We discuss the existence of foliations that are invariant under the dynamics for systems that are isotopic to Anosov diffeomorphisms. Specifically, we examine partially hyperbolic diffeomorphisms with one dimensional center that are isotopic to a hyperbolic toral automorphism and contained in a connected component. We show in this case there is a center foliation. We will also discuss more general cases where there is a weak form of hyperbolicity called a dominated splitting. This is joint work with Jerome Buzzi, Rafael Potrie, and Martin Sambarino.

This talk will introduce two well known methods for constructing minimal and not uniquely ergodic interval exchange transformations. Then it will discuss some recent results on minimal and not uniquely ergodic interval exchanges focussing on diophantine properties and connections to geometry. Joint work with J. Athreya, M. Boshernitzan, Y. Cheung, H. Masur, P. Reynolds and M. Wolf.

We consider a negatively curved closed Riemannian manifold. The "stochastic entropy" and the "linear drift" are two numbers which describe how the heat diffuses at infinity on the universal cover in large times. We recall some of their properties and discuss their regularity as functions of the metric.

I will review Riley's construction of a complete real hyperbolic metric on the figure eight knot complement, and explain how it can be modified to show that the figure eight knot complement is also spherical CR uniformizable (i.e. it occurs as the manifold at infinity of a well chosen discrete subgroup of automorphisms of the ball).

This talk will be about complete, nonpositively curved, Riemannian manifolds that are not compact, but do have finite volume. The topology of ends of such manifolds is well understood for the ``extreme cases'' of manifolds of pinched negative curvature and higher rank locally symmetric spaces, but appears to be more mysterious in general. I will describe examples of such manifolds and discuss a recent result showing in the four dimensional case that the end is aspherical. Joint work with T. Nguyen Phan and Y. Wu.

Finite translation surfaces are closed surfaces which carry outside of finitely many cone points an atlas all of whose transition maps are translations. They are obtained by gluing finitely many polygons along parallel edges of the same length via translations. Their natural automorphism group consists of the affine homeomorphisms of the surface with trivial derivative, i.e. which are translations. The number of translations of a translation surface of genus g is naturally bounded by the Riemann-Hurwitz formula. We study for which g the bound is achieved.

It is obvious that the usual Euclidean norm is strictly convex, by which we mean that, for all x and all nonzero y, either |x + y| > |x|, or |x - y| > |x|. We will discuss the existence of such a norm on an abstract (countable) group G. A sufficient condition is the existence of a faithful action of G by orientation-preserving homeomorphisms of the real line. No examples are known to show that this is not a necessary condition, and we will combine some elementary measure theory and dynamics with the theory of orderable groups to show that the condition is indeed necessary if G is amenable. This is joint work with Peter Linnell of Virginia Tech.

Some hyperbolic metrics are smooth, and some are not. We extend Wolf's parametrization of Teichmüller space by harmonic maps to the space of hyperbolic surfaces with cone singularities. These surfaces give rise to representations into the non-Hitchin components of the representation space Hom(π_1(M), PSL(2,R)).

In this talk, I will first present joint work with Piotr Przytycki and Richard Webb giving a new short proof of uniform hyperbolicity of curves and arc graphs. Namely, I will describe unicorn paths in arc and curve graphs and show that they form 1-slim triangles. Using this, one can deduce that arc graphs are 7-hyperbolic (and curve graphs are 17-hyperbolic) I will then overview some other results which, in a similar vein, give quick and purely topological-combinatorial proofs of curve graph results. If time permits, I will explain how such proofs can sometimes be adapted to work in the Out(F_n) setting and allow new insights there.

The space of pairs (Riemann surface, holomorphic 1-form) admits an action of the group SL(2,R). This generalizes the genus one case of the action on the tangent bundle of the modular surface. This action is intimately related to more classical dynamical systems, such as billiards in polygons and interval exchanges. Many questions about these systems can be addressed using the SL(2,R) action. Eskin and Mirzakhani recently proved a number of rigidity properties of SL(2,R)-invariant measures. In particular, these are of Lebesgue-class on complex manifolds. In this talk I will explain the proof that these manifolds are in fact algebraic varieties, with interesting arithmetic properties. I will begin by introducing some motivating examples from Teichmuller dynamics. I will then introduce the necessary concepts from algebraic geometry. No background in either subject will be assumed.

The stable ratio set of a nonsingular action is a notion introduced by Bowen and Nevo to prove pointwise ergodic theorems for measure preserving actions of certain nonamenable groups. I prove that stable ratio set of the action of a discrete subgroup of isometries of a CAT(-1) space with finite Bowen-Margulis measure on its boundary with the Patterson-Sullivan measure has numbers other than 0, 1 and \infinity, extending techniques of Bowen from the setting of hyperbolic groups. I also prove the same result for the action of the mapping class group on the sphere of projective measured foliations with the Thurston measure, using some "statistical hyperbolicity" properties for the (non hyperbolic) Teichmueller metric on Teichmueller space.

