Monday, 01 January 2024

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little wa known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.

In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.

1. Global geometry: Properness criterion and its quantitative estimate for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.

2. Spectral analysis: Construction of periodic L2-eigenfunctions for the Laplacian with indefinite signature, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition on the locally homogeneous space of indefinite metric.

Let X be a complex toric variety equipped with the action of an algebraic torus T, and let G be a complex linear algebraic group. We classify all T-equivariant principal G-bundles E over X and the morphisms between them. We then give a criterion for the equivariant reduction of the structure group of E to a Levi subgroup of G in terms of the automorphism group of the bundle. (With A. Dey, J. Dasgupta, B. Khan and M. Poddar.)

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/H be a homogeneous space. If a discrete subgroup of G acts properly and freely on G/H, the quotient space becomes a manifold locally modelled on G/H, and called a Clifford-Klein form. Since the late 1980s, the existence problem of compact Clifford-Klein forms of pseudo-Riemannian homogeneous spaces has been studied by various methods (e.g. Cartan projection of reductive Lie groups, rigidity theory in homogeneous dynamics, tempered unitary representations, de Rham and relative Lie algebra cohomology, Anosov representations). In this talk, I will explain my joint work-in-progress with Fanny Kassel (IHES) and Nicolas Tholozan (ENS) on a new necessary condition for the existence of compact Clifford-Klein forms. It is formulated in terms of the homotopy theory of sphere bundles and hence related to the Adams operations in KO-theory.

Tuesday, 02 January 2024

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little wa known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.

In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.

1. Global geometry: Properness criterion and its quantitative estimate for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.

2. Spectral analysis: Construction of periodic L2-eigenfunctions for the Laplacian with indefinite signature, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition on the locally homogeneous space of indefinite metric.

We discuss the moduli of binary cubic forms. We give a description of this moduli space in terms triples of an associated CM elliptic curve E, a degree-3 isogeny from E to E, and a point on E. We will also discuss an application of our construction.

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).

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.

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.

Wednesday, 03 January 2024

The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little wa known about global properties of the geometry even if we impose a locally homogeneous structure. This theme has been developed rapidly in the last three decades.

In the series of lectures, I plan to discuss two topics by the general theory and some typical examples.

1. Global geometry: Properness criterion and its quantitative estimate for the action of discrete groups of isometries on reductive homogeneous spaces, existence problem of compact manifolds modeled on homogeneous spaces, and their deformation theory.

2. Spectral analysis: Construction of periodic L2-eigenfunctions for the Laplacian with indefinite signature, stability question of eigenvalues under deformation of geometric structure, and spectral decomposition on the locally homogeneous space of indefinite metric.

Talk 1 (Kaluba) We will discuss the notion of positivity in rings, understood as being a sum of squares. This problem is classically relevant for polynomials, and in the context of property (T) it appears in the group ring and its augmentation ideal. We will discuss a computer-assisted approach, via semidefinite programming, of verifying whether an element of the group ring can be expressed as a sum of squares. An important feature of this method is that despite using numerical information, in certain situations it provides a rigorous proof of positivity. Together with a characterization of property (T) via positivity in the group ring this provides an approach to proving Kazhdan’s property (T) for some groups.

Talk 2 (Nowak) We will use the methods described in the first talk to prove property (T) for automorphism groups of free groups. First we will consider a singular case of Aut(F5) but the main focus will be the infinite case. We will describe an inductive procedure that allows to deduce property (T) for families of groups like SLn(Z) and Aut(Fn) from positivity of a single group ring element. Using this inductive approach we will prove property (T) for Aut(Fn) for all n ≥ 6, and reprove property (T) for SLn(Z), n ≥ 3. We will also discuss several consequences and applications, in particular we will show new estimates for Kazhdan constants.

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.)

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.

Thursday, 04 January 2024

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.

I will speak on work with F. Bleher and A. Lubotzky arising from a construction by Lubotzky and Grunewald of linear representations of the automorphism group Aut(Fn) of a free group on n elements. It turns out that if one replaces Fn by it’s profinite completion ˆ Fn, the construction of linear representations with large image of Aut( ˆ Fn) becomes simpler and more general than the discrete case. This leads to a solution of an open problem from the 1990’s having to do with constructing natural non-abelian representations of the so-called Grothendieck-Teichm¨uller group GT inside Aut( ˆ F2). I will discuss where GT comes from and current open problems in this area.

