# Past Junior Topology and Group Theory Seminar

Motivation for the study of fusion categories is twofold: Fusion categories arise in wide array of mathematical subjects, and provide the necessary input for some fascinating topological constructions. We will carefully define what fusion categories are, and give representation theoretic examples. Then, we will explain how fusion categories are inherently finite combinatorial objects. We proceed to construct an example that does not come from group theory. Time permitting, we will go some way towards introducing so-called modular tensor categories.

Much progress in the study of 3-manifolds has been made by considering the geometric structures they admit. This is nowhere more true than for 3-manifolds which admit a hyperbolic structure. However, in the land of algorithms a more combinatorial approach is necessary, replacing our charts and isometries with finite simplicial complexes that are defined by a finite amount of data.

In this talk we'll have a look at how in fact one can combine the two approaches, using the geometry of hyperbolic 3-manifolds to assist in this more combinatorial approach. To do so we'll combine tools from Hyperbolic Geometry, Triangulations, and perhaps suprisingly Polynomial Algebra to find explicit bounds on the runtime of an algorithm for comparing Hyperbolic manifolds.

The mapping class group of a surface with boundary acts freely and properly discontinuously on the fatgraph complex, which is a contractible cell complex arising from a cell decomposition of Teichmuller space. We will use this action to get a presentation of the mapping class group in terms of fat graphs, and convert this into one in terms of chord diagrams. This chord slide presentation has potential applications to computing bordered Heegaard Floer invariants for open books with disconnected binding.

Let X_1 and X_2 be finite graphs with isomorphic universal covers.

Leighton's graph covering theorem states that X_1 and X_2 have a common finite cover.

I will discuss recent work generalizing this theorem and how myself and Sam Shepherd have been applying it to rigidity questions in geometric group theory.

Gromov hyperbolicity is a property to metric spaces that generalises the notion of negative curvature for manifolds.

After an introduction about these spaces, we will explain the construction of horocyclic products related to lamplighter groups, Baumslag solitar groups and the Sol geometry.

We will describe the shape of geodesics in them, and present rigidity results on their quasi-isometries due to Farb, Mosher, Eskin, Fisher and Whyte.

Let phi be an outer automorphism of a free group. A topological representative of phi is a marked graph G along with a homotopy equivalence f: G → G which induces the outer automorphism phi on the fundamental group of G. For any given outer automorphism, the choice of topological representative is far from unique. Handel and Bestvina showed that sufficiently nice automorphisms admit a special type of topological representative called a train track map, whose dynamics can be well understood.

In this talk I will outline the definition and motivation for train tracks, and give a sketch of Handel and Bestvina’s algorithm for finding them.

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.

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.