AG2025

Summer School on Arithmetic and Random Groups

May 19–23, Northwestern University

All lectures will be held at the Great Room in Seabury Hall, 626 Haven St.

Mini-courses:
Michael Chapman : A complexity theoretic resolution of the Aldous–Lyons conjecture.
Michelle Chu : Arithmetic hyperbolic manifolds of simplest type.
Yair Glasner : Boomerang subgroups.
Homin Lee : Measure rigidity in higher rank lattice actions.
Matthew Stover : Geodesic submanifolds and dynamics.
Following the school, there will be a conference honoring Alex Furman’s 60th birthday: See the website.
Organizers: Nir Avni, Uri Bader, and Tsachik Gelander.

Loading...

Taking too long?

Reload document
| Open in new tab

Download
Abstracts:

Chapman:
In these 4 talks, we will go over the main ingredients in the recent refutation of the Aldous–Lyons conjecture due to Bowen, Lubotzky, Vidick and myself. The plan is as follows:
1. First topic: Soficity and the Aldous–Lyons conjecture
  Many open problems in group theory are known to be true for the subclass of sofic groups, which raises the following question: Are all groups sofic?
  We begin the lecture by providing several examples of open problems that are true for sofic groups. Then, we define soficity of finitely generated groups through the notion of cosoficity of invariant random subgroups (IRSs) of the free group. The Aldous–Lyons conjecture (ALC) asks: Are all IRSs of the free group cosofic? These problems are closely related, as a positive answer to ALC implies a positive answer to the group soficity problem. Though there are still no known non-sofic groups (which, as we just mentioned, would resolve both problems), we finally have a negative answer to ALC, namely, there exist non-cosofic IRSs of the free group.
2. Second topic: Subgroup Tests
  Subgroup Tests are (rational) convex combinations of continuous functions from the Chabauty space (subgroups of the free group equipped with the product topology) to {0,1}. These objects can be seen as the rules of a “test” that a “referee” runs on an invariant random subgroup, where the sampled subgroup may either “pass” or “fail” the test. Given a Subgroup Test T and an IRS pi, one can integrate T versus pi, which yields the same quantity as the probability that a subgroup sampled according to pi passes the referee’s test induced by T. This setup gives rise to the following two computational problems: Given a Subgroup Test T, what is the optimal passing probability of an IRS against it? And, what is this optimum if we are only allowed to use cosofic IRSs? Clearly, if all IRSs are cosofic, these two quantities are the same. The first quantity, where one optimizes over all IRSs, is called the “ergodic value of T”, while the second quantity which optimizes over cosofic IRSs is called the “sofic value of T”. It turns out that there is an algorithm that, given a Subgroup Test, approximates its sofic value from below, and another algorithm that approximates its ergodic value from above. In particular, if ALC has a positive answer, then the sofic value of a Subgroup Test can be approximated to any predetermined accuracy. The rest of this mini-course is devoted to proving that approximating the sofic value of a Subgroup Test is as hard as the Halting Problem, and thus there is no algorithm that is able to do it. In particular, ALC must have a negative answer.
3. Third topic: Subgroup Tests and Tailored Non-Local Games
  Towards our goal, we need to detour through quantum information theory. Non-local games are the combinatorial blueprints of certain experiments that can be run with two labs, and that aim to test that these labs share “entangled particles”. These objects arose as part of John Bell’s solution to the Einstein–Podolsky–Rosen paradox. Similar to Subgroup Tests, one can associate a value to every such non-local game, which is called its “quantum value”. After we provide several examples of non-local games, we will discuss a certain subclass of non-local games, called Tailored Games. There is a (essentially) value preserving map from this subclass of Tailored Games to Subgroup Tests, which implies that approximating the quantum value of a game is easier than approximating the sofic value of a test. In a breakthrough result, Ji–Natarajan–Vidick–Wright--Yuen showed (in 2019) that approximating the quantum value of a general non-local game is as hard as the Halting Problem. This result is known as MIP*=RE. So, it remains to show that their techniques can be transferred to the subclass of Tailored Games.
4. Fourth topic: Compression of Games
  The main tool in proving MIP*=RE is called Compression. The idea behind Compression is, given a game G, to provide an exponentially smaller game G’ which simulates G (at least, in the sense of its value). It turns out that if Compression can be done in polynomial time (in the description length of the game G), then approximating the value of G is as hard as the Halting Problem — similar results appear already in tiling theory, and recently Marks–Nezhadi–Yuen showed that this is a very general phenomenon. We will try to give some flavor of how one can prove Compression in the general setup of non-local games, while remarking along the way what needs to be changed so that it works for Tailored Games.
Chu:
This course will center around the study of arithmetic subgroups of isometries of (real) hyperbolic space of simplest type and their quotients which are finite volume hyperbolic orbifolds. These groups are constructed from quadratic forms defined over totally real number fields and the geometry of their quotients reflects the arithmetic properties of the associated forms. I introduce the construction and commensurability of these arithmetic groups and discuss several geometric properties of arithmetic hyperbolic manifolds such as their length spectrum and their totally geodesic submanifolds.
Glasner:

