ICTS

Lattices in high-rank semisimple groups enjoy a number of rigidity properties like super-rigidity, quasi-isometric rigidity, first-order rigidity, and more. In these lectures, we will add another one: uniform ( a.k.a. Ulam) stability. Namely, it will be shown that (most) such lattices D satisfy: every finite-dimensional unitary ”almost-representation” of D ( almost w.r.t. to a sub-multiplicative norm on the complex matrics) is a small deformation of a true unitary representation. This extends a result of Kazhdan (1982) for amenable groups and or Burger-Ozawa-Thom (2013) for SL(n,Z), n > 2. The main technical tool is a new cohomology theory (”asymptotic cohomology”) that is related to bounded cohomology in a similar way to the connection of the last one with ordinary cohomology. The vanishing of H2 w.r.t. to a suitable module implies the above stability. The talks are based on a joint work of the speakers with L. Glebsky and N. Monod. See arXiv:2301.00476.

Let G be a non-compact linear simple Lie group and H ⊂ G a reductive subgroup. We say that the homogeneous space G/H admits a standard compact Clifford-Klein form if there exists a reductive subgroup L ⊂ G such that L acts properly and co-compactly on G/H. I will describe relations between real root decompositions of Lie triples (g, h, l) corresponding to standard compact Clifford-Klein forms, under the assumption that g is not equal to h + l. This gives new classes of homogeneous spaces G/H which do not admit standard compact Clifford-Klein forms. For instance proper regular subalgebras h of g never generate homogeneous spaces G/H which admit standard compact Clifford-Klein forms (other than g = h + l).

One of the major results in the arithmetic theory of algebraic groups is the validity of the cohomological local-global (or Hasse) principle for simplyconnected and adjoint semisimple groups over number fields. Over the last several years, there has been growing interest in studying Hasse principles for reductive groups over arbitrary finitely generated fields with respect to suitable sets of discrete valuations. In particular, we have conjectured that for divisorial sets, the corresponding Tate-Shafarevich set, which measures the deviation from the localglobal principle, should be finite for all reductive groups. I will report on recent progress on this conjecture, focusing in particular on the case of algebraic tori as well as on connections to groups with good reduction.

Lindenstrauss and Varju asked the following questions: for every prime p, let Sp be a symmetric generating set of Gp := SL(2, Fp)×SL(2, Fp). Suppose the family of Cayley graphs {Cay(SL(2, Fp), pri(Sp))} is a family of expanders, where pri is the projection to the i-th component. Is it true that the family of Cayley graphs {Cay(Gp, Sp)} is a family of expanders? We answer this question and go beyond that by describing random-walks in various group extensions. (This is a joint work with Srivatsa Srinivas.)

In this talk we survey results on Hasse principle for homogeneous spaces under reductive groups over semiglobal fields, i.e., function fields of curves over complete discrete valued fields. We state some conjectures and explain recent progress on these conjectures for function fields of p-adic curves.

Given a field with a set of discrete valuations, we show how the genus of any division algebra over certain fields is related to the genus of some residue algebras at various valuations and the ramification data. Applications include showing triviality of the genus of quaternions over many fields such as higher local fields, function fields of curves over higher local fields and function fields of curves over real closed fields. When the base field is a function field of a curve over a global field with a rational point, the genus of any quaternion is related to the 2-torsion of the Tate-Shafarevich group of the Jacobian. As a consequence, when the curve is elliptic, the size of the genus can be computed directly using arithmetic data of the elliptic curve.

Extension of a Property (T) group by another is a Property (T) group. It follows that Property (T) is closed under product. The same is true for fiber products of two homomorphisms from Property (T) groups, provided one of them splits. In general there are counterexamples. The special case of fiber product of two copies of a homomorphism reduces to the question of Property (T) for a semidirect product given the same for the non-normal factor. Here necessary and sufficient conditions can be given. This is a joint work with M. Mj. The general case of relative Property (T) for semidirect products, where the normal factor need not be abelian, is an ongoing work with S. Nayak.

