Institute for Computational and Experimental Research in Mathematics (ICERM)

Abstract

The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasi-conformal deformation, and the gluing constructions of non-arithmetic lattices by Gromov and Piatetski-Shapiro.

• Video

Abstract

In this Mini Course, following a joint work with E.Ullmo, I will explain how (integral) Hodge theory naturally comes up in the study of totally geodesic subvarieties of a complex hyperbolic ball quotient S. From such a point of view, the finiteness of the maximal totally geodesics of S becomes a consequence of a very general conjecture about 'unlikely intersections'. The two lectures will give some motivations and an introduction to such techniques (no prior knowledge in Hodge theory will be assumed).

Abstract

Anosov representations provides a rich class of discrete embeddings of hyperbolic groups into semisimple Lie groups, which generalizes the classes of convex cocompact subgroups to the setting of higher rank Lie groups. In this talk I will discuss some results (old and new) regarding geometric, dynamical and arithmetic properties.

• Video

Abstract

The aim of this minicourse is to introduce fundamental concepts in hyperbolic geometry, such as limit sets, geometric finiteness, and critical exponent, and Mostow rigidity. We will discuss examples of arithmetic hyperbolic manifolds, and illustrate flexible geometric constructions like Dehn filling, quasi-conformal deformation, and the gluing constructions of non-arithmetic lattices by Gromov and Piatetski-Shapiro.

• Video

Abstract

The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic n-manifolds of Fuchsian ends which was proved by McMullen-Mohammadi-O. (for n=3) and by Minju Lee-O. (for n>3).

• Video
• Slides

Abstract

In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–Piatetski-Shapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this mini-course will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting.

• Video

Abstract

The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic n-manifolds of Fuchsian ends which was proved by McMullen-Mohammadi-O. (for n=3) and by Minju Lee-O. (for n>3).

• Video
• Slides

Abstract

In this Mini Course, following a joint work with E.Ullmo, I will explain how (integral) Hodge theory naturally comes up in the study of totally geodesic subvarieties of a complex hyperbolic ball quotient S. From such a point of view, the finiteness of the maximal totally geodesics of S becomes a consequence of a very general conjecture about 'unlikely intersections'. The two lectures will give some motivations and an introduction to such techniques (no prior knowledge in Hodge theory will be assumed).

Abstract

The frame bundle of an n-dimensional hyperbolic manifold X is the homogeneous space Γ\SO(n, 1)° for some discrete subgroup Γ and the frame flow is given by the right translation action by a one-parameter diagonalizable subgroup. We assume that Γ is Zariski dense and X is geometrically finite, i.e., it need not be compact but has at most finitely many ends consisting of cusps and funnels. We endow the frame bundle with the unique probability measure of maximal entropy called the Bowen-Margulis-Sullivan measure. In a joint work with Jialun Li and Wenyu Pan, we prove that the frame flow is exponentially mixing. The proof uses a countably infinite coding and the latest version of Dolgopyat's method. To overcome the difficulty in applying Dolgopyat's method due to the cusps of non-maximal rank, we prove a large deviation property for symbolic recurrence to certain large subsets of the limit set of Γ.

• Video
• Slides

Abstract

We will discuss the notion of length functions on groups, focusing on lattices in Lie groups of higher rank, and discuss how dynamics and ergodic theory can help to understand some questions about them.

Abstract

The aim of this mini course is to discuss some basic theory of homogeneous dynamics on the quotient of SO(n,1) by a discrete subgroup and to explain the topological rigidity of geodesic planes in hyperbolic n-manifolds of Fuchsian ends which was proved by McMullen-Mohammadi-O. (for n=3) and by Minju Lee-O. (for n>3).

• Video
• Slides

Abstract

"In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–Piatetski-Shapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this mini-course will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting."

• Video

Abstract

Let G be a connected semisimple real algebraic group and D be its Zariski dense discrete subgroup. We prove that if D\G admits any finite Bowen-Margulis-Sullivan measure, then D is virtually a product of higher rank lattices and discrete subgroups of rank one factors of G. This may be viewed as a measure-theoretic analogue of classification of convex cocompact actions by Kleiner-Leeb and Quint, which was conjectured by Corlette in 1994. This is joint work with Mikolaj Fraczyk. We will then discuss its application on the bottom of the L^2 spectrum, in joint work with Samuel Edwards, Mikolaj Fraczyk and Hee Oh.

• Video

Abstract

We will survey new results and open problems on triangle groups in SL_2(R), and their connections to Abelian varieties, Teichmueller curves and billiards in polygons.

