Monday, 27 February 2023

Let X be a nice topological space with fundamental group G. Using the correspondence between N-degree covering spaces of X and homomorphisms f:G-->S_N to the symmetric group, we may study random covers of X by analyzing random homomorphisms f:G-->S_N. In particular, we consider the random permutation f(g) given by the image of a fixed element g in G.

The mini-course will focus on random covers of hyperbolic surfaces - so G here is a surface group. I will describe a joint work with Michael Magee in which we develop a method to study the local statistics of the random permutation f(g), such as its number of fixed points, and derive several results which are analogous to well-known results when G is replaced with a free group.

I'll motivate the question, introduce the results, give a flavor of the techniques - both in representation theory and in combinatorial group theory - and present some open problems.

Let M be a compact interval or a circle. We study algebraic properties of Diff(M;r), the group of Cr diffeomorphisms of M. In particular, we focus on the following question: which finitely generated groups arise as subgroups of Diff(M;r)? We survey classical theories of Hölder, Denjoy, Kopell and Thurston regarding this matter, and give a detailed dynamical account of a finitely generated subgroup of Diff(M;r) that does not admit any injective homomorphism into Diff(M;s) for all s>r. The heart of the proof is a probabilistic argument that controls the dynamics of one-dimensional smooth diffeomorphisms (joint with Thomas Koberda).

Tuesday, 28 February 2023

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.

Let X be a nice topological space with fundamental group G. Using the correspondence between N-degree covering spaces of X and homomorphisms f:G-->S_N to the symmetric group, we may study random covers of X by analyzing random homomorphisms f:G-->S_N. In particular, we consider the random permutation f(g) given by the image of a fixed element g in G.

The mini-course will focus on random covers of hyperbolic surfaces - so G here is a surface group. I will describe a joint work with Michael Magee in which we develop a method to study the local statistics of the random permutation f(g), such as its number of fixed points, and derive several results which are analogous to well-known results when G is replaced with a free group.

I'll motivate the question, introduce the results, give a flavor of the techniques - both in representation theory and in combinatorial group theory - and present some open problems.

Let M be a compact interval or a circle. We study algebraic properties of Diff(M;r), the group of Cr diffeomorphisms of M. In particular, we focus on the following question: which finitely generated groups arise as subgroups of Diff(M;r)? We survey classical theories of Hölder, Denjoy, Kopell and Thurston regarding this matter, and give a detailed dynamical account of a finitely generated subgroup of Diff(M;r) that does not admit any injective homomorphism into Diff(M;s) for all s>r. The heart of the proof is a probabilistic argument that controls the dynamics of one-dimensional smooth diffeomorphisms (joint with Thomas Koberda).

Wednesday, 01 March 2023

Let X be a nice topological space with fundamental group G. Using the correspondence between N-degree covering spaces of X and homomorphisms f:G-->S_N to the symmetric group, we may study random covers of X by analyzing random homomorphisms f:G-->S_N. In particular, we consider the random permutation f(g) given by the image of a fixed element g in G.

The mini-course will focus on random covers of hyperbolic surfaces - so G here is a surface group. I will describe a joint work with Michael Magee in which we develop a method to study the local statistics of the random permutation f(g), such as its number of fixed points, and derive several results which are analogous to well-known results when G is replaced with a free group.

I'll motivate the question, introduce the results, give a flavor of the techniques - both in representation theory and in combinatorial group theory - and present some open problems.

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.

Thursday, 02 March 2023

We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a graph to the minimal number of cells in a simplicial classifying space for the group. As a corollary we prove that any two isomorphic finite-index subgroups of a non-elementary hyperbolic group have the same index.

The mini-course will focus on random covers of hyperbolic surfaces - so G here is a surface group. I will describe a joint work with Michael Magee in which we develop a method to study the local statistics of the random permutation f(g), such as its number of fixed points, and derive several results which are analogous to well-known results when G is replaced with a free group.

I'll motivate the question, introduce the results, give a flavor of the techniques - both in representation theory and in combinatorial group theory - and present some open problems.