I will describe joint work with Domingo Toledo on residual finiteness for cyclic central extensions of fundamental groups of aspherical manifolds, its application to central extensions of certain arithmetic lattices, and discuss some open problems on residual properties of central extensions of lattices and their connections to the geometry of locally symmetric spaces.

We will discuss some recent progress on problems that can be reformulated as Diophantine questions on orbits of ”thin” groups.

For every surface S, the pure mapping class group GS acts on the SL2-character variety ChS of the fundamental group P of S. The character variety ChS is a scheme over the ring of integers. Classically this action on the real points ChS(R) of the character variety has been studied in the context of the Teichmuller theory and SL(2,R)-representations of P. In a seminal work, Goldman studied this action on a subset of ChS(R) which comes from SU(2)-representations of P. In this case, Goldman showed that if S is of genus g > 1 and zero punctures, then the action of GS is ergodic. Previte and Xia studied this question from topological point of view, and when g > 0, proved that the orbit closure is as large as algebraically possible. Bourgain, Gamburd, and Sarnak studied this action on the Fp-points ChS(Fp) of the character variety where S is a puncture torus. They conjectured that in this case, this action has only two orbits, where one of the orbits has only one point. Recently, this conjecture is proved for large enough primes by Chen. When S is an n-punture sphere, the finite orbits of this action on ChS(C) are connected to the algebraic solutions of Painleve differential equations.

We study actions of algebraic tori on homogeneous spaces. A conjecture of Cassels and Swinnerton-Dyer about the purely real (homogeneous) forms formulated in 1955 and still open can be translated in terms of such actions. We prove that the natural generalization of the conjecture to the non-purely real forms is not valid. The proof is based on the complete description of the forms with discrete set of values at the integer vectors and non-representing non-trivially zero over the rational numbers.

We shall present different notions of bounded generation for Zariskidense subgroups of linear algebraic groups; in particular, we shall treat the notion of exponential and purely exponential parametrization. After surveying on classical results, we shall show recent applications of Diophantine approximation techniques in the theory of linear groups which lead to a classification of groups admitting purely exponential parametrization or bounded generation.

We study the G-character variety of a random group, where G is a semisimple complex Lie group. The typical behavior depends on the defect of the random presentation. We are able to describe what happens for all but an exponentially small probability of exceptions. In particular we compute the dimension and show the absolute irreducibility of the character variety in defect at least two. In defect one we also exhibit a phenomenon of Galois rigidity showing the finiteness of the character variety and proving lower bounds on its cardinality. The proofs are conditional on GRH via the use of effective Chebotarev type theorems and uniform mixing bounds for Cayley graphs of finite simple groups. This is joint work with Peter Varju and Oren Becker.

A finitely generated residually finite group G is called profinitely rigid if for any other finitely generated residually finite group H, whenever the profinite completions of H and G are isomorphic, then H is isomorphic to G. In this talk we will discuss some recent work that constructs finitely presented groups that are profinitely rigid amongst finitely presented groups but not amongst finitely generated one. This builds on previous work that used ”controlled SL(2)-representations” to construct examples of arithmetic lattices in PSL(2,R) and PSL(2,C) that are profinitely rigid.

We shall present different notions of bounded generation for Zariskidense subgroups of linear algebraic groups; in particular, we shall treat the notion of exponential and purely exponential parametrization. After surveying on classical results, we shall show recent applications of Diophantine approximation techniques in the theory of linear groups which lead to a classification of groups admitting purely exponential parametrization or bounded generation.

The systole of a compact Riemannian manifold M is the least length of a non-contractible loop on M. In this talk I will survey some recent work with S. Lapan and J. Meyer on the systolic geometry of arithmetic locally symmetric spaces, emphasizing lower bounds for systoles and systole growth along congruence covers.

We develop algorithms and computer programs which verify criteria of properness of discrete group actions on semisimple homogeneous spaces. We apply these algorithms to find new examples of non-virtually abelian discontinuous group actions on homogeneous spaces which do not admit proper $SL(2,\mathbb{R})$-actions.

