Tag - Arithmetic groups

Richard Ng: Discovering modular data from congruence representations

14th May, 2023

Modular fusion categories (MFCs) arise naturally in many areas of mathematics and physics. Associated with an MFC is a pair of complex matrices, called modular data, which are arguably the most important invariants of an MFC. The modular data of an MFC generate some uncanonical congruence representations of SL2(ℤ). In this talk, we will discuss how modular data could be reconstructed or discovered from congruence representations of SL2(ℤ). The talk is based on a joint work with Eric Rowell, Zhenghan Wang and Xiao-Gang Wen.

Shai Evra: Optimal strong approximation and the Sarnak-Xue density hypothesis

3rd November, 2022

It is a classical result that the modulo map from SL2(ℤ) to SL2(ℤ/qℤ), is surjective for any integer q. The generalization of this phenomenon to other arithmetic groups goes under the name of strong approximation, and it is well understood. The following natural question was recently raised in a letter of Sarnak: What is the minimal exponent e, such that for any large q, almost any element of SL2(ℤ/qℤ) has a lift in SL2(ℤ) with coefficients of size at most qe? A simple pigeonhole principle shows that e is strictly greater than 3/2. In his letter Sarnak proved that this is in fact tight, namely e = 3/2, and call this optimal strong approximation for SL2(ℤ). The proof relies on a density theorem of the Ramanujan conjecture for SL2(ℤ). In this talk we will give a brief overview of the strong approximation, a quantitative strengthening of it called super strong approximation, and the above mentioned optimal strong approximation phenomena, for arithmetic groups. We highlight the special case of p-arithmetic subgroups of classical definite matrix groups and the connection between the optimal strong approximation and optimal almost diameter for Ramanujan complexes. Finally, we will present the Sarnak-Xue density hypothesis and describe recent ongoing works on it relying on deep results coming from the Langlands programme.

Chen Meiri: Word Width in Higher Rank Arithmetic Groups

18th October, 2022

A word on d letters is an element of the free group of rank d, say, with basis x1,…,xd. Given a word w=w(x1,…,xd) on d letters, for every group G, there is a word map w:Gd→ G given by substituting the xi with elements of G. We say that a word w has a finite width n in the group G if any element in the subgroup generated by w(G) is a product of at most n element of w(G) or their inverses. In this talk, I will survey results about word width in several families of groups and then restrict the focus to the family of higher rank arithmetic groups. I will present a conjecture about word width in higher-rank arithmetic groups and explain some consequences, most notably, to the Congruence Subgroup Problem.

François Thilmany: Uniform discreteness of arithmetic groups and the Lehmer conjecture

1st November, 2021

The famous Lehmer problem asks whether there is a gap between 1 and the Mahler measure of algebraic integers which are not roots of unity. Asked in 1933, this deep question concerning number theory has since then been connected to several other subjects. After introducing the concepts involved, we will briefly describe a few of these connections with the theory of linear groups. Then, we will discuss the equivalence of a weak form of the Lehmer conjecture and the 'uniform discreteness' of cocompact lattices in semisimple Lie groups (conjectured by Margulis). Joint work with Lam Pham.

Alex Lubotzky: First-order rigidity of high-rank arithmetic groups

16th July, 2021

The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SLn(ℤ), for n ≥ 3, SLn(ℤ[1/p]) for n ≥ 2, their finite-index subgroups and many more. A number of remarkable results about them have been proven including: Weil local rigidity, Mostow strong rigidity, Margulis superrigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. We will add a new type of rigidity: 'first order rigidity'. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to D then L is isomorphic to D. This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non-isomorphic finitely generated groups which are elementarily equivalent to them.

Uri Onn: Base change and representation growth of arithmetic groups

3rd June, 2021

A group is said to have polynomial representation growth if the sequence enumerating the isomorphism classes of finite-dimensional irreducible representations according to their dimension is polynomially bounded. The representation zeta function of such group is the associated Dirichlet generating series. In this talk I will focus on representation zeta functions of arithmetic groups and their properties. I will explain the ideas behind a proof of a variant of the Larsen-Lubotzky conjecture on the representation growth of arithmetic lattices in high rank semisimple Lie groups (joint with Nir Avni, Benjamin Klopsch and Christopher Voll) and analogous results for arithmetic groups of type A2 in positive characteristic

Alexander Hulpke: Index computations in arithmetic groups

30th April, 2020

The question of whether a subgroup, given by generators, has finite (and then which) index is a natural question in group theory. Unfortunately, for natural groups such as SLn(ℤ) and Sp2n(ℤ), this question cannot have a general algorithmic solution. Nevertheless it is often possible to determine this information in many cases using a computer. I will describe some approaches to this problem and illustrate these in examples.

This is joint work with Alla Detinko (Hull) and Dane Flannery (Galway).

Mikolaj Fraczyk: Density conjecture for horizontal families of lattices in SL(2)

2nd April, 2020

Let G be a real semisimple Lie group with an irreducible unitary representation π. The non-temperedness of π is measured by the parameter p(π), which is defined as the infimum of p≥2 such that π has matrix coefficients in Lp(G). Sarnak and Xue conjectured that for any arithmetic lattice Γ⊂G and principal congruence subgroup Γ(q)⊂Γ, the multiplicity of π in L2(G/Γ(q)) is at most O(V(q)2/p(π)+ε) where V(q) is the covolume of Γ(q). In some contexts such estimate is a decent substitute for the Ramanujan conjecture. For G of real rank 1 Sarnak and Xue translate the estimate into a Diophantine counting problem which they managed to solve for SL2(ℝ) and SL2(ℂ).

In this talk I will explain how one can get the same multiplicity bounds for families of pairwise non-commensurable lattices in G=SL2(ℝ), SL2(ℂ) given as unit groups of maximal orders of quaternion algebras over number fields (“horizontal families”). Namely: m(π,Γ)≪V2/p(π)+ε, where V is the covolume of Γ. I will also discuss similar bounds on multiplicities of representations π1×π2 of G=SL2(ℝ)2 where π1 is fixed non-tempered but π2 is allowed to vary together with the lattice.