Abstracts

Concise formulae in groups of non-positive curvature

A word w is concise in a class of groups C if the verbal subgroup w(G) is finite for every group G in C in which w takes only finitely many values. Concise words were introduced by Philip Hall, who conjectured that every word is concise. Although this conjecture has been disproved, it is still an open problem to determine whether every word is concise in the class of residually finite groups.

In joint work with Moritz Petschick we considered a natural extension of the notion of conciseness to first-order formulae in the language of groups and established this property for various classes of formulae and groups. In this talk I will give an overview of this problem and present joint work with Laura Ciobanu where we explore conciseness of formulae in acylindrically hyperbolic groups and in related classes of groups.

Poset condition in the lattice of subgroups of a group

This is a standard approach in group theory: given a subgroup property P — for instance, normality — one asks what can be said about the structure of groups in which all subgroups, or at least sufficiently many of them, satisfy that property.

I will present both classical results and more recent developments, which place these results in a broader framework by considering different possible meanings of “sufficiently many” subgroups.

Divisor class group of Kummer curves

In a previous paper we gave a basis of the Riemann-Roch space L(D) of a divisor of the shape D = P1 + … + Pt + qO on a hyperelliptic curve H of equation y^2 = f(x), with deg(f) = 2g+1 (g being the genus of H, and O being its point at infinity), linking it to the Mumford representation of D. In that occasion, the referee kindly suggested to extend the investigation to the case of a superelliptic curve C (sometimes called a  Kummer curve in the literature) of equation y^n = f(x), with deg(f) = 2g/(n-1) + 1 (g being the genus of C). The problem turned out, indeed, to show much better than the previous case n = 2 how involved such computation can be. We will give an account of it, under the least possible assumptions.

On the conjugacy problem in Aut(F2) and related problems

The aim of this talk is to present practical algorithms for the automorphism group of the free group of rank two, Aut(F_2). Central to our approach is a method for computing canonical normal forms of automorphisms. Then we show how to compute a basis for the fixed point subgroup of an automorphism and how to solve the conjugacy problem in Aut(F_2). Finally, we introduce a new algorithm for computing generating sets of centralizers.

Malcev-like anti-commutative algebras

The power series terms of local loop multiplications at the unit element define several multilinear operations in the tangent space of the local loop. Particularly interesting is the case when these multilinear operations are determined only by a bilinear operation. The simplest such loops are the Moufang loops and the more general diassociative  loops (any two elements generate a subgroup). The corresponding generalizations of Lie algebras are the Malcev algebras and the binary Lie algebras (any two elements generate a Lie subalgebra), which play an important role in non-associative Lie theory. In the talk we study anti-commutative algebras, which are extensions of a one-dimensional algebra by  nilpotent Lie algebra and at the same time extensions of the two-dimensional non-abelian Lie algebra by an abelian algebra. These algebras, just like the five-dimensional solvable Malcev algebras, have a flag of subalgebras and can therefore be considered their closest relatives.

We find normal forms of the multiplications, and determine the isomorphism classes of solvable anti-commutative algebras that have the same decomposition properties as solvable Malcev or binary Lie algebras.

On splendid Morita equivalences between blocks of tame representation type

In the modular representation theory of finite groups, it is common to "compare" block algebras (i.e. the indecomposable algebra factors of the group algebras)  using different notions of categorical equivalences, preserving different global and/or local, structural or numerical invariants (module categories, derived and stable categories, Hochschild cohomology, the numbers of ordinary and modular irreducible characters, defect groups, fusion system, (generalised) decomposition matrices, specific classes of modules, etc).  One may use the standard notion of a Morita equivalence for finite-dimensional algebras (i.e. an equivalence of the module categories over these algebras). It is, however, too weak to detect invariants such as the vertices and sources of modules. In this respect, it is better to consider the stronger notion of a splendid Morita equivalence, which takes the action of the group in consideration in a subtle way. 

Puig's Finiteness Conjecture asserts that if a defect group is fixed, then, up to splendid Morita equivalence, there are only finitely many equivalence classes of blocks of group algebras with this defect group. The aim of this talk is to survey background ideas, methods and strategies to verify the validity of this conjecture for blocks of tame representation type.

Free semigroups in the group of affine transformations of a field

The existence of free subgroups and free semigroups is a classical theme in group theory, going back to a question posed by von Neumann. Beyond existence, one can ask whether such free objects are typical: for instance, whether randomly chosen elements generate a free subgroup or a free semigroup with high probability. A natural probabilistic model for this question is given by random walks on the ambient group.

In this talk, I will discuss this problem for groups of affine transformations of a field. More precisely, we will study conditions under which random elements sampled by a random walk generate a free semigroup. This is based on joint work with Richard Aoun.

Some general considerations on Chermak's partial groups

Chermak's partial groups have been a key tool for proving, through a constructive argument, the existence and uniqueness of a certain category, satisfying some very specific properties, associated to any saturated fusion system. This was a fundamental step in achieving a proof of the Martino-Priddy conjecture independent of the classification of the finite simple groups.

The talk mostly originates from two questions: What are the essential, underlying properties that are involved in Chermak´s construction? And is it possible to generalize it?

The main topic will then consist in recent advances about our understanding of partial groups, of their relation to groups and of what makes Chermak's construction so special.

This is joint work with Justin Lynd and Philip Hackney.

Fusion and Subgroup Lattices

In general, the subgroup lattice of a finite group does not suffice to determine whether or not two subgroups are conjugate.

In particular, there exist both abelian and non-abelian groups with isomorphic subgroup lattices.

In this talk we will discuss various approaches to colour the associated Hasse-diagramm of a subgroup lattice in order to visualise fusion in the group.