Let M be a compact interval or a circle. We study algebraic properties of Diff(M;r), the group of Cr diffeomorphisms of M. In particular, we focus on the following question: which finitely generated groups arise as subgroups of Diff(M;r)? We survey classical theories of Hölder, Denjoy, Kopell and Thurston regarding this matter, and give a detailed dynamical account of a finitely generated subgroup of Diff(M;r) that does not admit any injective homomorphism into Diff(M;s) for all s>r. The heart of the proof is a probabilistic argument that controls the dynamics of one-dimensional smooth diffeomorphisms (joint with Thomas Koberda).

Friday, 03 March 2023

Monday, 06 March 2023

We will discuss the connections between Kazhdan (also called property T) groups and expander graphs, Bernoulli percolation and some other statistical physics models on Kazhdan groups, and measurable cost.

Measure equivalence was introduced by Gromov as a measurable analogue to quasi-isometry. It is related, on the geometric side, to the study of lattices in locally compact groups. It also has strong connections, on the ergodic side, to the notion of orbit equivalence, i.e. studying when two measure-preserving actions of countable groups on standard probability spaces have the same orbits. I will first introduce and motivate the study of measure equivalence, and then explain how negative curvature in groups can be leveraged in various contexts to prove rigidity theorems from this viewpoint.

Tuesday, 07 March 2023

The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). Since the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices.

In this course, I will consider random walks on groups of isometries of hyperbolic spaces, and study the hitting measure of the random walk on the boundary.

In particular, I will discuss recent progress on the following two problems:

1) Can this measure be absolutely continuous with respect to Lebesgue? What is its dimension?

2) Can the hyperbolic boundary be identified with the Poisson boundary?

We will discuss the connections between Kazhdan (also called property T) groups and expander graphs, Bernoulli percolation and some other statistical physics models on Kazhdan groups, and measurable cost.

Measure equivalence was introduced by Gromov as a measurable analogue to quasi-isometry. It is related, on the geometric side, to the study of lattices in locally compact groups. It also has strong connections, on the ergodic side, to the notion of orbit equivalence, i.e. studying when two measure-preserving actions of countable groups on standard probability spaces have the same orbits. I will first introduce and motivate the study of measure equivalence, and then explain how negative curvature in groups can be leveraged in various contexts to prove rigidity theorems from this viewpoint.

Wednesday, 08 March 2023

The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). Since the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices.

In this course, I will consider random walks on groups of isometries of hyperbolic spaces, and study the hitting measure of the random walk on the boundary.

In particular, I will discuss recent progress on the following two problems:

1) Can this measure be absolutely continuous with respect to Lebesgue? What is its dimension?

2) Can the hyperbolic boundary be identified with the Poisson boundary?

We will discuss the connections between Kazhdan (also called property T) groups and expander graphs, Bernoulli percolation and some other statistical physics models on Kazhdan groups, and measurable cost.

Thursday, 09 March 2023

The distribution of sums of real-valued random variables is determined by the classical theorems of probability (law of large numbers, central limit theorem). Since the 1960’s Furstenberg, Oseledets and others have generalized such results tor noncommutative groups, e.g. groups of matrices.

In this course, I will consider random walks on groups of isometries of hyperbolic spaces, and study the hitting measure of the random walk on the boundary.

In particular, I will discuss recent progress on the following two problems:

1) Can this measure be absolutely continuous with respect to Lebesgue? What is its dimension?

2) Can the hyperbolic boundary be identified with the Poisson boundary?

Measure equivalence was introduced by Gromov as a measurable analogue to quasi-isometry. It is related, on the geometric side, to the study of lattices in locally compact groups. It also has strong connections, on the ergodic side, to the notion of orbit equivalence, i.e. studying when two measure-preserving actions of countable groups on standard probability spaces have the same orbits. I will first introduce and motivate the study of measure equivalence, and then explain how negative curvature in groups can be leveraged in various contexts to prove rigidity theorems from this viewpoint.

Friday, 10 March 2023

In this course, I will consider random walks on groups of isometries of hyperbolic spaces, and study the hitting measure of the random walk on the boundary.

In particular, I will discuss recent progress on the following two problems:

1) Can this measure be absolutely continuous with respect to Lebesgue? What is its dimension?

2) Can the hyperbolic boundary be identified with the Poisson boundary?