We shall present different notions of bounded generation for Zariskidense subgroups of linear algebraic groups; in particular, we shall treat the notion of exponential and purely exponential parametrization. After surveying on classical results, we shall show recent applications of Diophantine approximation techniques in the theory of linear groups which lead to a classification of groups admitting purely exponential parametrization or bounded generation.

A hypergeometric group is a subgroup of GLn(C) generated by the companion matrices of two monic coprime polynomials of degree n. It arises as the monodromy group of a hypergeometric differential equation, and if the defining polynomials are also self-reciprocal and form a primitive pair, then its Zariski closure  inside GLn(C) is either a symplectic or an orthogonal group. In this talk, we will discuss the arithmeticity and thinness of the hypergeometric groups whose defining polynomials also have integer coefficients.

The aim of this talk will be to construct functions on a cyclic group of odd order d whose ”convolution square” is proportional to their square. For that, we will have to interpret the cyclic group as a subgroup of an abelian variety with complex multiplication, and to use the modularity properties of their theta functions.

The theory of Bruhat and Tits opened the door to studying and constructing representations of general p-adic reductive groups. I will give an overview of our understanding of the construction and category of representations of p-adic groups and indicate how it crucially relies on Bruhat–Tits theory and the Moy–Prasad filtration.

In this talk I will introduce Hitchin representations and show that only a finite part of the Jordan-Lyapunov spectra of a Hitchin representation is enough to completely determine it up to conjugacy classes.

Let G be the group of rational points of a connected reductive linear algebraic group defined over a nonarchimedean local field k. Under some conditions that are satisfied when the residue characteristic of k is large, S. DeBacker used Moy-Prasad theory to sharpen, refine and generalize some results of Waldspurger, establishing an explicit form of the Howe conjecture for G and proving the conjecture of T. Hales, A. Moy and G. Prasad on the range of validity for the Harish-Chandra–Howe local expansion for characters of irreducible admissible representations of G. We will report on joint work with J. Adler and E. Sayag, in which we pursue similar questions for p-adic symmetric spaces.

The theory of invariant random subgroups (IRS), which has been developed quite rapidly during the last decade, has been very fruitful to the study of lattices and their asymptotic invariants. However, restricting to invariant measures limits the scope of problems that one can approach (in particular since the groups involved are highly non-amenable). It was recently realised that the more general notion of stationary random subgroups (SRS) is still very effective and opens paths to deal with questions which previously seemed out of reach.

In the talk I will describe various old and new results concerning arithmetic groups and general locally symmetric manifolds of finite as well as infinite volume that can be proved using ‘randomness’, e.g.:
1. Kazhdan-Margulis minimal covolume theorem;
2. Most hyperbolic manifolds are non-arithmetic (joint work with A. Levit);
3. Higher rank manifolds of large volume have a large injectivity radius (joint with Abert, Bergeron, Biringer, Nikolov, Raimbault and Samet); 
4. Margulis’ infinite injectivity radius conjecture: For manifolds of rank at least 2, finite volume is equivalent to bounded injectivity radius (joint with M. Fraczyk).

A well-known result of Hasse states that the local-global principle holds for norms over number fields for cyclic extensions. In other words, if L/F is a cyclic extension of number fields then an element λ ∈ F× is in the image of norm map NL/F : L× → F× if and only if λ is in the image of the norm map locally everywhere,  i.e., for completions associated to all archimedean and non-archimedean places of F. In this talk, we would consider local-global principles for norms and product of norms over fields which are function fields of curves over complete discretely valued fields, for example, C((t))(x).

To any group G is associated the space Ch(G) of all characters on G. After defining this space and discussing its interesting properties, I’ll turn to discuss dynamics on such spaces. Our main result is that the action of any arithmetic group on the character space of its amenable/solvable radical is stiff, i.e, any probability measure which is stationary under random walks must be invariant. This generalizes a classical theorem of Furstenberg for dynamics on tori. Relying on works of Bader,  Boutonnet, Houdayer, and Peterson, this stiffness result is used to deduce dichotomy statements (and ’charmenability’) for higher rank arithmetic groups pertaining to their normal subgroups, dynamical systems, representation theory and more. The talk is based on a joint work with Uri Bader.