Asymptotic dimension of a metric space is a large scale analog of the usual covering dimension. I will talk about a proof that the curve complex of a surface has asymptotic dimension bounded by a certain linear function of the complexity of the surface. Previously, the known bound was exponential. This is joint work with Ken Bromberg.

A famous theorem of S. Mozes implies that a mixing action of a semisimple group on a probability space $X$ is mixing of all orders. Prompted by a question of D. Dolgopyat, we have studied the quantitative behavior of multiple correlation integrals and provided explicit exponential rates of decay for the Weyl chamber action of higher rank groups. In this talk, we will discuss these results and their significance, the major techniques we used (especially the use of semi-direct products, which provides a gateway for similar results in much more general settings) and applications to equidistribution and the search for limit laws.

A quasi-Fuchsian group is a discrete group of Mobius transformations of the Riemann sphere which is isomorphic to the fundamental group of a compact surface and acts properly on the complement of a Jordan curve: the limit set. In 1979, Bowen proved a remarkable rigidity theorem on the Hausdorff dimension of the limit set of a quasi-Fuchsian group: it is equal to 1 if and only if the limit set is a round circle. This theorem now has many generalizations. We will present a new proof of Bowen's result as a by-product of a new lower bound on the Hausdorff dimension of the limit set of a quasi-Fuchsian group. This lower bound is in terms of the differential geometric data of an immersed, incompressible minimal surface in the quotient manifold. If time permits, generalizations of this result to other convex-co-compact surface groups will be presented.

Kerckhoff's solution to the Nielsen realization problem showed that the action of any finite subgroup of the mapping class group on Teichmüller space has a fixed point. The set of fixed points is a totally geodesic submanifold. We study the coarse geometry of the set of points which have bounded diameter orbits in the Teichmüller metric. We show that each such almost-fixed point is within a uniformly bounded distance of the fixed point set, but that the set of almost-fixed points is not quasiconvex. In addition, the orbit of any point is shown to have a fixed barycenter. In this talk, I will discuss the machinery and ideas used in the proofs of these theorems.

Abstract

An invariant random subgroup (IRS for short) of a countable group G is a conjugation invariant probability measure on the (compact metric) space of all subgroups of G.
Theorem: Let G < \GL_n(F) be a countable non-amenable linear group with a simple center free Zariski closure. Then.
1. There is a topology on G such that for every IRS, almost every non-trivial subgroup of G is open.
2. G contains a dense free subgroup F in this topology.
3. For every IRS, the map taking a subgroup H < G to its intersection with F becomes an F-invariant isomorphism of probability spaces.
We say that an action of a group G on a probability space is a.s.n.f if almost all point stabilizers are non-trivial.
Corollary: Let G as above acts on two probability spaces X and Y. If both actions are a.s.n.f then so is the diagonal action on the product.

I’ll explain why many finite covers of SL(m,Z)\SL(m,R)/SO(m) have non-trivial homology classes generated by totally geodesic flat (m-1)-tori. This is joint work with Tam Nguyen Phan.

Let N = X / H be the quotient of a product of symmetric spaces and buildings by a cocompact S-arithmetic lattice in a reductive group G. Does H have a generating set whose associated geodesics on N are, say, bounded above by a polynomial in vol(N) depending only on dim(X)? Conjectures in number theory imply the answer is no even in the case where H is the unit group of a number field k (i.e., G = GL_1 and S is the archimedean places). I will talk about joint work with Ted Chinburg on the case where G is the multiplicative group of a division algebra D over a number field k, generalizing work of H. W. Lenstra when D = k, saying that, once S is sufficiently large, the answer is actually yes. Even stronger, we prove the existence of a matrix representation where the generators have `small' entries.

A typical example of a residually free group is a subgroup of a direct product of surface groups. We are interested in how the rank of homology and the cellular volume of classifying spaces grow as one passes to subgroups of increasing finite index in a fixed group of geometric interest.
I shall introduce an invariant that measures the number of cells that one needs in the k-skeleta of classifying spaces for a tower of finite-index subgroups in a group with a finite K(G,1), explaining a connection to L^2 invariants. I shall then use the structure theory of residually-free groups to calculate asymptotic volume-invariants and L^2 betti numbers for such groups. This is joint work with Desi Kochloukova (Campinas).

Over the past years CAT(0) cube complexes have played a major role in geometric group theory and have provided many examples of interesting group actions on CAT(0) spaces. In the search for highly symmetric CAT(0) cube complexes -- just as for their 1-dimensional analogues, trees -- it is natural to consider the sub-class of "regular" CAT(0) cube complexes, i.e., cube complexes with the same link at each vertex. However, unlike regular trees, general regular CAT(0) cube complexes are not necessarily uniquely determined by their links. In this talk, we will discuss a necessary and sufficient condition for uniqueness. We will then explore some examples of unique regular cube complexes and the properties of their automorphism groups.

We will investigate a dynamical system that comes from geodesic trajectories on flat surfaces. We will start with known results for the square torus and the double pentagon surface, and then discuss new results for Bouw-Möller surfaces, made from many polygons.