Based on two joint works one with Waltraud Lederle, one with Tobias Hartnick and Waltraud Lederle.

We will focus on discrete countable group $\Gamma$ and the dynamics of its action (by conjugation) on its Chabauty space: The (compact metrizable) space $Sub(\Gamma)$ of all of its subgroups. An invariant Borel probability measure on this space is known as an IRS. A boomerang subgroup is a subgroup $\Delta \in Sub(\Gamma)$ that is recurrent with respect to the action of each group element separately.

Lecture 1: Boomerang subgroups and their basic properties. We will discuss the structure of boomerangs in many special cases including Nilpotent groups, hyperbolic groups and (dense subgroups of) simple p-adic Lie groups. We will prove a version of Borel density for boomerang subgroups in linear groups: showing how such subgroups interact nicely with the Zariski topolgoy.

Lecture 2: Ubequity of boomerangs in free groups. The Chabauty space serves as a universal factor for measure class perserving dynamical systems via the stabilizer map

$Stab: (X,\mathcal{B},[\mu],\Gamma) \longrightarrow (Sub(\Gamma),\Gamma).$
If the action is probability preserving it gives rise to a IRS, $Stab_*(\mu)$ which is always supported on boomerang subgroups. We will discuss the combinatorics of boomerangs in nonabelian free groups establish the ubequity of boomerangs that do not come from IRSs. This contrasts with the situation in higher rank lattices, which is the subject of the third lecture.

Lecture 3: Boomerangs in lattices of higher $\mathbb{Q}$ -rank. Our main theorem says that in a lattice $\Gamma$ of $\mathbb{Q}$ -rank at least $2$ in a connected (absolutely) simple Lie group $G$ every boomerang subgroup is either finite and central or of finite index. I will show how this theorem implies, and generalizes, the Nevo-Stuck-Zimmer theorem as well as Margulis’ normal sugroup theorem for these lattices. I will demonstrate the proof on an array of examples, each one of which demonstrates a certain aspect of the proof.
Lee

In this mini-course, we will discuss about actions of higher rank lattices, focusing on how measures and a measure rigidity scheme play important roles in various settings.

First, we introduce some definitions and properties related to actions of higher rank lattices and measure rigidity scheme.

Then, using the measure rigidity scheme, we will discuss classical theorems, Margulis’ normal subgroup theorem and Margulis’ superrigidity theorem, as well as, recent works on smooth actions of higher rank lattices on manifolds, so-called Zimmer program.
Stover
This will be a gentle introduction to how geodesic submanifolds of locally symmetric spaces can be understood as orbit closures of dynamical systems, and how this point of view plays a role in proofs of a wide range of results from number theory to differential geometry. This course should be accessible to any PhD student that has completed a standard course in differential topology.

Lecture notes:

Chapman

Loading...

Taking too long?

Reload document

|

Open in new tab

Download

Stover