An element \(x\) of a group \(G\) is called a test element if for any endomorphism \(\phi\) of \(G\), \(\phi(x) = x\) implies that \(\phi\) is an automorphism. The first non-trivial example of a test element was given by Nielsen in 1918, when he proved that every endomorphism of a free group of rank 2 that fixes the commutator \([x, y]\) of a pair of generators must be an automorphism. A subgroup \(R\) of a group \(G\) is said to be a retract of \(G\) if there is a homomorphism \(r : G \to R\) that restricts to the identity on \(R\).

In this talk I will discuss the distribution of test elements in free groups and the role of retracts in the study of endomorphisms of free groups. This is a joint work with Slobodan Tanushevski.

Zeta functions of groups are analytic functions encoding information on groups and group-related objects. In this talk, I will define two bivariate zeta functions related to representations and to conjugacy classes of finitely generated nilpotent groups. We shall see how they can be used to understand other (univariate) zeta functions of groups.
I will present some of their arithmetic and analytic properties, such as Euler decompositions, rationality and functional equations of local factors, and domains of convergence and meromorphic continuation, pointing out differences and similarities between the bivariate and the univariate worlds.

Coxeter groups are groups with 'nice and symmetric' presentations. In my talk we want to study the automorphism group of a Coxeter group which is isomorphic to a free product \(\frac{\mathbb{Z}}{2\mathbb{Z}} \ast \cdots \ast \frac{\mathbb{Z}}{2\mathbb{Z}}\). In the first part of the talk we will look at algebraic properties of the automorphism group. In particular we will give an embedding into another automorphism group and we will study representations. The second part will focus on geometric properties–more precisely fixed point properties–of the automorphism group. We will investigate the question whether 'nice' actions on trees and \(\mathrm{CAT}(0)\) spaces have a fixed point.

In the structure theory of totally disconnected, locally compact groups started in [6], the iteration of automorphisms plays an essential role. Several subgroups can be associated to an automorphism \(f\) of a locally compact group \(G\), like the contraction group \(con(f)\) of all group elements whose forward orbit under \(f\) converges to the neutral element \(e\), or the group \(par(f)\) of all group elements whose forward orbit is relatively compact. For totally disconnected \(G\), the study of such subgroups was started in [1], and connections were established there to other notions from the structure theory of totally disconnected groups (like tidy subgroups and the scale). In the talk, I'll give an introduction to this area of research, including some recent results both in the general case and for the special case of automorphisms (and endomorphisms) of Lie groups over local fields

One approach to study infinite groups is to treat them as geometric objects. In the talk we will refine this approach by associating a geometric object to an action on a compact space.
The geometry of the associated invariant turns out to reflect topological and ergodic properties of the action (amenability, Haagerup propery, spectral gap) and may even remember the action completely (Fisher et al.), but it may also not distinguish between actions of the integers and of the trivial group (joint work with Kielak).
The construction also yields exotic examples of interest in computer science and large scale geometry (super-expanders and counterexamples to the coarse Baum–Connes conjecture).

Let \(G_1\) and \(G_2\) be either compact simple groups or finite simple groups of Lie type. Let \(\mu_1\) and \(\mu_2\) be symmetric probability measures on \(G_1\) and \(G_2\), respectively. Under some mild conditions on \(\mu_1\), \(\mu_2\), one knows that the distribution of the random walk on \(G_i\) driven by \(\mu_i\) converges to the uniform distribution, and the speed of convergence is governed by the spectral gap of \(\mu_i\). A coupling of \(\mu_1\) and \(\mu_2\) is a probability measure \(\mu\) on \(G_1 \times G_2\) with marginal distributions \(\mu_1\) and \(\mu_2\), respectively. Under what conditions does \(\mu\) have a spectral gap depending on the gaps of \(\mu_1\) and \(\mu_2\)?

In this talk I will first review some of the old and new methods for establishing spectral gaps, mainly based on pioneering works of Kazhdan and Bourgain-Gamburd and then discuss the question stated in the previous paragraph. This talk is based on a joint work with Amir Mohammadi a

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In joint work with Robert Young, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When \(n\) is less than the rank of the associated symmetric space, we show that the \(n\)-dimensional filling volume function of the lattice grows at the same rate as that of the associated symmetric space, and when \(n\) is equal to the rank, we show that the \(n\)-dimensional filling volume function grows exponentially. This broadly generalizes a theorem of Lubotzky-Mozes-Raghunathan on length distortion in lattices and confirms conjectures of Thurston, Gromov, and Bux-Wortman.

