Arbeitsgemeinschaft : Thin Groups and Super-approximation ( hybrid meeting )

The aim of this workshop was to discuss the super-approximation of thin groups, its dynamical implications in terms of the mixing of geodesic flows, and applications to various problems in arithmetic, geometry, and dynamics. Mathematics Subject Classification (2010): 22E40, 37F35, 11N45. Introduction by the Organizers The workshop Thin groups and super-approximation, was well attended with over 50 participants (in person + zoom participants) with broad geographic representation from all continents. This workshop was a nice blend of researchers with various backgrounds. Let us recall that a thin group refers to a finitely generated Zariski dense subgroup of an S-arithmetic subgroup of a (semisimple real) algebraic group that is of infinite covolume. Given a thin group Γ of a semisimple algebraic group G, we consider a family, say F , of its congruence subgroups. Fixing a finite symmetric generating subset Ω of Γ, the super-approximation for Γ with respect to F means that the family of Cayley graphs (Γ/Γn,Ω), Γn ∈ F forms an expander family. The main theme of this workshop as explained in our program was divided into the following three topics: 2 Oberwolfach Report 50/2021 (1) Overview the super-approximation; (2) Dynamical implications in terms of uniform exponential mixing of geodesic flows; (3) Applications to problems in arithmetic, geometry, and dynamics. The lectures followed carefully the scheduled program: 6 of the lectures were devoted to the first topic, 7 devoted to the second, and 5 devoted to the third. The first theme of this meeting was to understand what the super-approximation is, and go over proofs of some of the main results on this topic. There have been many advances in our understanding of the spectral gap properties of random walks in (locally) compact groups, and some of the progress manifested itself in super-approximation property of thin groups. In the first lecture, Pham went over the basics of random walk in a compact group and mentioned some of the main results of super-approximation due to Bourgain-Gamburd, Bourgain-Varjú, Golsefidy-Varjú, and Golsefidy. In the second lecture on this topic, Sert went over the Bourgain-Gamburd machine and showed how these techniques answer Lubotzky’s 1-2-3 problem. As part of the Bourgian-Gamburd machine, one needs to understand the approximate subgroups. The first result of this type is due to Helfgott. Dona explained the main ideas of Helfgott’s proof. The BourgainGamburd machine works in a single scale setting. In a multiscale setting, one needs to focus on various scales separately. Kogler explained how Bourgain-Gamburd and Benoit-de Saxcé overcame these difficulties in the Archimedean setting: compact semisimple Lie groups. Winkel went through Varjú’s thesis and explained how he dealt with the direct product of certain finite groups. Finally, Machado explained how Golsefidy used understanding of approximate submodules and a quantitative p-adic open function theorem to prove the super-approximation for powers of primes. The second topic is to understand the implication of super-approximation in terms of the uniform exponential mixing of the geodesic flow on congruence covers of a hyperbolic manifold of the form Γ\Hn where Γ < SO(n, 1) is a thin geometrically finite subgroup. Thanks to the well-developed theory of Patterson-Sullivan, there has been quite a lot of progress on this. After one lecture on preliminaries on hyperbolic geometry given by Islam, the second lecture on this topic by Luethi discussed the mixing and its consequence on matrix coefficients on a fixed geometrically finite manifold. There are two approaches to prove exponential mixing: one using the representation theory, following Edwards-Oh paper, on which we had 2 lectures by Dabeler and Han and the other using the so-called Dolgopyat machine, following Oh-Winter and Sarkar, on which we had 2 lectures by Chow and Corso. As an application of mixing, the last lecture on this topic by Lee was devoted to the discussion on circle-counting problem for Apollonian circle packings, following the works of Kontorovich-Oh, and Oh-Shah. The third topic concerns the combination of the previous two with other techniques (most notably, the affine sieve and the orbital circle method) to make progress on (or solve outright) a wide range of (initially unrelated) problems spanning arithmetic, geometry, and dynamics. In the first lecture on these topics, Arbeitsgemeinschaft: Thin Groups and Super-approximation 3 Sarkar gave a basic overview of the types of problems that are amenable to attack, setting up the next few lectures. Jones explained the local-global problem for integral Apollonian packings and generalizations, and sketched how the orbital circle method is used in work of Bourgain-Kontorovich and others to show a density-one analogue. Zaremba’s conjecture, as well as its origins in and applications to numerical integration and pseudorandom sequences, was explained by Zhang, who also discussed some of the ideas in the proof of a density-one analogue. This lecture and the previous one exhibited how the orbital circle method, while broadly applicable as a “tool,” is not a quotable “theorem,” as its actual use in practice varies greatly, with rather different machinery responsible for both the major arcs and minor arcs analyses in these two settings. The last two lectures in this series concerned applications of the affine sieve. Kim’s lecture stated the Einsiedler-Lindenstrauss-Michel-Venkatesh conjecture on lowlying, closed, fundamental geodesics on the modular surface, and sketched its resolution by Bourgain-Kontorovich, using methods from the “Beyond Expansion” program; the latter gives exponents of distribution greatly exceeding those coming from spectral gap methods alone. The final lecture, by Litman, discussed progress on the Pythagorean Prime Triples problems, spanning work of Kontorovich-Oh, Hong-Kontorovich, and Bourgain-Kontorovich, each reducing the number of prime factors of, say, the hypotenuse, in a thin (but not too thing) Zariski-dense orbit of Pythagorean triples. In order to accommodate the different time zone, each day began only at 3 pm and ended at 10 pm at the MFO and the participants devoted the morning time for group discussion and to go over lectures from previous days. There were a lot of topics which were covered, and we felt we had a very good panoramic view of the recent developments of these topics. Acknowledgement: The MFO and the workshop organizers would like to thank the National Science Foundation for supporting the participation of junior researchers in the workshop by the grant DMS-1641185, “US Junior Oberwolfach Fellows”. Arbeitsgemeinschaft: Thin Groups and Super-approximation 5 Arbeitsgemeinschaft (hybrid meeting): Thin Groups and Superapproximation