilings, and especially quasiperiodic ones, such as Penrose tilings, are not only beautiful but crucially important in crystallography. A very powerful tool to study such tilings is cohomology. In order to define it, the first approach is to define a metric on the set of tilings and then define the hull of a tiling as the closure of its orbit under translations. The cohomology of a tiling is then defined as the Cech cohomology of its hull. A more direct (and recent) definition involves treating a tiling as a CW-structure and considering the "pattern-equivariant" subcomplex of the cellular cochain complex. These two definitions yield isomorphic results (J. Kellendonk, 2002) We'll also see some applications of tiling cohomology to the study of shape deformations, and compute some examples.

*Abstract:*

We will study n-dimensional badly approximable points on curves. Given an analytic non-degenerate curve in R^n, we will show that any countable intersection of the sets of weighted badly approximable points on the curve has full Hausdorff dimension. This strengthens a previous result of Beresnevich by removing the condition on weights. Compared with the work of Beresnevich, we study the problem through homogeneous dynamics. It turns out that the problem is closely related to the study of distribution of long pieces of unipotent orbits in homogeneous spaces.

*Abstract:*

Abstract: We provide explicit Diophantine conditions on the coefficients of degree 2 polynomials under which the limit of an averaged pair correlation density is consistent with the Poisson distribution, using a recent effective Ratner equidistribution result on the space of affine lattices due to Strömbergsson. This is joint work with Jens Marklof.

*Abstract:*

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

*Abstract:*

Let $b$ be a positive integer larger than or equal to two. A real number $x$ is called normal to base $b$, if in its base-$b$ expansion all finite blocks of digits occur with the expected frequency. Equivalently, $x$ is normal to base $b$ if the orbit of $x$ under the multiplication-by-$b$ map is uniformly distributed in the unit interval with respect to Lebesgue measure.While there are many explicit constructions of normal numbers to a single base it remains an open problem going back to Borel in 1909 to exhibit an easy example of an absolutely normal number (i.e. a real number that is normal to all integer bases simultaneously). In this talk I will explain algorithms by Sierpinski and Becher-Heiber-Slaman that produce absolutely normal numbers one digit after the other. I will show how these algorithms can be extended to give computable constructions of absolutely normal numbers that also have a normal continued fraction expansion, or are normal with respect to expansions to non-integer bases. Some ideas from ergodic theory will occur, but the proofs are based on large deviation theorems from probability theory for sums of dependent random variables. This allows to make certain constants implied by the Shannon-McMillan-Breimann theorem in special cases explicit so we can in fact avoid ergodic theory. If time permits, I will also say something about the trade-off between time-complexity and speed of convergence to normality for normal numbers.

*Abstract:*

I will discuss a convolution operator associated with Harish-Chandra’s Schwartz space of discrete groups of any semisimple Lie group. I will show that the latter space carries a natural structure of convolution algebra. Besides, a control of the l^2 convolutor norm by the norm of this space holds. I will explain how this inequality is related to property RD and I will make a connection with the Baum-Connes conjecture.

*Abstract:*

We consider the orbits {pu(n^{1+r})} in Γ∖PSL(2,R), where r>0, Γ is a non-uniform lattice in PSL(2,R) and u(t) is the standard unipotent group in PSL(2,R). Under a Diophantine condition on the intial point p, we prove that such an orbit is equidistributed in Γ∖PSL(2,R) for small r>0, which generalizes a result of Venkatesh. Also we generalize this Diophantine condition to any finite-volume homogeneous space G/Γ, and compute Hausdorff dimensions of Diophantine points of various types in a rank one homogeneous space G/Γ. In particular, this gives a Jarnik-Besicovitch theorem on Diophantine approximation in Heisenberg groups.

*Abstract:*

For the abstract see the attached .pdf

*Abstract:*

A lattice is topologically locally rigid (t.l.r) if small deformations of it are isomorphic lattices. Uniform lattices in Lie groups were shown to be t.l.r by Weil [60']. We show that uniform lattices are t.l.r in any compactly generated topological group.

A lattice is locally rigid (l.r) is small deformations arise from conjugation. It is a classical fact due to Weil [62'] that lattices in semi-simple Lie groups are l.r. Relying on our t.l.r results and on recent work by Caprace-Monod we prove l.r for uniform lattices in the isometry groups of proper geodesically complete CAT(0) spaces, with the exception of SL_2(\R) factors which occurs already in the classical case.

