Tag - Infinite groups

Jingyin Huang: The Helly geometry of some Garside and Artin groups

15th November, 2021

Artin groups emerged from the study of braid groups and complex hyperplane arrangements. Artin groups have very simple presentation, yet rather mysterious geometry with many basic questions widely open. I will present a way of understanding certain Artin groups and Garside groups by building geometric models on which they act. These geometric models are non-positively curved in an appropriate sense, and such curvature structure yields several new results on the algorithmic, topological and geometric aspects of these groups. No previous knowledge on Artin groups or Garside groups is required.

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.

John Mackay: Conformal dimension and decompositions of hyperbolic groups

22nd October, 2021

The boundary of a Gromov hyperbolic group carries a canonical family of metrics which determine the quasi-isometry type of the group. Pansu's conformal dimension of the boundary gives a natural and important quasi-isometric invariant. I will discuss how this invariant behaves when the group splits over two-ended subgroups (i.e. when the boundary has local cut points), and applications to the question of Bonk and Kleiner asking for a characterization of when this dimension equals one.

Oksana Bezushchak: Locally matrix algebras and algebras of Mackey

21st October, 2021

n this talk we will discuss:

1. Tensor decompositions of locally matrix algebras and their parametrization by Steinitz numbers.

2. Automorphisms and derivations of locally matrix algebras.

3. Automorphisms and derivations of Mackey algebras and Mackey groups. In particular, we describe automorphisms of all infinite simple finitary torsion groups (in the classification of J.Hall) and derivations of all infinite-dimensional simple finitary Lie algebras (in the classification of A.Baranov and H.Strade).

Chloe Papin: A Whitehead Algorithm for Generalized Baumslag-Solitar Groups

15th October, 2021

Baumslag-Solitar groups BS(p,q) =< a,t | tapt-1 = aq > were first introduced as examples of non-Hopfian groups. They may be described using graphs of cyclic groups. In analogy with the study of Out(Fn) one can study their automorphisms through their action on an "outer space". After introducing generalized Baumslag-Solitar groups and their actions on trees, I will present an analogue of a Whitehead algorithm which takes an element of a free group and decides whether there exists a free factor which contains that element.

George Willis: Constructing groups with flat-rank greater than 1

30th August, 2021

The contraction subgroup for x in the locally compact group, G,

con(x)={ g ∈ G ∣ xngx−n → 1 as n → ∞ },

and the Levi subgroup is

lev(x)={ g ∈ G ∣ {xngx−n}n∈ℤ has compact closure }.

The following will be shown. Let G be a totally disconnected, locally compact group and x ∈ G. Let y ∈ lev(x). Then there are x′ ∈ G and a compact subgroup, K ≤ G such that:
-K is normalized by x′ and y,
-con(x′)=con(x) and lev(x′)=lev(x) and
-the group ⟨x′,y,K⟩ is abelian modulo K, and hence flat.
If no compact open subgroup of G normalized by x and no compact open subgroup of lev(x) normalized by y, then the flat-rank of ⟨x′,y,K⟩ is equal to 2.

James Parkinson: Automata for Coxeter groups

9th August, 2021

In 1993 Brink and Howlett proved that finitely generated Coxeter groups are automatic. In particular, they constructed a finite state automaton recognizing the language of reduced words in the Coxeter group. This automaton is constructed in terms of the remarkable set of "elementary roots" in the associated root system.

In this talk we outline the construction of Brink and Howlett. We also describe the minimal automaton recognizing the language of reduced words, and prove necessary and sufficient conditions for the Brink-Howlett automaton to coincide with this minimal automaton. This resolves a conjecture of Hohlweg, Nadeau, and Williams, and is joint work with Yeeka Yau.

Sven Raum: Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

9th August, 2021

In the 90s, Nebbia conjectured that a group of tree automorphisms acting transitively on the tree's boundary must be of type I, that is, its unitary representations can in principal be classified. For key examples, such as Burger-Mozes groups, this conjecture is verified. Aiming for a better understanding of Nebbia's conjecture and a better understanding of representation theory of groups acting on trees, it is natural to ask whether there is a characterization of type I groups acting on trees. In 2016, we introduced in collaboration with Cyril Houdayer a refinement of Nebbia's conjecture to a trichotomy, opposing type I groups with groups whose von Neumann algebra is non-amenable. For large classes of groups, including Burger-Mozes groups, we could verify this trichotomy.

In this talk, I will motivate and introduce the conjecture trichotomy for groups acting on tress and explain how von Neumann algebraic techniques enter the picture.

Alex Lubotzky: First-order rigidity of high-rank arithmetic groups

16th July, 2021

The family of high-rank arithmetic groups is a class of groups playing an important role in various areas of mathematics. It includes SLn(ℤ), for n ≥ 3, SLn(ℤ[1/p]) for n ≥ 2, their finite-index subgroups and many more. A number of remarkable results about them have been proven including: Weil local rigidity, Mostow strong rigidity, Margulis superrigidity and the Schwartz-Eskin-Farb Quasi-isometric rigidity. We will add a new type of rigidity: 'first order rigidity'. Namely if D is such a non-uniform characteristic zero arithmetic group and L a finitely generated group which is elementary equivalent to D then L is isomorphic to D. This stands in contrast with Zlil Sela's remarkable work which implies that the free groups, surface groups and hyperbolic groups (many of which are low-rank arithmetic groups) have many non-isomorphic finitely generated groups which are elementarily equivalent to them.