Waldhausen defined higher K-groups for categories with certain extra structure. In this talk I will define categories with cofibrations and weak equivalences, outline Waldhausen's construction of the associated K-Theory space, mention a few important theorems and give some examples. If time permits I will discuss the infinite loop space structure on the K-Theory space.

# Past Junior Topology and Group Theory Seminar

Subgroup separability is a group-theoretic property that has important implications for geometry and topology, because it allows us to lift immersions to embeddings in a finite sheeted covering space. I will describe how this works in the case of graphs, and go on to motivate the construction of special cube complexes as an attempt to generalise the technique to higher dimensions.

As the title says, in this talk I will be giving a casual introduction to higher categories. I will begin by introducing strict n-categories and look closely at the resulting structure for n=2. After discussing why this turns out to be an unsatisfying definition I will discuss in what ways it can be weakened. Broadly there are two main classes of models for weak n-categories: algebraic and geometric. The differences between these two classes will be demonstrated by looking at bicategories on the algebraic side and quasicategories on the geometric.

Brady showed that there are hyperbolic groups with non-hyperbolic finitely presented subgroups. I will present a new construction of this kind using Bestvina-Brady Morse theory.

*n*≥ 3, SL

_{n}(ℤ) has (T). A nice feature of the approach I will follow is that it works entirely within the world of discrete groups: this is in contrast to the classical method, which relies on being able to embed a group as a lattice in an ambient Lie group.

I will explain why one can symplectically embed closed symplectic
manifolds (with integral symplectic form) into CP^{n} and compute the
weak homotopy type of the space of all symplectic embeddings of such a
symplectic manifold into CP^{∞}.

The notion of automatic groups emerged from conversations between Bill
Thurston and Jim Cannon on the nice algorithmic properties of Kleinian
groups. In this introductory talk we will define automatic groups and
then discuss why they are interesting. This centres on how automatic
groups subsume many other classes of groups (e.g. hyperbolic groups,
finitely generated Coxeter groups, and braid groups) and have good
properties (e.g. finite presentability, fast solution to the word
problem, and type FP_{∞}).