Hyperbolic groups are groups with ‘negative curvature’. Although their definition is geometric in nature, hyperbolic groups satisfy various algebraic rigidity properties. One example is Paulin’s theorem, which sharply characterizes torsion-free hyperbolic groups with finitely many outer automorphisms.

In this talk I will introduce the notion of hyperbolic groups (with lots of examples), and walk through the proof of Paulin’s theorem, which uses a limiting argument to extract group actions on spaces called ‘R-trees’. If time permits, I will explain additional applications of this technique.