Any algebra A is defined by generators and relations. Some relations, e.g. commutativity (xy-yx=0) and nilpotency of degree n (xⁿ=0), are polynomials and valid for all the elements of the algebra. These relations are called polynomial identities and are the equivalent of laws/group identities in group theory. An algebra satisfying such a polynomial is called PI. For example all f.d. algebras are PI.

In this talk we will give an introduction into the algebraic information that polynomial identities delivers. More precisely we will explain the so called (c_n(A))_n codimension sequence, which will serve as an invariant. The exponential and polynomial growth rate of this sequence are of special interest. We will present recent results concerning a conjecture of Amitsur and Regev asserting that the exponential growth rate is always an integer and the polynomial growth rate an half-integer. Moreover, as will be illustrated through examples, this values are computable and tightly connected with the algebraic structure of A. Along the way we will mention results showing that the aforementioned values can be used to distinguish varieties of algebras. Some similarities (or rather differences) with the growth sequence of a group will also be pointed out.

Massey products are, originally, operations defined in the context of algebraic topology, on cohomology rings. However, when one specializes to group cohomology, work of Dwyer shows that the study of these operations amounts to that of elementary extension problems, involving the group of unipotent matrices over a finite field. When we specialize further, and consider Galois groups in particular, an ambitious conjecture predicts that Massey products always vanish. This has been actually proved for triple Massey products, in complete generality. In this talk, I will describe joint work with Minac, Topaz and Wittenberg, showing that the conjecture is true for fourfold Massey products in the cohomology of number fields.

Für eine endlich erzeugte Gruppe G mit Erzeugendensystem X und einer beliebigen Untergruppe H, bezeichne b(H,n) die Anzahl der Elemente in H, welche bezüglich der Wortmetrik d_X höchstens die Länge n haben. Dann existiert der Grenzwert lim b(H,n)^(1/n) für alle Untergruppen H aller hyperbolischen Gruppen G.

Weiter möchte ich zeigen, wie sich die Technik auf den Fall relativ hyperbolischer Gruppen erweitern werden lässt, um ein analoges Resultat für Limesgruppen zu erhalten.

Algebraic groups over local fields, as PSL(n,K), provide many examples of non-discrete compactly generated locally compact groups that are (topologically) simple. Let us denote by S the set of all topological groups with these properties. Groups acting on trees provide another main source of examples of groups in S: for a fixed locally finite semiregular tree T, there are infinitely many (pairwise non-isomorphic) closed subgroups of Aut(T) which belong to S. The set of closed subgroups of Aut(T) carries a natural compact topology (called the Chabauty topology), so a natural question arises: Which groups appear as Chabauty limits of simple subgroups of Aut(T)? Can we obtain new examples of groups in S by observing accumulation points of (infinitely many) well-known examples? The talk will discuss those questions and related topics, without assuming that the audience is familiar with topological groups.

Let G be a group generated by a set S⊆ G that is closed under G-conjugation. For each g∈ G we can define a partially ordered set whose maximal chains correspond to the S-reduced words for g. In this talk we disuss several notions of connectivity of the corresponding poset diagram, such as Hurwitz-transitivity or lexicographic shellability. The goal of this talk is to present a framework which (conjecturally) enables us to locally check for properties. This is joint work with Vivien Ripoll.

This talk will deal with the general problem of compactifying Bruhat-Tits buildings, which are cellular spaces associated to reductive algebraic groups over local fields. The latter spaces are classically used to better understand the structure of the groups under consideration because, thanks to their nice symmetry properties, they admit very transitive actions. Still, the main part of the talk will be dedicated to introducing techniques from analytic non-archimedean geometry, after V. Berkovich, at least those that are useful to the problem. One of the main ideas is to be able to use, at several intermediate steps, some (possibly huge) valued ground fields as is in algebraic geometry. One of the outcomes is the possibility to extend the geometric parametrization of maximal compact subgroups using buildings, by parameterizations of remarkable subgroups using their compactifications.