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.

We will present our joint work with I. Biswas and C. Maity on certain explicit descriptions of lower dimensional Betti numbers of homogeneous spaces of Lie groups. We will begin by recalling some of the earlier works in this subject and give our motivation. We will then describe our results and some of the applications in special cases.

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.

A basic question in group theory is what can we learn about a finitely generated and residually finite group from its profinite completion? In this talk, we will focus on the relation between the widths of conjugacy classes in a higher rank arithmetic group to the corresponding widths in the profinite completion of this group and explain the connection to the Congruence Subgroup Property. This is joint work with Nir Avni.

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.

Friday, 05 January 2024

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

Let K be an imaginary quadratic field and E/K be an elliptic curve with good ordinary reduction at an odd prime number p. We study the rational points of E along the anticyclotomic Zp-extension of K.

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.

Given a finite set of elements in a semisimple algebraic group over the field of algebraic numbers, one can define its normalized height as a weighted sum over all places of its associated joint spectral radii. The height gap theorem asserts that this height is bounded away from zero provided the subgroup generated by the finite set is Zariski-dense. This result can be seen as a non-commutative analogue to the Lehmer or Bogomolov problem in diophantine number theory. In recent joint work with Oren Becker, we show how this can be used to deduce uniform spectral

gaps for actions of Zariski-dense subgroups on homogeneous spaces of algebraic groups and establish the existence of a uniform lower bound on the first eigenvalue of Cayley graphs of certain finite simple groups of Lie type.

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.

Monday, 08 January 2024

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 give examples of non-commensurable but isospectral locally symmetric spaces, thereby completing the work of Lubotzky, Samuels and Vishne. The main step is to show that adelic conjugation of lattices in SL(1,D) by the adjoint group preserves the spectrum, where D is a division algebra over a number field F (under some additional hypothesis), extending the work done by one of us, when D is quaternion. This is joint work with Sandeep Varma.

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.

For connected reductive groups G over non-archimedean local fields k, Bruhat-Tits theory provides a metric space B(G) (called the building of G) equipped with a G(k)-action that is a valuable tool for working with“large” compact open subgroups of G(k). In particular, the conjugacy classes of maximal compact open subgroups (there is often more than one, unlike for connected Lie groups) can be understood through the study of G(k)-stabilizers of certain points in B(G). The understanding of the subgroup structure of G(k) provided by Bruhat-Tits theory has made it a very powerful tool in number theory and representation theory. In these lectures, after some preliminaries about local fields and BN-pairs, we’ll give an axiomatic overview of the general theory (including some examples), discuss the utility of passage to “large” residue fields, and deduce some applications (such as useful group-theoretic decompositions). Further applications in representation theory will be discussed separately by Fintzen.

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.

Tuesday, 09 January 2024

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.

In 2005, Lubotzky, Samuels and Vishne provided the first examples of noncommensurable compact isospectral locally symmetric spaces. These examples were constructed as quotients of the symmetric space SLn(R)/K associated to the group SLn(R) by suitable arithmetic subgroups, and the proof of their isospectrality relied on some results of Harris and Taylor. In this talk, I will introduce a simpler proof of isospectrality of the relevant locally symmetric spaces which is based on a direct comparison of the Selberg trace formulas. The method is likely to be applicable to other types of locally symmetric spaces. This is joint work with Andrei Rapinchuk.

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.

For connected reductive groups G over non-archimedean local fields k, Bruhat-Tits theory provides a metric space B(G) (called the building of G) equipped with a G(k)-action that is a valuable tool for working with“large” compact open subgroups of G(k). In particular, the conjugacy classes of maximal compact open subgroups (there is often more than one, unlike for connected Lie groups) can be understood through the study of G(k)-stabilizers of certain points in B(G). The understanding of the subgroup structure of G(k) provided by Bruhat-Tits theory has made it a very powerful tool in number theory and representation theory. In these lectures, after some preliminaries about local fields and BN-pairs, we’ll give an axiomatic overview of the general theory (including some examples), discuss the utility of passage to “large” residue fields, and deduce some applications (such as useful group-theoretic decompositions). Further applications in representation theory will be discussed separately by Fintzen.

