Symmetric spaces and generalizations

Abstract: In 2015 Gelander described how to compactify the moduli space of finite-area hyperbolic surfaces using invariant random subgroups. The goal of my two talks is to describe „what“ this IRS compactification is. 

In my first talk I will introduce invariant random subgroups and the moduli space of finite-area hyperbolic surfaces. At the end I will describe Gelander’s construction of the IRS compactification.

In my second talk I will relate the IRS compactification to the augmented moduli space, also known as the Deligne—Mumford compactification. We will see that there is a natural map from the latter to the former. Moreover, we will see, that this map is a continuous finite-to-one surjection and that the cardinalities of its fibers admit a uniform upper bound. It will turn out, that this upper bound depends only on the topology of the underlying surface.

Abstract: This is a two-part talk. In part 1, we introduce the space of pointed hyperbolic surfaces. Some facts about hyperbolic surfaces are recalled. We also draw a connection to Chabauty topology. In part 2, we sketch proofs for path-connectivity and local connectivity of the space at some particular points.

Abstract: The group of automorphisms of a bi-regular tree contains a rich class of non-linear subgroups G that share similar properties as the linear ones. Given that, classical questions from homogeneous dynamics can be examined and proved. For example, if H is a discrete subgroup of G, recent results show there is a classification of ergodic probability measures on G / H that are invariant under horospherical subgroups. When H is moreover a cocompact lattice, the horospherical action is uniquely ergodic. The talk will contain a gentle introduction to classical homogeneous dynamics, a brief overview of our main results, as well as some of the main ingredients of the proofs. This is a joint project with Vladimir Finkelshtein and Cagri Sert.

Abstract: I introduce Kac--Moody symmetric spaces and present a theorem about them being colimits of a certain diagram consisting of their rank-1 and rank-2 subspaces.

Abstract: Symmetric spaces arise in many areas of mathematics and physics and have been extensively studied. There are for example several compactifications of a symmetric space on non-compact type. In the first part of this talk (video 1), we will give some basics about symmetric spaces and then introduce the Satake and the Martin compactification of symmetric spaces and show how they are related. In the second part (video 2), we will explain the horofunction compactification for the general setting of a metric space and give an explicit description of it for a normed vector space with a "nice" norm. We will then use this result to determine the horofunction compactification of a symmetric space X of non-compact type. Finally we show how any Satake or Martin compactification of X can be realized as a horofunction compactification with respect to an approriate Finsler metric on X. 

Abstract: Let G be a simple higher rank Lie group and let X be the associated symmetric space. Margulis conjectured that any discrete subgroup Gamma of G such that X/Gamma has uniformly bounded injectivity radius must be a lattice. I will present the proof of this conjecture and explain how stationary random subgroups play the central role in the argument.x

Abstract: In the first talk I introduce buildings and Kac-Moody groups. In the second talk, I introduce the Davis complex of a building and present a formula for L^2-Betti numbers of groups acting on buildings. The goal is to exhibit a family of Coxeter diagrams from which we produce finitely presented simple Kac-Moody groups that lie in different measure equivalence classes. 

Abstract: In this talk we present the idea that the volume of locally symmetric spaces controls their complexity. We will then show that the minimal number of generators of certain lattices in higher rank semi-simple Lie groups is sub-linear in their co-volume. Finally, we will present the situation in other types of Lie groups.

Abstract: We introduce the notion of systolic complexes and give conditions on presentations to construct such complexes using Cayley graphs. We give examples of groups admitting such a presentation.

The Lehmer problem (1933), also referred to as the `Lehmer conjecture', asks whether there is a uniform lower bound on the Mahler measure of algebraic integers which are not roots of unity. Although rooted in number theory, many interesting connections between the Lehmer problem and other fields, including combinatorics in finite fields and geometric group theory. Following his celebrated Arithmeticity Theorem, Margulis conjectured in his book (1991) the uniform discreteness of cocompact lattices in higher rank semisimple Lie groups and observed that it would follow from a weak form of the Lehmer conjecture. We will discuss the equivalence of these conjectures and some refinements. This is based on joint work with François Thilmany.