Convergence groups are a generalization of (relatively) hyperbolic groups: the action of a locally compact group G on a compactum T is convergence if it satisfies a dynamical property, or equivalently if it is 3-proper. On the other hand, a compactification of a locally compact group is called a Specker compactification if it satisfies natural properties linked to the group. We first show that, given a convergence action of G on T, there is a natural way to "almost" get a Specker compactification of G. Conversely, when G is compactly generated we show that the action of G on any Specker compactification is convergence. This gives the existence of "pullback spaces" for convergence actions in some cases.

Curtis-Tits groups are groups defined as non-trivial completions of Curtis-Tits amalgams. Via the Curtis-Tits theorem they generalize groups of Lie type and groups of Kac-Moody type. We'll describe all Curtis-Tits groups with 3-spherical diagram and explore some geometric and algebraic properties.

Braid groups (more generally spherical type Artin groups) can be viewed as groups of fractions of (at least) two monoids: the Artin monoid and the dual monoid, the latter containing the former. These monoids have a Garside structure, hence simple elements. They are generated by atoms. The atoms of the Artin monoid are the canonical lifts of the simple reflexions.

A natural question is to decompose dual simple elements into products (with signs) of atoms of the Artin monoid, in particular to find shortest such decompositions. I will present some results and conjectures in this direction involving conjectures on root systems. This is work in progress with Thomas Gobet.

I will talk about connections between dynamical systems, geometry, and algebra which are caused by an algebraic object, called iterated monodromy group (IMG). IMG is a self-similar group associated to every branched covering map f of the 2-sphere (in particular, to every rational map). It was introduced by Nekrashevych and encodes combinatorial information about the map f and its dynamics in a computationally efficient way. While properties of the IMGs of polynomial maps have been widely explored in the last years, the properties of the IMGs of non-polynomial rational maps still need to be investigated.

I will present a new method for proving exponential growth of IMGs and illustrate it for a rational map f whose Julia set is the entire Riemann sphere. The proof is based on the “geometry” of the tilings associated to the map f, which are obtained by Bonk and Meyer. Moreover, the method can be adjusted for the IMGs of some infinite families of rational maps with Julia set given by the whole Riemann sphere or a Sierpinski carpet. This is a joint project with Mario Bonk (University of California, Los Angeles) and Daniel Meyer (University of Jyväskylä).

For every finite Coxeter group W, the noncrossing partitions NC(W) are a lattice in the absolutely ordered group W. In the case of W being the symmetric group, T. Brady and J. McCammond showed that the order complex of NC(W) embeds into a spherical building. In this talk, I will explain an embedding defined in a different way. One advantage of the new embedding is that it can be generalized to noncrossing partitions of type B. In joint work with P. Schwer, we show that the noncrossing partitions of type B also embed into spherical buildings in a nice way. As an application, we compute the radius of the Huwritz graph of type B(n), which answers an open question of Adin and Roichman.

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The handlebody group is the subgroup of the mapping class group of a surface formed by those mapping classes which extend to a handlebody. We will study rigidity phenomena of this group as a subgroup of the mapping class group. On the one hand, the group is rigid: any inclusion (of a finite index subgroup of) the genus g handlebody group in the genus g mapping class group is simply a conjugation. On the other hand, once we consider inclusions in other mapping class groups, rigidity ceases to hold.

The well-known Jacobson--Morozov Theorem states that every nilpotent element of a complex semisimple Lie algebra Lie(G) can be embedded in an sl_2-subalgebra. Moreover, a result of Kostant shows that this can be done uniquely, up to G-conjugacy. Much work has been done on extending these fundamental results to the modular case when G is a reductive algebraic group over an algebraically closed field of characteristic p > 0. I will discuss joint work with David Stewart, proving that the uniqueness statement of the theorem holds in the modular case precisely when p is larger than h(G), the Coxeter number of G. In doing so, we consider complete reducibility of subalgebras of Lie(G) in the sense of Serre/McNinch.