Moreover we are able to extend certain finiteness results due to Wang to this more general context of CAT(0) groups.

In the talk I will explain the above notions and results, and present some ideas from the proofs.

This is a joint work with Tsachik Gelander.

*Abstract:*

Manifolds of negative sectional curvature are an object of interest and their study goes back to Cartan and Hadamard.

It is well known that the topology of such manifolds is controlled, to some extent, by their volume. This is best illustrated in dimension 2: the homemorphism type of a compact orientable surface is determined by its volume (suitably normalized) - this follows from the celebrated Gauss-Bonnet theorem. Gromov proved in 1978 that the Betti numbers of negatively curved manifolds are bounded by means of the volume in every dimension, but also provided an example of a sequence of negatively curved 3-manifolds of uniformly bounded volume and pairwise different first integral homology. A crucial tool in Gromov's proof is the famous "thin-thick decomposition" of a manifold.

In my talk I will report on a joint work with Gelander and Sauer, in which we introduce a modification of this decomposition that gives a better model for the topology of a manifold: a negatively curved manifold is homotopic to a simplicial complex with handles, where the number of simplices is bounded by means of the volume of the manifold. This shows in particular that Gromov's 3d example could not be given in higher dimensions and that in dimension 5 and more the number of homeomorphism types of manifolds is bounded by means of the volume.

*Abstract:*

We present a new approach (joint with M. Bjorklund (Chalmers)) for finding new patterns in difference sets E-E, where E has a positive density in Z^d, through measure rigidity of associated action.

By use of measure rigidity results of Bourgain-Furman-Lindenstrauss-Mozes and Benoist-Quint for algebraic actions on homogeneous spaces, we prove that for every set E of positive density inside traceless square matrices with integer values, there exists positive k such that the set of characteristic polynomials of matrices in E - E contains ALL characteristic polynomials of traceless matrices divisible by k.

By use of this approach Bjorklund and Bulinski (Sydney), recently showed that for any quadratic form Q in d variables (d >=3) of a mixed signature, and any set E in Z^d of positive density the set Q(E-E) contains kZ for some positive k. Another corollary of our approach is the following result due to Bjorklund-Bulinski-Fish: the discriminants D = {xy-z^2 , x,y,z in B} over a Bohr-zero non-periodic set B covers all the integers.

*Abstract:*

The subject of harmonic analysis on Lie groups is well studied but can be rather opaque for non-experts. For the Heiseberg Lie group, or more specifically its Lie algebra, there exists the so-called Weyl transform: a linear map that allows one to define functions on the Lie algebra in a straightforward manner. However abstract the original Lie algebraic definitions might be, it will be shown that all objects of interest can be brought into the form of explicit orthogonal function expansions on concrete spaces. The focus of this talk will be to describe a short path from foundational principles to a kind of noncommutative polar coordinates on the Heisenberg Lie algebra, during which many interesting connections to spectral and representation theory will be manifest.

*Abstract:*

In the paper ``Formal noncommutative symplectic geometry'', Maxim Kontsevich introduced three versions of cochain complexes $\GC_{\Com}$, $\GC_{\Lie}$ and $\GC_{\As}$ ``assembled from'' graphs with some additional structures. The graph complex $\GC_{\Com}$ (resp. $\GC_{\Lie}$, $\GC_{\As}$) is related to the operad $\Com$ (resp. $\Lie$, $\As$) governing commutative (resp. Lie, associative) algebras. Although the graphs complexes $\GC_{\Com}$, $\GC_{\Lie}$ and $\GC_{\As}$ (and their generalizations) are easy to define, it is hard to get very much information about their cohomology spaces. In my talk, I will describe the links between these graph complexes (and their modifications) to the cohomology of the moduli spaces of curves, the group of outer automorphisms $\Out(F_r)$ of the free group $F_r$ on $r$ generators, the absolute Galois group $\Gal(\overline{\bbQ}/\bbQ)$ of rationals, finite type invariants of tangles, and the homotopy groups of embedding spaces.

*Abstract:*

The Out(G)-action on the group cohomology H^n(G) of a group G is an important object of study in group theory. On the contrary, almost nothing is known about the corresponding Out(G)-action on the bounded group cohomology H^n_b(G). This talk will introduce bounded group cohomology and then look at the case of G=F_2 and n=2. There the dynamics of the unipotent elements in Out(F_2) on a dense subset C(F_2) of H^2_b(F_2) will be presented concretely and visualized. In particular we will show that no element of C(F_2) is fixed by the Out(F_2)-action, partly answering a question of Miklós Abért.

