| Résumé: | Let $K$ be a non-archimedean local field with residue field of characteristic $p$. We give necessary and sufficient conditions for a two-generator subgroup $G$ of $\mathrm{PSL}_2(K)$ to be discrete, where either $K=\mathbb{Q}_p$ or $G$ contains no elements of order $p$. We give a practical algorithm to decide whether such a subgroup $G$ is discrete. We also give practical algorithms to decide whether a two-generator subgroup of $\mathrm{SL}_2(\mathbb{Q}_p)$ or $\mathrm{SL}_2(\mathbb{R})$ is dense. A crucial ingredient for this work is a structure theorem for two-generator groups acting by isometries on a $\Lambda$-tree. |
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At the intersection of data science and algebraic topology, topological data analysis (TDA) is a recent field of study, which provides robust mathematical, statistical and algorithmic methods to analyze the topological and geometric structures underlying complex data. TDA has proved its utility in many applications, including biology, material science and climate science, and it is still rapidly evolving. Barcodes are frequently used invariants in TDA. They provide topological summaries of the persistent homology of a filtered space. Understanding the structure and geometry of the space of barcodes is hence crucial for applications.
In this talk, we use Coxeter complexes to define new coordinates on the space of barcodes. These coordinates define a stratification of the space of barcodes with n bars where the highest dimensional strata are indexed by the symmetric group. This creates a bridge between the fields of TDA, geometric group theory and permutation statistics, which could be exploited by researchers from each field. No prerequisite on TDA or Coxeter complexes are required. |
| Résumé: | A Hadamard manifold – or more generally a CAT(0) space – is said to have higher rank if every geodesic line lies in a flat plane. If a higher rank Hadamard manifold admits finite volume quotients, then it has to be a symmetric space or split as a direct product. This is the content of Ballmann’s celebrated Rank Rigidity Theorem, proved in the 80s. It has been conjectured by Ballmann that his theorem generalizes to the synthetic setting of CAT(0) spaces. In the talk I will discuss Ballmann’s conjecture and report on recent progress. |
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Groups that act faithfully on rooted trees can be studied via their finite quotients. There are several natural collections of finite quotients that can be chosen for this. The mathematical object that encodes all the finite quotients in a particular collection and the maps between them is the profinite completion of the group (with respect to the chosen collection). Taking all possible finite quotients of the group gives *the* profinite completion of the group, and this maps onto each of the other completions. Determining the kernels of these maps is known as the congruence subgroup problem. This has been studied by various authors over the last few years, most notably for self-similar groups and (weakly) branch groups. In the case of self-similar regular branch groups, much insight can be gained into this problem using a symbolic-dynamical point of view. After reviewing the problem and previous work on it, I will report on work in progress with Zoran Sunic on determining the dynamical complexity of these completions and calculating some of these kernels with relative ease.
Examples will be given. No previous knowledge of profinite, self-similar or branch groups is required. |
| Résumé: | I will present the possible Gromov-Hausdorff limits of geodesically complete, CAT(0)-spaces admitting a discrete group of isometries of bounded codiameter and the structure of the possible limit groups. The focus will be on the collapsing case: namely when the injectivity radius of the quotient space goes to zero along the sequence. Joint work with A.Sambusetti. |
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Crossed products arising from topological dynamical systems are an important source of examples of C*-algebras and form ground for interaction between dynamics and operator algebras. Included in this class are reduced group C*-algebras which code representation theoretic information of a group.
Operator algebraists are interested in the ideal structure of C*-algebras, and developed sophisticated tools to prove (non-)simplicity of the examples described above. But there are few results about the ideal structure of crossed product C*-algebras unless the dynamical system and the group it is constructed from are well-behaved. I will report on joint work with Are Austad (University of Southern Denmark) in which we introduce the $\ell^1$-ideal intersection property. All non-zero ideals in the crossed product C*-algebra of a dynamical system satisfying this property can be detected already inside the much smaller and more concrete $\ell^1$-crossed product. We prove that large classes of groups, such as lattices in connected Lie groups and linear groups over algebraic integers in a number field have this property for ANY action on a locally compact Hausdorff space. The proof combines the theory of twisted groupoid C*-algebras and C*-simplicity with structure results about amenable subgroups. I will introduce a general audience with a background in group theory to these results and outline some of our methods. |
| Résumé: | The Bohr compactification of a topological group $G$ is a compact group naturally associated to $G$ which is defined by a universal property with respect to all continuous homomorphisms of $G$ to compact groups. Given an arithmetic group $G$ in an algebraic group $H$, we show how the description of the Bohr compactification of $G$ reduces to the case where $H$ is semisimple. In this latter case, we describe the Bohr compactification of $G$, identifying in particular its connected component. |
| Résumé: | A locally compact group is said to be irreducibly represented if it admits a continuous unitary representation that is both irreducible and faithful. A classical problem coming back to Burnside is to find an algebraic characterisation of the finite groups that are irreducibly represented. Such a characterisation was obtained by Gashütz in the 60's and was extended to the framework of countable discrete groups by Bekka and de la Harpe in 2008. However, no such characterisation seems to be known beyond this frame. The purpose of this talk is to present an algebraic characterisation of the connected amenable algebraic Lie groups that are irreducibly represented. |
| Résumé: | Let $G$ be a finitely generated group. To every action of $G$ is associated a graph, called the Schreier graph of the action. We consider all Schreier graphs of $G$ as a whole, and we are interested in geometric properties common to all of them. In the talk we will be interested in lower bounds for their growth. For large classes of solvable groups, we are able to give explicit lower bounds, that are sharp for many motivating examples. Our approach consists in exhibiting certain subsets $L$ of $G$, that roughly speaking have the property that every finite configuration of $L$ appears somewhere in the graph of every faithful $G$-action. The study of such subsets is closely related to the study of confined subgroups of $G$, which are a natural generalization of uniformly recurrent subgroups (URS) introduced by Glasner-Weiss. This is joint work with Nicolas Matte Bon. |
| Résumé: | We introduce the notion of boomerang subgroups of a discrete group. Those are subgroups satisfying a strong recurrence property, when we consider them as elements of the space of all subgroups with the conjugation action. We prove that every boomerang subgroup of $\mathrm{SL}(n,\mathbf{Z})$ is finite and central or of finite index. Thus we give a new and simple proof of the Nevo-Stuck-Zimmer rigidity theorem for $\mathrm{SL}(n,\mathbf{Z})$ avoiding almost all measure theory. This is joint work with Yair Glasner. |