ICTS

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.)

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.

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.

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.

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.

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.

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.

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.

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 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.

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.

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.

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.

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.

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.

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.

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.

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.

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.