*Abstract:*

I will discuss a work in progress on a problem which lies in the intersection of Diophantine approximation and Geometry of Numbers. The solution involves homogeneous dynamics. Here is a brief intro to the simplest instance of the problem:

A 2-dimensional grid is a set of the form L + v where L is a lattice in R^2 and v is a vector. A grid is called t-bad if for any x = (x_1,x_2) in it |x_1x_2|>t. It is known that for any given lattice L the set {v : L+v is t-bad for some t>0} is 2-dimensional (i.e. has maximal possible dimension).

Can it happen that for for a fixed t>0 the set {v : L+v is t-bad} has dimension 2? The answer is yes but it is very rare.

*Abstract:*

Let A be a finite-dimensional Lie algebra of vector fields on R^n which contains vector fields \partial /\partial x_i+h.o.t , i = 1,...,n (such algebras are called transitive) and let I = {V\in A: V(0)=0} be the isotropy subalgebra of A. The linear approximations at 0 of the vector fields of I form a Lie algebra j^1I. Assume that dim I = dim j^1I so that j^1I is a faithful representation of I in gl(n). Under which condition are I and j^1I diffeomorphic, i.e. can be sent one to the other by a local diffeomorphism of R^n? I will discuss this question from various points of view and will formulate and explain some unexpected theorems, for example that I and j^1 I are diffeomorphic if dim I = dim j^1\I = 1, without any restrictions on the eigenvalues of a vector field which span I.

*Abstract:*

In joint work with David Simmons, we show that the set of badly approximable vectors in R^d, are a measure zero set with respect to the natural self-similar measures on sufficiently regular fractals, such as the Koch snowflake or Sierpinski gasket. The proof uses a classification result for stationary measures on homogeneous spaces, extending work of Benoist and Quint. I will try to give an outline of the proof in the simplest case.

*Abstract:*

The Mozes-Shah theorem states that the weak star limit of algebraic measuresof semi-simple groups without compact factors is again an algebraic measure.Work of Einsiedler, Margulis and Venkatesh quantifies this result, describing howwell a closed orbit of a subgroup is equidistributed in an ambient homogeneous space.In joint work with Einsiedler and Wirth, we consider a special situation in the S-adic world to solve a remaining case in the problem on joint equidistribution of primitive points on spheres and their orthogonal lattices initiated by Shapira.

*Abstract:*

In this talk I will describe some new arithmetic invariants for pairs of torus orbits on inner forms of PGLn and SLn. These invariants generalize a work of Linnik in rank one and allow us to significantly strengthen results towards the equidistribution of packets of periodic torus orbits on higher rank S-arithmetic quotients. An important aspect of our method is that it applies to packets of periodic orbits of maximal tori which are only partially split.

Packets of periodic torus orbits are natural collections of torus orbits coming from a single rational adelic torus and are closely related to class groups of number fields. This is a generalization due to Einsiedler, Lindenstrauss, Michel and Venkatesh of the natural grouping of periodic geodesics and Hecke points on the modular surface by their discriminant.

A novel aspect of our method is that we are able to utilize the action of the Galois group of the splitting field of the torus.

*Abstract:*

``Topological structures'' associated to a topological dynamical system are recently developed tools in topological dynamics. They have several applications, including the characterization of topological dynamical systems, computing automorphisms groups and even the pointwise convergence of some averages. In this talk I will discuss some developments of this subject,emphasizing applications to the pointwise convergence of some averages.

*Abstract:*

Quasi-isometric embeddings is the key feature that we look for we study geometry of spaces on large scales. Generally, there is nothing much we can say about embeddings. But when spaces are symmetric spaces of non-compact type or lattices, we can say a lot more. I will discuss about examples and rigidity phenomenon of embeddings between symmetric spaces and lattices. Part of the talk is from a joint work with David Fisher.

*Abstract:*

The relation between continued fraction expansions and the geodesic flow on the quotient space SL_2(R)/SL_2(Z) is well studied and understood and dates back to Artin. In this talk we will discuss its positive characteristic analogue, for its similarities to the real case, and its surprising differences. Time permitting, we shall discuss some recent results.