The coclass of a finite \(p\)-group of order \(p^n\) and class \(c\) is defined as \(n-c\). Using coclass as the primary invariant in the investigation of finite \(p\)-groups turned out to be a very fruitful approach.
Together with Bettina Eick, we have developed a coclass theory for nilpotent associative algebras over fields. A central tool are the coclass graphs associated with the algebras of a fixed coclass. The graphs for coclass zero are well understood. We give a full description for coclass one and explore graphs for higher coclasses.
We prove several structural results for coclass graphs, which yield results in the flavor of the coclass theorems for finite \(p\)-groups. The most striking observation in our experimental data is that for finite fields all of these graphs seem to exhibit a periodic pattern. We want to prove and exploit this periodicity in order to describe the infinitely many nilpotent associative \(F\)-algebras of a fixed coclass by finitely many p

In 1998, in view of defining generalized Hecke algebras associated to an arbitrary complex reflection group, Broué, Malle and Rouquier investigated generalized braid groups, which provide a further generalization of Artin groups of finite Coxeter type. They stated a number of theorems and conjectures, which are all proved now. In this talk I will recall this background material, and describe some of the tools involved in the proofs. In particular, the determination of the center of the generalized pure braid groups made use of special monoids associated to these groups, whose (Garside) properties are similar to the properties of more classical Artin monoids. bitrary complex reflection group, Broué, Malle and Rouquier investigated generalized braid groups, which provide a further generalization of Artin groups of finite Coxeter type. They stated a number of theorems and conjectures, which are all proved now. In this talk I will recall this background material, and describe some of the

Let \(l\) be a prime number. I shall explain a recent joint work with P. Koymans where, conditionally on the Generalized Riemann Hypothesis, we establish a conjecture of F. Gerth on the statistical behavior of the \(l\)-Sylow of the class group of a cyclic degree \(l\) extension of \(\mathbb{Q}\). Our work builds on a recent breakthrough of A. Smith that covered the case \(l=2\) and the quadratic fields to be imaginary. Altogether this settles a set of conjectures advanced by F. Gerth and directly inspired on the Cohen--Lenstra heuristics. In this talk I shall also give a brief introduction to these heuristics. I shall explain how using large random matrices over suitable DVR's one can explain intuitively the main term in our asymptotic result and its difference with the main term in Smith's theorem. These heuristics models have recently been generalized in several directions. I shall explain a recent joint work with E. Sofos where, using homological algebra, we extended these heuristi

The study of toric varieties is a wonderful part of algebraic geometry that has deep connections with polyhedral geometry. There are elegant theorems, unexpected applications, and marvelous examples. [Cox, Little, Schenck]

This talk is an introduction to this rich subject. As an example of the wonderfulness claimed above, we will look at generalizations of the celebrated Bernstein–Kushnirenko Theorem. It expresses the number of common zeros of \(n\) polynomials in \((\mathbb{C}^*)^n\) as the mixed volume of a polytope – a convex-geometric invariant.

If we have only \(k < n\) equations, the set of solutions will no longer be finite. However, there are still formulas relating cohomological invariants of the zero set to lattice point counts in Minkowski sums of polytopes.

This is based on joint projects with S. Di Rocco, M. Juhnke-Kubitzke, B. Nill, R. Sanyal, and T. Theobald.

Consider the geometry of the action of Bianchi groups \(\mathrm{SL}(2,\mathcal{O}_d)\) on the hyperbolic space \(\mathbb{H}^3\), where \(\mathcal{O}_d\) is the ring of integers of the imaginary quadratic field \(K = \mathbb{Q}(\sqrt{-d})\). We obtain, for some values of \(d\), an upper estimate for the height of some matrix \(M\) that takes a given point \((z,t) \in \mathbb{H}^3\) into the fundamental domain of the Bianchi group. This generalizes a lemma of Habegger and Pila about the action of the modular group on \(\mathbb{H}^2\). We use coarse fundamental domains that look like the so-called Siegel sets to make computations easier.

I will show how Karl Lorensen and I now have simplified our arguments for establishing the connection between rank conditions on rings that are group graded and amenability of the corresponding groups.