Limit groups are a powerful tool in the study of free and hyperbolic groups (and even broader classes of groups). I will define limit groups in various ways: algebraic, logical and topological, and draw connections between the different definitions. We will also see how one can equip a limit group with an action on a real tree, and analyze this action using the Rips machine, a generalization of Bass-Serre theory to real trees. As a conclusion, we will obtain that hyperbolic groups whose outer automorphism group is infinite, split non-trivially as graphs of groups.

# Past Junior Topology and Group Theory Seminar

On any hyperbolic surface, the number of curves of length at most L is finite. However, it is not immediately clear how quickly this number grows with L. We will discuss Mirzakhani’s breakthrough result regarding the asymptotic behaviour of this number, along with recent efforts to generalise her result using currents.

Right-angled Artin groups (RAAGs) were first introduced in the 70s by Baudisch and further developed in the 80s by Droms.

They have attracted much attention in Geometric Group Theory. One of the many reasons is that it has been shown that all hyperbolic 3-manifold groups are virtually finitely presented subgroups of RAAGs.

In the first part of the talk, I will discuss some of their interesting properties. I will explain some of their relations with manifold groups and their importance in finiteness conditions for groups.

In the second part, I will focus on my PhD project concerning subgroups of direct products of RAAGs.

The automorphism group of a d-regular tree is a topological group with many interesting features. A nice thing about this group is that while some of its features are highly non-trivial (e.g., the existence of infinitely many pairwise non-conjugate simple subgroups), often the ideas involved in the proofs are fairly intuitive and geometric.

I will present a proof for the fact that the outer automorphism group of (Aut(T)) is trivial. This is original joint work with Gil Goffer, but as is often the case in this area, was already proven by Bass-Lubotzky 20 years ago. I will mainly use this talk to hint at how algebra, topology and geometry all play a role when working with Aut(T).

Whitehead published two papers in 1936 on free groups. Both concerned decision problems for equivalence of (sets of) elements under automorphisms. The first focused on primitive elements (those that appear in some basis), the second looked at arbitrary sets of elements. While both of the resulting algorithms are combinatorial, Whitehead's proofs that these algorithms actually work involve some nice manipulation of surfaces in 3-manifolds. We will have a look at how this works for primitive elements. I'll outline some generalizations due to Culler-Vogtmann, Gertsen, and Stallings, and if we have time talk about how it fits in with some of my current work.

Knotoids are a generalisation of knots that deals with open curves. In the past few years, they’ve been extensively used to classify entanglement in proteins. Through a double branched cover construction, we prove a 1-1 correspondence between knotoids and strongly invertible knots. We characterise forbidden moves between knotoids in terms of equivariant band attachments between strongly invertible knots, and in terms of crossing changes between theta-curves. Finally, we present some applications to the study of the topology of proteins. This is based on joint works with D.Buck, H.A.Harrington, M.Lackenby and with D. Goundaroulis.

A group splits as an HNN-extension if and only if the rank of its abelianisation is strictly positive. If we fix a class of groups one may ask a few questions about these splittings: How distorted are the vertex and edge groups? What form can the vertex and edge groups take? If they remain in our fixed class, do they also split? If so, under iteration will we terminate at something nice? In this talk we will answer all these questions for the class of one-relator groups and go through an example or two. Time permitting, we will also discuss possible generalisations to groups with staggered presentations.

An important property of Gromov hyperbolic spaces is the fact that every path for which all sufficiently long subpaths are quasi-geodesics is itself a quasi-geodesic. Gromov showed that this property is actually a characterization of hyperbolic spaces. In this talk, we will consider a weakened version of this local-to-global behaviour, called the Morse local-to-global property. The class of spaces that satisfy the Morse local-to-global property include several examples of interest, such as CAT(0) spaces, Mapping Class Groups, fundamental groups of closed 3-manifolds and more. The leverage offered by knowing that a space satisfies this property allows us to import several results and techniques from the theory of hyperbolic groups. In particular, we obtain results relating to stable subgroups, normal subgroups and algorithmic properties.

We will discuss what it means to study the homology of a group via the construction of the classifying space. We will look at some examples of this construction and some of its main properties. We then use this to define and study the homology of the mapping class group of oriented surfaces, focusing on the approach used by Harer to prove his Homology Stability Theorem.

When Gromov defined non-positively curved cube complexes no one knew what they would be useful for.

Decades latex they played a key role in the resolution of the Virtual Haken conjecture.

In one of the early forays into experimenting with cube complexes, Aitchison, Matsumoto, and Rubinstein produced some nice results about certain "cubed" manifolds, that in retrospect look very prescient.

I will define non-positively curved cube complexes, what it means for a 3-manifold to be cubed, and discuss what all this Haken business is about.