To a CAT(0) group one can attach the visual boundary. For hyperbolic groups this boundary is independent of the space on which the group acts. Furthermore, if the hyperbolic group is 1-ended, then the boundary is locally connected and hence path-connected. For 1-ended CAT(0) groups neither of these results is true. For instance, Croke and Kleiner give an example of a RAAG which acts on two spaces with non homeomorphic boundaries, this example can also be shown to have a non path-connected boundary. We build on these results to show that certain CAT(0) groups do not have path-connected visual boundary and give more examples of groups without unique boundary. This is joint work with Michael Ben-Zvi.

In 20th century, there were two big projects of group theory. The first one was involved the classical, special and restricted versions of the Burnside Problem. The original problem asked whether every periodic group is locally finite. Indeed, Novikov and Adian answered this problem negatively by constructing the first infinite finitely generated examples of periodic groups. Later, Grigorchuk, Gupta and Sidki and Olshanskii also constructed other infinite finitely generated periodic groups. On the other hand, these kind of examples motivated the questions of the following type: What kind of structural information forces a periodic group to be locally finite? We will give examples of these kind of theorems related to centralizers. On the other hand, the second very important project was the Classification of Finite Simple Groups, which was completed by using information about the centralizers of involutions. These results on centralizers lead questions in infinite group theory about centralizers. In this talk we will prove a theorem about centralizers of p-subgroup in simple locally finite groups and we will give a characterization for $PSL_{p}(k)$ where $k$ is an infinite algebraic extension of a finite field of characteristic different from $p$.

When do the maps onto alternating groups separate elements and subgroups of a right-angled Coxeter group? More precisely, which RACGs have the property that for any infinite-index quasiconvex subgroup and a finite collection of elements not in the subgroup, there exists a surjection onto an alternating group such that no element lands in the image of the subgroup. Obviously, if the RACG is $\mathbb{Z}$ or a direct product, this fails. I'll show that all other RACGs have this property by constructing a suitable finite-sheeted cover of the presentation complex (and hence the permutation action on the cosets of the associated subgroup). Similar result holds for hyperbolic orientable surface groups and also for right-angled Artin groups.

Given any topological manifold, the Gromov norm is a semi-norm defined on any homology class that measures how efficiently the representing cycles can be expressed as the linear combination of singular simplicies. While being topological, the Gromov norm is closely related to the geometry of the manifold. In this talk, we will focus on higher rank locally symmetric spaces. We show that any non-trivial class whose degree is large enough has positive Gromov norm. We also discuss potential examples of non-trivial class whose Gromov norm is zero. This shows our result is very close to the optimal.

Es werden Ergebnisse diskutiert, die zeigen, inwiefern man Eigenschaften über endliche Gruppen durch Aussagen erster Stufe charakterisieren kann. (Alle nötigen Definitionen aus der Modelltheorie werden einfach erklärt.) Bei einigen der Ergebnisse stützt man sich auf Folgerungen der Klassifikation der endlichen einfachen Gruppen. Aber keineswegs tauchen nur endliche Gruppen auf: auch unendliche Modelle und Ultraprodukte werden benutzt, die gewonnenen und zu gewinnenden Ergebnisse zu beleuchten.

