Tag - Group theory

Mikhail Belolipetsky: Growth of lattices in semisimple Lie groups

22nd April, 2021

A discrete subgroup G of a Lie group H is called a lattice if the quotient space G/H has finite volume. By a classical theorem of Bieberbach we know that the group of isometries of an n-dimensional Euclidean space has only finitely many different types of lattices. The situation is different for the semisimple Lie groups H. Here the total number of lattices is infinite and we can study its growth rate with respect to the covolume. This topic has been a subject of our joint work with A. Lubotzky for a number of years. In the talk I will discuss our work and some other more recent related results.

Laura Ciobanu: Free group homomorphisms and the Post Correspondence Problem

19th April, 2021

The Post Correspondence Problem (PCP) is a classical problem in computer science that can be stated as: is it decidable whether, given two morphisms g and h between two free semigroups A and B, there is any non-trivial x in A such that g(x)=h(x)? This question can be phrased in terms of equalizers, asked in the context of free groups, and expanded: if the 'equalizer' of g and h is defined to be the subgroup consisting of all x where g(x)=h(x), it is natural to wonder not only whether the equalizer is trivial, but what its rank or basis might be. While the PCP for semigroups is famously insoluble and acts as a source of undecidability in many areas of computer science, the PCP for free groups is open, as are the related questions about rank, basis, or further generalizations. However, in this talk we will show that there are links and surprising equivalences between these problems in free groups, and classes of maps for which we can give complete answers.

Tamar Bar-On: Segal and Nikolov Theorem and applications

18th April, 2021

A profinite group is called strongly complete if every subgroup of finite index is open. Strongly complete groups are very useful, since in such groups the algebra determines the topology. For example, every homomorphism from a strongly complete group to any profinite group is continuous, and thus a homomorphism in the category of profinite groups. For many years it was an open question, whether every finitely generated profinite group is strongly complete. In 2000 Segal and Nikolov published a positive proof for this conjecture. In the talk we present the general idea of the proof, and show some nice results relying on this theorem.

Tim Susse: Geometric Properties of Random Right-angled Coxeter Groups

15th April, 2021

Given a finite simplicial graph, we can generate a right-angled Coxeter group (RACG): each vertex corresponds to a generator and two generators commute if and only if the corresponding vertices are adjacent. This assignment is unique up to isomorphism, which allows us to characterize geometric and algebraic properties of the RACG using combinatorial properties of its generating graph. This also allows us to use a model of random graphs to study random RACGs.

In this talk we will focus on the divergence of a RACG. Using the Erdös-Renyi random graph model, we show that at a wide range of densities a random RACG asymptotically almost surely has quadratic divergence. We will also give a sharp threshold, below which a random RACG has (almost surely) at least cubic divergence, and above which its divergence is (almost surely) at most quadratic. This is joint work with Jason Behrstock, Victor Falgas-Ravry and Mark Hagen.

Pham Huu Tiep: Character bounds for finite simple groups

12th April, 2021

Given the current knowledge of complex representations of finite simple groups, obtaining good upper bounds for their characters values is still a difficult problem, a satisfactory solution of which would have significant implications in a number of applications. We first discuss some such applications. Then we will report on recent results that produce such character bounds.

Oren Becker: Stability of approximate group actions

11th April, 2021

An approximate unitary representation of a group G is a function f from G to U(n) such that f(gh) is close to f(g)f(h) for all g,h. Is every approximate unitary representation just a slight deformation of a unitary representation? The answer depends on G and on the norm on U(n). If G is amenable, the answer is positive for the operator norm on U(n) (Kazhdan '82). The answer remains positive if we use the normalized Hilbert-Schmidt norm and allow a slight change in the dimension n (Gowers-Hatami '15, De Chiffre-Ozawa-Thom '17). For both norms, the answer is negative if G is a non-abelian free group (or a non-elementary word-hyperbolic group). In this talk we shall discuss a similar notion where U(n) is replaced by Sym(n) with the normalized Hamming metric. We study the cases where G is either free, amenable or equal to SLr(ℤ), r ≥ 3. When G is finite, a slight variation of our main theorem provides an efficient probabilistic algorithm to determine whether a function f from G to Sym(n) is close to a homomorphism when |G| and n are both large.

Cornelius Pillen: On the Humphreys-Verma conjecture and Donkin’s tilting module conjecture

29th March, 2021

This talk focuses on the Humphreys-Verma conjecture about the lifting of principal indecomposable modules for restricted Lie algebras to their ambient algebraic groups and on Donkin's tilting module conjecture. Several techniques that are needed to tackle the conjectures for small primes will be introduced and results for the exceptional group G2 will be discussed in some detail.

Apoorva Khare: Polymath14: Groups with norms

25th March, 2021

Consider the following three properties of a general group G:

Algebra: G is abelian and torsion-free.

Analysis: G is a metric space that admits a 'norm', namely, a translation-invariant metric d( . , . ) satisfying: d(1,gn) = |n| d(1,g) for all g ∈ G and integers n.

Geometry: G admits a length function with 'saturated' subadditivity for equal arguments: l(g2) = 2 l(g) for all g ∈ G.

While these properties may a priori seem different, in fact they turn out to be equivalent (and also to G being isometrically and additively embedded in a Banach space, hence inheriting its norm). The non-trivial implication amounts to saying that there does not exist a non-abelian group with a 'norm'. We will discuss motivations from analysis, probability, and geometry; then the proof of the above equivalences; and finally, the logistics of how the problem was solved, via a PolyMath project that began on a blog post of Terence Tao.

Arman Darbinyan: Subgroups of left-orderable groups

18th March, 2021

A recent advancement in the theory of left-orderable groups is the discovery of finitely generated left-orderable simple groups by Hyde and Lodha. We will discuss a construction that extends this result by showing that every countable left-orderable group is a subgroup of such a group. In conjunction with this construction, we will also discuss computability properties of left-orders in groups. Based on a joint work with M. Steenbock.