*Abstract:*

In a recent publication, R. Willett, E. Guentner, and G. Yu develop a new concept which predicates connections between asymptotic dimension, amenable actions, transformation groupoids, nuclear dimension of C*-algebras and more. Our aim in giving these two talks is to go over what we think are the main points of this paper and to provide the proper context for these results.

For the most part, the talks are independent of each other.

*Abstract:*

A rational function defined over the rationals has only finitely many rational preperiodic points by Northcott's classical theorem. These points describe a finite directed graph (with arrows connecting between each preperiodic point and its image under the function). We give a classification, up to a conjecture, of all possible graphs of quadratic rational functions with a rational periodic critical point. This generalizes the classification of such graphs for quadratic polynomials over the rationals by Poonen (1998). This is a joint work with Jung Kyu Canci (Universität Basel).

*Abstract:*

I will discuss the mean square of sums of the generalised divisor function over arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q tends to infinity we establish a relationship with a matrix integral over the unitary group, and analyse the integral. This is a joint work with Jon Keating, Brad Rodgers and Zeev Rudnick.

*Abstract:*

Abstract is attached.

*Abstract:*

It is well known that for actions of amenable groups entropy is monotone decreasing under factor maps. In this talk, I will show that this fails in a very strong way for actions of non-amenable groups. Specifically, if G is a countable non-amenable group then there exists a finite integer n with the following property: for every pmp action of G on (X, \mu) there is a G-invariant probability measure \nu on n^G such that the action of G on (n^G, \nu) factors onto the action of G on (X, \mu). For many non-amenable groups, n can be chosen to be 4 or smaller. We also obtain a similar result for continuous actions on compact metric spaces and continuous factor maps.

*Abstract:*

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Is there a point set Y in R^d, and C>0, such that every convex set of volume 1 contains at least one point of Y and at most C? This discrete geometry problem was posed by Gowers in 2000, and it is a special case of an open problem posed by Danzer in 1965. I will present two proofs that answers Gowers' question with a NO. The first approach is dynamical; we introduce a dynamical system and classify its minimal subsystems. This classification in particular yields the negative answer to Gowers' question. The second proof is direct and it has nice applications in combinatorics. The talk will be accessible to a general audience. [This is a joint work with Omri Solan and Barak Weiss].

--- Note the special date and time! ---

*Abstract:*

We will start by recalling the definition of diffraction of a quasi-crystal explained in last week’s talk. The majority of the talk is then devoted to the question how to construct explicit examples of quasi-crystals with pure point diffraction.

We will introduce cut-and-project schemes and define the notion of a model set. It turns out that model sets are examples of quasi-crystals with “mostly” pure-point diffraction, and in some sense they are the only such examples (Meyer's theorem).

Using the dynamical system on the hull explained in last week’s talk, we will derive an explicit formula for the diffraction of a regular model set. The key ingredient is Schlottmann’s torus parametrisation, which provides a measurable isomorphism between the dynamical system on the hull of a regular model set and an almost homogeneous system, which in the case of R^n is simply an irrational rotation on a torus.

Time permitting we will discuss possible generalizations to non-commutative (and in particular arithmetic) quasi-crystals in the sense of our recent work with Björklund and Pogorzelski.

*Abstract:*

In 1982, the Technion physicist Dan Shechtman verified the existence of physical quasicrystals via diffraction experiments - an observation for which he was awarded the Nobel prize in chemistry in 2011.In the past decades, mathematical diffraction theory evolved into a beautiful and rich research topiccombining various disciplines such as functional analysis, fourier analysis and ergodic theory.

This talk introduces the notion of autocorrelation for Delone sets in locally compact groups in terms of dynamical systems.A formula involving the Siegel transform is derived. To draw the connection to the classical theory, we stick to theframework of abelian groups giving rise to a uniquely ergodic dynamical system. Using the uniform ergodic theorem,it is shown that the (unique) autocorrelation measure coincides with the classical notion via a Folner limit ofnormalized difference Dirac combs. We conclude the talk by defining the diffraction measure as the Fouriertransform of the autocorrelation measure.

*Abstract:*

The geometry of complex hyperbolic space has been well studied from several viewpoint and using tools from several branches of mathematics. For its quaternionic counterpart on the other hand, much less is known.

In an upcoming series of seminars, we will study complex and quaternionic hyperbolic spaces from several perspectives and see which tools from the complex setting generalize to the quaternionic.

In this talk I will give you a glance of what is to come in the series and present some recent results, joint with T. Hartnick, about a potential for the invariant four form of quaternionic hyperbolic space.