Brady and Watt have constructed combinatorial $K(\pi,1)$'s for the spherical Artin groups $B(W)$, relying on their dual braid presentation and the lattice structure of the set $NC(W)$ of noncrossing partitions of the associated reflection group $W$. Moreover, after Deligne's proof of Brieskorn's $K($\pi$,1)$ conjecture these models are homotopy equivalent to the space of regular orbits of $W$. They are remarkably simple to describe, as (topological) quotients of the order complex of $NC(W)$, yet their relation to the geometry of $B(W)$ seems to lack a good explanation. In particular, the fact that the dual and standard presentations agree was until recently proven only in a case-by-case way (relying on the classification of finite Coxeter groups). In a seminal work, Bessis related the dual presentation of $B(W)$ with the study of the braid monodromy of its discriminant hypersurface. Then, he used the combinatorics of $NC(W)$ as a recipe to construct the universal covering space of the discriminant complement and thus prove the $K(\pi,1)$ conjecture for all $\textbf{complex}$ reflection groups. We will explain how this study of the braid monodromy of the discriminant, along with some of our recent enumerative results gives a uniform proof of the dual braid presentation for $B(W)$. More importantly, and following a suggestion of Bessis, we will describe a way to build via this geometry a CW-complex that is a priori homeomorphic to the discriminant complement but also in fact a realization of the Brady complex. This was proposed originally as a way to simplify part of Bessis' work, but seems also well-suited for the study of CAT(0) properties of $B(W)$ via Brady and McCammond's orthoschemes. This is work in progress.

The integral geometry formula relates the volume of a real projective hypersurface to the expected number of real intersections with a random line. By random, we mean with respect to a probability measure invariant under the action of the rotation group $SO(n+1)$ on real projective $n$-space. In this talk, I will discuss the $p$-adic analogue of this formula and the connection to zeta functions. (Joint work with Antonio Lerario.) $\\$ I will also discuss some new computational techniques for solving 0-dimensional polynomial systems of equations over $\mathbb{Q}_p$ in finite precision, through which one can obtain an estimate for the $p$-adic volume of a variety.

AC: I will give a survey on Euler-Mahonian distributions on sets of permutations. Based on recent work of Gessel and Zhuang, I will then discuss a relationship between Hadamard products of rational generating functions, joint distributions of permutation statistics, and so-called shuffle algebras. $\\$ TR: We consider the enumeration of conjugacy classes within families of unipotent groups associated with graphs. In the case of so-called cographs, this problem turns out to be closely related to the study of rank distributions in spaces of generic matrices with support constraints. This connection has consequences for analytic properties of associated zeta functions. This is joint work with Christopher Voll.

A classical problem in discrete and combinatorial geometry asks the following: Does there exist a (finite) set of local operations such that any two (nice enough) triangulations of the same manifold can be transformed one into the other using just those local moves? In general, an answer to this question has been provided by Pachner in terms of so-called bistellar flips, shellings and inverses. In this talk, we consider d-dimensional triangulations that have a combinatorial property, called balanced, which means that they can be colored (in the graph-theoretic sense) with just d+1 colors. We ask the same question as before but, in addition, we want the local operations to be such that coloring (and balancedness) are preserved in each step. For manifolds without boundary, this question was answered in the positive by Izmestiev, Klee and Novik using so-called cross-flips. For manifolds with boundary, we show that as in the classical situation, shellings and inverses suffice for this aim. All necessary notions will be explained throughout this talk. This is joint work with Lorenzo Venturello.

For a closed manifold M with contractible universal cover, the Singer conjecture predicts that the \(L^2\)-Betti numbers of M are concentrated in the middle dimension. In this talk I will discuss what is known and unknown about this conjecture and explain why it does not have a rational analogue. I will also explain how (following Davis, Okun, and Schreve) to think of it as a conjecture about arbitrary discrete groups and describe a variant relating mod 2 \(L^2\)-Betti numbers of groups to van Kampen embedding obstructions of their boundaries that fits especially well with computations in right angled Artin groups.

The Erdős-Ko-Rado theorem gives an upper-bound on the size of a pairwise intersecting family of small subsets of [n]. If the size of the family is near the upper bound, then the family is a star. A theorem of Gerstenhaber gives an upper-bound on the dimension of a space of nilpotent matrices. There are generalizations to other Lie algebras, and if the dimension of the space achieves the upper-bound, then the space is the nilradical of a Borel subalgebra. I'll talk about how to adapt a linear algebraic groups approach of Draisma, Kraft, and Kuttler to theorems of Gerstenhaber type for the Erdős-Ko-Rado situation.