Seminars in Lie Groups

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Jonas Deré: Simply transitive NIL-affine actions of soluble Lie groups

26th August, 2024

Although not every 1-connected soluble Lie group G admits a simply transitive action via affine maps on ℝn, it is known that such an action exists if one replaces ℝn by a suitable nilpotent Lie group H, depending on G. However, not much is known about which pairs of Lie groups (G,H) admit such an action, where ideally you only need information about the Lie algebras corresponding to G and H. In recent work with Marcos Origlia, we show that every simply transitive action induces a post-Lie algebra structure on the corresponding Lie algebras. Moreover, if H has nilpotency class 2 we characterize the post-Lie algebra structures coming from such an action by giving a new definition of completeness, extending the known cases where G is nilpotent or H is abelian.

Yifan Jing: Measure Doubling for Small Sets in SO3(ℝ)

30th May, 2023

Let SO3(ℝ) be the 3D-rotation group equipped with the real-manifold topology and the normalized Haar measure μ. Confirming a conjecture by Breuillard and Green, we show that if A is an open subset of SO3(ℝ) with sufficiently small measure, then μ(A2) > 3.99 μ(A).

Emma Brink: Condensed Group Cohomology

18th April, 2023

The theory of condensed sets, developed by Dustin Clausen and Peter Scholze, provides a framework well-suited to study algebraic objects that carry a topology. In my talk, I will discuss the basic properties of the cohomology of condensed groups and its relation to continuous group cohomology. Johannes Anschütz and Arthur-César le Bras showed that for locally profinite groups and solid (e.g. discrete) coefficients, condensed group cohomology agrees with continuous group cohomology. On the other hand, if G is a locally compact and locally contractible topological group (e.g., a Lie group), and M is a discrete group with trivial G-action, then the condensed group cohomology of G, M (the sheaves of continuous functions into G and M) is isomorphic to the singular cohomology of the classifying space of G with coefficients in M, whereas the continuous group cohomology of G with coefficients in M is isomorphic to the singular cohomology of the classifying space of π0(G) with coefficients in M.

Generalizing results of Johannes Anschütz and Arthur-César le Bras on locally profinite groups, I will explain that continuous group cohomology with solid coefficients can be described as a cohomological δ-functor in the condensed setting for a large class of topological groups.

Zhiren Wang: Classification of Smooth Actions by Higher-Rank Lattices in Critical Dimensions

17th April, 2023

The Zimmer programme asks how lattices in higher-rank semisimple Lie groups may act smoothly on compact manifolds. Below a certain critical dimension, the recent proof of the Zimmer conjecture by Brown-Fisher-Hurtado asserts that, for SLn(ℝ) with n ≥ 3 or other higher rank ℝ-split semisimple Lie groups, the action is trivial up to a finite group action. In this talk, we will explain what happens in the critical dimension for higher rank ℝ-split semisimple Lie groups. For example, non-trivial actions by lattices in SLn(ℝ), n ≥ 3, on (n-1)-dimensional manifolds are isomorphic to the standard action on ℝPn-1 up to a finite quotient group and a finite covering.

Yifan Jing: Measure Growth in Compact Simple Lie Groups

8th November, 2022

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| ≫ min ( |A|1+c, |G| ). In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset of a compact simple Lie group G, then μ(AAA) > min ( (3+c)μ(A), 1 ), where μ is the normalized Haar measure on G. I will also talk about how to use this result to solve the Kemperman Inverse Problem, and discuss what will happen when G has high dimension or when G is non-compact.

Matthew Conder: Discrete 2-generator subgroups of PSL2(ℚp)

10th October, 2022

Discrete 2-generator subgroups of PSL2(ℝ) have been extensively studied by investigating their action by Möbius transformations on the hyperbolic plane. Due to work of Gilman, Rosenberger, Purzitsky and many others, there is a complete classification of such groups by isomorphism type, and an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℝ) is discrete.

Here we completely classify discrete 2-generator subgroups of PSL2(ℚp) over the p-adic numbers ℚp by studying their action by isometries on the corresponding Bruhat-Tits tree. We give an algorithm to decide whether or not a 2-generator subgroup of PSL2(ℚp) is discrete, and discuss how this can be used to decide whether or not a 2-generator subgroup of SL2(ℚp) is dense.

Matthew Conder: Discrete two-generator subgroups of PSL2(ℚp)

1st July, 2022

Due to work of Gilman, Rosenberger, Purzitsky and many others, discrete two-generator subgroups of PSL2(ℝ) have been completely classified by studying their action by Möbius transformations on the hyperbolic plane. Here we aim to classify discrete two-generator subgroups of PSL2(ℚp) by studying their action by isometries on the Bruhat-Tits tree. We first give a general structure theorem for two-generator groups acting by isometries on a tree, which relies on certain Klein-Maskit combination theorems. We will then discuss how this theorem can be applied to determine discreteness of a two-generator subgroup of PSL2(ℚp).

Dipendra Prasad: Branching laws: homological aspects

19th May, 2022

This lecture will partly survey branching laws for real and p-adic groups which often is related to period integrals of automorphic representations, discuss some of the more recent developments, focusing attention on homological aspects and the Bernstein decomposition.

Lisa Carbone: A Lie Group Analogue for the Monster Lie Algebra

29th November, 2021

The Monster Lie algebra 𝔪 is an infinite-dimensional Lie algebra constructed by Borcherds as part of his programme to solve the Conway-Norton Monstrous Moonshine Conjecture. We describe how one may approach the problem of associating a Lie group analogue for 𝔪 and we outline some constructions.

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.