# Fields Academy Shared Graduate Course: Randomness in Groups

## Description

**Instructor:** Professor Kasra Rafi, University of Toronto

**Course Description**

There are several methods for choosing a generic element in a group. One can equip the group with the word metric associated to some finite generating set and then take an element at random in a ball of radius $R$ with respect to this metric. Alternatively, one can choose an element using a random walk process on the group. That is, let $\mu$ be a probability measure on a $\Gamma$. Then $\mu$ defines a random walk process where a sample path is a sequence of element $\omega_{n}$ $\in$ $\Gamma$ with the property that in each step the probability of transition from $\omega_{n}$ to $\omega_{n+1}$ $=$ $\omega_{n}\gamma$ is $\mu(\gamma)$. We can then consider $\omega_{n}$, for a large $n$, to be a randomly chosen element.

It is often easier to prove properties of elements chosen using a random walk process because there is an explicit probabilistic description of $\omega_{n}$. For example, a random mapping class chosen using a random walk process is known to be pseudo-Anosov but the same is not known if the mapping class is chosen using the first method. One way to answer such questions is to consider if the two notion of randomness can ever be asymptotically the same.

Here, asymptotic means that we should take the limit of each process at infinity. Consider the following simple case where $M$ is a compact manifold of negative curvature. The Gromov boundary at infinity of the universal cover $\partial\tilde{M}$ of $M$ is a topological sphere. This sphere carries two types of measures corresponding to two notions of randomness in the fundamental group of $M$. First, we can consider a weak limit of uniform average of orbit points contained in large balls in $\tilde{M}$. This limit coincides with the conformal measure built from the visual metric and is referred to as the Patterson-Sullivan measure. On the other hand, for almost every sample path $\omega_{n}$ and every point $x$ $\in$ $\tilde{M}$, the sequence $\omega_{n}(x)$ converges to a point on $\partial\tilde{M}$. Hence, we can consider the corresponding hitting measure on $\partial\tilde{M}$ where the measure of a set $E$ $\subset$ $\partial$$T$ is the probability that, for $x$ $\in$ $\tilde{M}$ and a sample path {$\omega_{n}$} of the random walk, $\omega_{n}(x)$ converges to a point of $E$. The above question translate to:

**Question 1.** *Can the Patterson-Sullivan measure be the hitting measure of a random walk?*

In the setting of lattices in semi-simple Lie groups acting on the associated symmetric spaces, Question 0.1 is answered by a celebrated theorem of Furstenberg. He showed that the Lebesgue measure on the Furstenberg boundary of a symmetric space is the hitting measure of some random walk with finite first moment.

In this class, we will go over the proof of Furstenberg's theorem. We then attemp to generalize it to other settings, most importantly, the setting of the action of Mapping class group on Teichmüller space.

More precisely, let $\mathcal{T}$ be the Teichmüller space of a surface $S$, that is the space of marked Riemann surfaces homeomorphic to $S$, and let Map($S$) be the mapping class group, the group of isotopy class orientation preserving homeomorphisms of $S$. Then Map($S$) acts on $\mathcal{T}$ by changing the marking. In this setting, we can use the Thurston boundary of Teichmüller space which can be identified with the space of projectivized measured foliations, $\partial\mathcal{T}$ $=$ $\mathcal{P}\mathcal{M}\mathcal{F}$, and is equipped with a natural *Lebesgue measure* $\mu_{Th}$ known as the Thurston measure. It follows from the work of Athreya-Bufetov-Eskin-Mirzakhani that the Patterson-Sullivan measure is in the Lebesgue class and Kaimanovich-Masur showed that the Thurston boundary can be used at the Poisson boundary for the action of mapping class group on Teichmüller space.

The goal of the class is to prove the following theorem.

**Theorem 2.** *There is a measure $\mu$ on the mapping class group with finite first-moment such that the corresponding hitting measure $\nu$ is absolutely continuous with respect to the Thurston measure $\mu_{Th}$ on $\partial\mathcal{T}(S)$.*

*(More logistical details are coming soon.)*