Program

Tentative Schedule

Kasia Jankiewicz: Artin and Coxeter groups

Coxeter groups and Artin groups play fundamental roles in geometric group theory, and other areas of mathematics. They originate as reflections groups and the fundamental groups of associated hyperplane arrangements. We will survey approaches of studying them using geometry and combinatorics, and discuss examples.

Claudio Llosa: Finiteness properties, Morse theory and subgroups of hyperbolic groups

We say that a group is of finiteness type $F_n$ if it admits a classifying space (or $K(G,1)$) with finite $n$-skeleton. These finiteness properties were introduced by Wall in the 1960s as a generalisation of finite generation and finite presentability, which are equivalent to $F_1$ and $F_2$. These homotopical finiteness properties have homological counterparts $FP_n(R)$, where $R$ is a unital abelian ring, defined in terms of the existence of finitely generated projective resolutions of the trivial $RG$-module $R$ up to dimension $n$. We will start this course with an introduction to homotopical and homological finiteness properties, their relations and properties. We will then explain how Morse theory can be used to construct groups with exotic finiteness properties, such as the Bestvina--Brady groups. They include groups that are $F_n$ and not $F_{n+1}$ for every positive integer $n$, as well as groups that are $FP_2(Z)$, but not $F_2$. Following these classical examples, we will look at more recent applications of Morse theory with a particular focus on subgroups of hyperbolic groups with exotic finiteness properties.

Macarena Arenas: (Classical & Cubical) Small Cancellation

Laura Ciobanu: Equations in groups

1st talk: For a group or semigroup or ring G, solving equations where the coefficients are elements in G and the solutions take values in G can be seen as akin to solving systems of linear equations in linear algebra, Diophantine equations in number theory, or more generally, polynomial systems in algebraic geometry.

In the first talk I will give an introduction to solving equations in infinite non-abelian groups, with emphasis on free groups and hyperbolic groups.

2nd talk: In the second talk I will show how imposing certain constraints on the solutions can tilt the balance between decidability and undecidability, and I will present some recent results on equations with constraints in one-relator groups.

Sam Fisher: Special coherent groups

A group is coherent if all its finitely generated subgroups are finitely presented. In cohomological dimension two, Wise conjectures that coherence is equivalent to the vanishing of the second L²-Betti number. There has been much recent progress in showing that interesting classes of groups (such as one-relator groups, due to Jaikin and Linton) are coherent, which all provide evidence for one implication of Wise's proposed classification of coherence. The goal of these two lectures is to discuss the converse implication, which predicts that coherence implies the vanishing of the second L²-Betti number, and prove Wise's conjecture for virtually special groups. We will stress the connection with Novikov (co)homology and free-by-cyclic groups, keeping the group cohomology prerequisites to a minimum. This is based on joint work with Pablo Sánchez-Peralta.

Rober Kropholler: Homological Dehn Functions

I will survey various definitions of homological Dehn functions as well as homotopical Dehn functions. I will then give an idea of which techniques from the homotopical version work in this algebraic setting as well as which don’t. Finally I will end with some open questions and directions for future study.

Olga Varghese: Algebraic, geometric and profinete properties of Dyer groups

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