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.

Wednesday, 10 January 2024

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.

Given a finite index subgroup Γ of G(Z), where G is a Q-simple algebraic group of Q-rank at least one and R-rank at least two, consider the group N generated by Γ ∩ U+(Z) and Γ ∩ U−(Z). Here U+ is the unipotent radical of a minimal parabolic Q-subgroup of G, and U− is its opposite. An old result of Tits, Vaserstein, Raghunathan and myself says that N has finite index in G(Z). The proof presented here is more streamlined and is much simpler. The same proof shows the centrality of the congruence subgroup kernel, showing that the group G(Z) has the congruence subgroup property.

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.

For connected reductive groups G over non-archimedean local fields k, Bruhat-Tits theory provides a metric space B(G) (called the building of G) equipped with a G(k)-action that is a valuable tool for working with“large” compact open subgroups of G(k). In particular, the conjugacy classes of maximal compact open subgroups (there is often more than one, unlike for connected Lie groups) can be understood through the study of G(k)-stabilizers of certain points in B(G). The understanding of the subgroup structure of G(k) provided by Bruhat-Tits theory has made it a very powerful tool in number theory and representation theory. In these lectures, after some preliminaries about local fields and BN-pairs, we’ll give an axiomatic overview of the general theory (including some examples), discuss the utility of passage to “large” residue fields, and deduce some applications (such as useful group-theoretic decompositions). Further applications in representation theory will be discussed separately by Fintzen.

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.

Thursday, 11 January 2024

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 1989, G. Prasad established a hands-on formula to compute the covolume of S-arithmetic subgroups of semisimple groups over global fields. The formula has since then found many applications in the theory of arithmetic groups, starting with the contemporary Borel–Prasad finiteness theorem. In the talk, we aim to describe Prasad’s volume formula and its contents. We will then survey some of its most striking applications, such as the finiteness theorem, the classification of fake projective planes, and the study of lattices of small covolume in various simple groups.

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 an algebraic group over a local field k. We show that the image of G(k) under every continuous homomorphism into a (Hausdorff) topological group is closed if and only if the center of G(k) is compact. This is joint work with Uri Bader.

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.

In a joint work with Nikolay Bogachev, Alexander Kolpakov and Leone Slavich we show that immersed totally geodesic m-dimensional suborbifolds of an n-dimensional arithmetic hyperbolic orbifold correspond to finite subgroups of the commensurator whenever m is sufficiently large. In particular, for n = 3 this condition includes all totally geodesic suborbifolds. We call such totally geodesic subspaces by finite centraliser subspaces (or fc-subspaces for short) and use them to formulate an arithmeticity criterion for hyperbolic lattices. We show that a hyperbolic orbifold is arithmetic if and only if it has infinitely many fc-subspaces, while in the non-arithmetic case the number of fc-subspaces is finite and bounded in terms of the volume. The case of special interest is that of exceptional trialitarian 7-dimensional orbifolds – we show that every such orbifold contains totally geodesic arithmetic hyperbolic 3-orbifolds of exceptional type.

Friday, 12 January 2024

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).

Bertrand Deroin and Sebastian Hurtado recently proved the 30-yearold conjecture that if G is an almost-simple algebraic Q-group, and the real rank of G is at least two, then no arithmetic subgroup of G is left-orderable. We will discuss this theorem, and explain some of the main ideas of the proof, by illustrating them in the simpler case where the real field R is replaced with a p-adic field. Harmonic functions and continuous group actions are key tools.

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).

Motivated by a recent result of M. F. Bourque and B. Petri, we constructed a sequence of hyperbolic manifolds with a large number of closed geodesics of shortest length. The aim of this talk is to explain what ”large” means, and how arithmetic groups enter in this context. This will involve results obtained in collaboration with Cayo D´oria and Emanoel Freire.

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.