Minicourse

Andrei Rapinchuk (University of Virginia, USA)

Lecture Note Download   Part1   Part2   Part3

- Title: Arithmetic and Zariski-dense subgroups: weak commensurability, eigenvalue rigidity and applications to locally symmetric spaces.

- Abstract: The goal of this lecture series is to present the techniques for analyzing arithmetic and general Zariski-dense subgroups of algebraic groups that were developed in a joint work with Gopal Prasad. This work was initially motivated by the long-standing problems in geometry concerning isospectral and iso-length-spectral Riemannian manifolds. We have been able to settle some of these problems for locally symmetric spaces associated with simple real algebraic groups through introducing the notion of weak commensurability for Zariski-dense subgroups of semi-simple algebraic groups and investigating when two arithmetic groups are weakly commensurable. In order to describe various ingredients of this approach, we will include a brief review of the required notions and results from the theory of algebraic groups.

The important feature of our analysis of weak commensurability is that many techniques, and also certain results, apply not only to arithmetic but in fact to arbitrary (finitely generated) Zariski-dense subgroups. In particular, there is growing evidence to support the expectation that two simple algebraic groups containing weakly commensurable Zariski-dense subgroups must be closely related. We call this phenomenon ``eigenvalue rigidity" since weak commensurability is based on matching the eigenvalues of the elements in the subgroups. It should be pointed out that this new form of rigidity is expected to hold even when the given Zariski-dense subgroups are free groups, and the classical rigidity theorems are inapplicable. We will report on the current work in this direction which is joint with Vladimir Chernousov and Igor Rapinchuk.

Talks

Mikhail Belolipetsky (IMPA, Brazil)

▷ Talk 1 and 2

- Title: Counting isospectral manifolds.

- Abstract: The main purpose of my two lectures is to discuss a recent joint work with Ben Linowitz of the same title. In the first lecture we will introduce the problem of isospectrality starting from the celebrated question of Mark Kac about if one can hear the shape of the drum, and then discuss in some detail Sunada's group theoretical method for constructing isospectral manifolds. This method gives many known examples of isospectral but not isometric manifolds and appears in our work as well. In the second lecture we will discuss our paper in which we show that surprisingly many higher rank locally symmetric spaces are mutually isospectral. I will review the main ingredients of the proof with an emphasis on some related open problems.

Vincent Emery (Universität Bern, Switzerland)

▷ Talk 1

- Title: Hyperbolic manifolds and pseudo-arithmeticity.
- Abstract: I will introduce and motivate a notion of pseudo-arithmeticity, which possibly applies to all lattices in PO(n,1) with n>3. Moreover I will explain that the covolumes of such lattices arise as rational linear
combinations of special values of L-values; this generalizes the case of arithmetic lattices. This is joint work with Olivier Mila.

▷ Talk 2

- Title: Salem numbers and arithmetic hyperbolic groups
- Abstract: I will explain some ideas of a recent paper, joint with Ratcliffe and Tschantz, where we show that (log of) Salem numbers can be realized as translation lengths of elements in higher dimensional arithmetic hyperbolic lattices. The work generalizes known results in dimension 2 and 3.

Alan Reid (Rice University, USA)

▷ Talk 1 and 2

- Title: Spectra of geodesics and geodesic surfaces in arithmetic hyperbolic 3-manifolds and beyond.
- Abstract: In these two lectures we will provide (in Lecture 1) an introduction to the connections between number theory and the spectra of geodesics and geodesic surfaces in arithmetic hyperbolic 3-manifolds. This will lead to (in Lecture 2) a more general discussion of the spectrum of incompressible surfaces in hyperbolic 3-manifolds.

Lola Thompson (Oberlin College, USA)

▷ Talk 1

- Title: Counting quaternion algebras.

- Abstract: We will introduce some classical techniques from analytic number theory and show how they can be used to count quaternion algebras over number fields subject to various constraints. Because of the correspondence between maximal subfields of quaternion algebras and geodesics on arithmetic hyperbolic manifolds, these counts can be used to produce quantitative results in spectral geometry. This talk is based on joint work with B. Linowitz, D. B. McReynolds, and P. Pollack.

▷ Talk 2

- Title: Bounded gaps between primes and volumes of manifolds.

- Abstract: In 1992, Reid posed the question of whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to Reid’s question, Futer and Millichap have recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first n geodesic lengths. In the present talk, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, we obtain infinitely many k-tuples of arithmetic, hyperbolic 3-orbifolds which are pairwise non-commensurable, have certain prescribed geodesic lengths, and have volumes lying in an interval of bounded length. One of the key ideas stems from the breakthrough work of Maynard and Tao on bounded gaps between primes. We will introduce the Maynard-Tao approach and then discuss how it can be applied in a geometric setting. This talk is based on a series of papers with B. Linowitz, D. B. McReynolds, and P. Pollack.

Jeffrey Meyer (California State University, San Bernardino USA)

▷ Talk 1

- Title: Systoles in Arithmetic Locally Symmetric Spaces.

- Abstract: The systole of a closed Riemannian manifold is the minimal length of a geodesic loop. This number measures how pinched the manifold is. For particularly symmetric manifolds, such as arithmetic hyperbolic manifolds, you would then expect this number to be relatively large. In this talk, I will go over foundational results in systolic geometry, including relationships between systole and volume, some low dimensional examples, and the Short Geodesic Conjecture.

▷ Talk 2

- Title: Systole Growth Up Congruence Towers.

- Abstract: In a cover of a Riemannian manifold, a geodesic loop may or may not unwrap, and consequently the systole may or may not increase. Given a tower of covers, how should one expect the systole to grow? In this talk, I will discuss how for congruence towers, which are particularly symmetric, the growth is at least logarithmic in volume. Many researchers, including Buser-Sarnak, Katz-Schaps-Vishne, Murillo, and Lapan-Linowitz-Meyer, have contributed results on this type of growth for a variety of spaces. I will discuss these results, sketch the key arguments in my joint work with Lapan and Linowitz, and draw analogies between systole growth results and property tau for congruence subgroups.