We begin with a discussion of the origins of geometric group theory: the importance of the core decision problems first formulated by Dehn; the emergence of classes of groups that demand special attention; and the emergence of negative and non-positive curvature from the core issues in group theory. This leads to a description of the geography of the universe of finitely presented groups, with emphasis on the role played by manifestations of non-positive curvature.
From this platform, the following topics will be developed.
The study and construction of group actions; complexes of groups, the role of non-positive curvature in devlopability. The fundamental groups of non-positively curved (orbi)spaces. Rigidity. Concepts of dimension.
Hyperbolic groups. Towards semihyperbolic groups: competing theories and the difficulties of discriminating between the associated classes of groups. The rich and diverse subgroup theory.
Algorithmic structures: connecting the complexity of the basic decision problems to the geometry of manifolds via diagrams; Dehn functions and the isoperimetric spectrum; language-theoretic complexity, the geometry of normal forms, automatic groups. Conjugacy problems and cryptography.
Isomorphism problems for groups and manifolds in the presence of non-positive curvature (positive and negative results).
Throughout, many examples will be given to complement the elegance of the general theory and to illustrate how ideas from non-positive curvature in group theory can be applied elsewhere, eg the solution to Grothendieck's problems concerning profinite completions.