Tag - Lie groups

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.

Yuri Berest: Spaces of quasi-invariants and homotopy Lie groups

22nd July, 2021

Quasi-invariants are natural algebraic generalizations of classical invariant polynomials of finite reflection groups. They first appeared in mathematical physics - in the work of O. Chalykh and A. Veselov on quantum integrable systems - in the early 1990s, and since then have found many interesting applications in other areas: most notably, representation theory, algebraic geometry and combinatorics. In this talk, I will explain how the algebras of quasi-invariants arise in topology: as cohomology rings of certain spaces naturally attached to compact connected Lie groups. Our main result is a generalization of a well-known theorem of A. Borel that realizes the algebra of classical invariant polynomials of a Weyl group W(G) as the cohomology ring of the classifying space BG of the corresponding Lie group G. Perhaps most interesting here is the fact that our construction of spaces of quasi-invariants is purely homotopy-theoretic. It can therefore be extended to some non-Coxeter (p-adic pseudo-reflection) groups, in which case the compact Lie groups are replaced by the so-called p-compact groups (a.k.a. homotopy Lie groups).

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.

Doron Puder: Word-Measures on Unitary Groups

2nd February, 2017

One approach to studying properties of random walks on groups with random generators is to study word-measures on these groups. This approach was proven useful for the study of symmetric groups and random regular graphs. In the current work we focus on the unitary groups U(n). For example, if w is a word in F2 = <x,y>, sample at random two elements from U(n), A for x and B for y, and evaluate w(A,B). The measure of this random element is called the w measure on U(n). We study the expected trace (and other invariants) of a random unitary matrix sampled from U(n) according to the w-measure, and find surprising algebraic properties of w that determine these quantities.

Hee Oh: Effective circle count for Apollonian circle packings, via spectral methods

6th February, 2012

We will describe a recent effective counting result for Apollonian circle packings. The main ingredient of this result is an effective equidistribution of closed horospheres in an infinite volume hyperbolic 3-manifold whose fundamental group has critical exponent bigger than one. We will explain how the spectral theory of Lax and Phillips can be used for such equidistribution results.