• Video
• Slides

Abstract

As a companion and complement to the mini-course on rigidity of geodesic planes in infinite volume hyperbolic manifolds, I will discuss several examples of rigidity and nonrigidity of geodesic planes in hyperbolic 3-manifolds. These results are orthogonal to the ones obtained by McMullen, Mohammadi and Oh, and illustrate what can happen if some of their assumptions are relaxed. Some of these examples were produced by explicit computations and can be visualized very nicely, and I will discuss these aspects as well.

• Video

Abstract

In the 1970s seminal work of Margulis showed that higher rank lattices have superrigid representations, which in particular implies that all such lattices are arithmetic. Since then Gromov–Piatetski-Shapiro and Deligne–Mostow have shown that a similar superrigidity theorem cannot hold for all lattices in the isometry group of real or complex hyperbolic space, i.e., in the rank 1 setting. However in recent work joint with Bader, Fisher, and Stover we show that one can prove certain superrigidity/arithmeticity theorems provided the associated manifold satisfies the geometric condition that it contains infinitely many (maximal) totally geodesic submanifolds. Specifically, we show that this latter criteria forces a hyperbolic manifold to be arithmetic.
The goal of this mini-course will be first to recount the work of Margulis on superrigidity of higher rank lattices and then to go on to discuss the proof of the aforementioned theorem. This will include a discussion of the connections between homogeneous dynamics and geodesic submanifolds, their interaction with superrigidity, and an introduction to techniques introduced by Bader and Furman for studying algebraic representations of ergodic actions. If time permits, we will also discuss the analogous theorem for complex hyperbolic manifolds and the key differences from the real hyperbolic setting.

• Video

Abstract

Strongly convergent sequences of hyperbolic manifolds arise naturally in the study of Kleinian group representations, for example, the Dehn surgeries on hyperbolic knots. It turns out that such sequences usually have uniform control on the geometry and dynamics, such as the uniform convergence of small eigenvalues of the Laplacian, and the Patterson-Sullivan measures. We will talk about the uniform convergence results in this talk and apply them to count uniformly along the sequence the number of simple closed geodesics and orthogeodesics. This is joint work with Franco Vargas Pallete.

• Video

• Video

Abstract

In this talk, I will discuss the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I talk about the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.

• Video

Abstract

As a companion and complement to the mini-course on rigidity of geodesic planes in infinite volume hyperbolic manifolds, I will discuss several examples of rigidity and nonrigidity of geodesic planes in hyperbolic 3-manifolds. These results are orthogonal to the ones obtained by McMullen, Mohammadi and Oh, and illustrate what can happen if some of their assumptions are relaxed. Some of these examples were produced by explicit computations and can be visualized very nicely, and I will discuss these aspects as well.

• Video

Abstract

We show that for a C^2 IFS on R, either up to smooth conjugacy the IFS has vanishing second derivative on its attractor, or the self-conformal measure has polynomial decay of Fourier coefficients. A key argument is a cocycle version of Dolgopyat's method and resulting spectral gap-type estimates and renewal theorem. This is a joint work with Amir Algom and Federico Rodriguez Hertz.

• Video

Abstract

I will describe joint work with Domingo Toledo on residual finiteness for cyclic central extensions of fundamental groups of aspherical manifolds, its application to central extensions of (arithmetic) real and complex hyperbolic lattices, and connections to several open problems.

• Video

Abstract

The cost of a probability measure preserving action of a countable group G on X is an invariant that generalizes the rank (minimal number of generators) of G and measures the “minimal average number of maps” needed to connect every pair of points of X in the same G orbit. The fixed price conjecture predicts that any two essentially free p.m.p. actions of the group G have the same cost. In my talk I will report on a joint work with Sam Mellick and Amanda Wilkens in which we prove fixed price one for higher ranks lattices in semisimple real or p-adic groups. As a corollary we obtain that the number of generators of index n subgroup of such a group grows like o(n) which implies new state of the art results on the growth of mod-p homology groups. The proof is based on certain miraculous properties of the Poisson-Voronoi tessellation of higher rank symmetric spaces (not present in rank 1) that might be of independent interest.

• Video

Abstract

Bianchi subgroups are cofinite non-cocompact lattices in PSL_2(C), defined by \Gamma_d = PSL_2(O_d), where O_d is the ring of integers of the imaginary quadratic field of discriminant -d. The Bianchi orbifold \Omega_d = \Gamma_d\H^3 is known to contain infinitely many immersed totally geodesic surfaces, which can be identified with integral binary hermitian forms over O_d. In this talk, I will show that some of these immersed totally geodesic surfaces are in fact orientable embedded closed totally geodesic surfaces using number theoretic ideas. I then present some numerical data concerning these surfaces. This talk is based on a joint work with Alan Reid and on an ongoing project with Sam Kim and James Rickards.

• Video